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Veterinary clinic business plan

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Argumentative Essay about Why Smoking Should be Banned. Extracts from this document. ?Page | ________________ Ban Smoking in Public Places Smoking is veterinary clinic business one of the leading causes of essay american preventable deaths in clinic, the United States, and doing it in public is even worse because smokers not only essays, harm themselves, but also those around them. I remember how my great uncle would always smoke in veterinary business, his house regardless of his surroundings. When his daughter visited him for a vacation, she brought her three year old daughter with her. The baby was healthy before coming to her grandpa's house, but in less than two weeks, she had developed ear infections and started to cough. Word. When they went to the doctors' office to find out how she had gotten it, the doctor linked it back to the second hand smoke that she was breathing in. When my great uncle found out about it, he finally started to care about where he was smoking and who he was with, and decided that he would smoke in business plan, a secluded place where no one but him may be affected. . read more. Though many smokers already know of the health risks that come with smoking cigarettes, but what most of them do not, is how secondhand smoking can affect other people. ?Breathing in good psychology, low doses of plan secondhand smoke can increase a person?s risk of essay proud heart attack? according to clinic business plan a study by Dr. Meyers, a professor of Cardiology and Preventive Medicine at University of Kansas (Sciencedaily.com).

Smokers should not risk non-smokers for essay proud something that they do. Business Plan. In continuation of Dr. Meyers? study is that, ?secondhand smoking increases the transition likelihood of a heart attack by making the blood ?sticky?, making it more prone to clotting, and reducing the amount of good cholesterol in the body? (Sciencedaily.com). Non-smokers should not be worrying about paying more medical bills, especially in today?s economic situation. Many businesses provide health insurance to their employees.

According to a statistical data collected by Live Strong, employers spend an extra $49 million in medical costs for employees who are exposed to secondhand smoking in their jobs (Nelson). . Veterinary Plan. read more. Smoking in public should be banned at a federal level without question. Smoking negatively affects people, the essay environment and businesses. There are no logical reasons to allow smoking in public places and doing so does not benefit anyone but cigarette makers. Smokers should have secluded areas where they could smoke where no one may be harmed. The government should provide trash cans in these areas so smokers would not just throw their cigarette butts on the floor. Anyone caught smoking in veterinary clinic plan, public places and littering their cigarettes should be fined. Smoking endangers people who chose not to engage in the same reckless behavior, like my baby cousin who had gotten so sick because of my great uncle?s smoking. It also dirties our environment and harms other animals that are in search of essay food.

Businesses who provide health insurance to clinic business plan their workers affected by second hand smoking are wasting money that could be used to hire more employees. Banning smoking in public may lead to a better way of life and bond metathesis, a healthier place to live. . Veterinary Clinic Plan. read more. This student written piece of essay on true work is one of many that can be found in our GCSE Writing to Argue, Persuade and Advise section. Start learning 29% faster today 150,000+ documents available Just £6.99 a month. Join over 1.2 million students every month Accelerate your learning by 29% Unlimited access from just £6.99 per month. Related GCSE Writing to Argue, Persuade and Advise essays.

Should smoking be banned in public places? Putting a ban on smoking in public places would do great things for the economy and help businesses to attract more people. Veterinary Business Plan. Non-smokers would feel less threatened and not have to sit and worry about the odd smoke cloud hitting them in the face while enjoying a drink or a meal. Sigma. Pip was a common working boy; he was used to veterinary business his status and was happy. But after he met the cold and proud Estelle, at Satis house, he fell foolishly in love with her. He became ashamed of himself, ashamed of practice ap english essays his family and ashamed of his work. Because of the effects smoking has on you and others should it be banned . They have not totally blocked out smokers but have catered for them in veterinary plan, a way of putting benches and a heater outside every bar and restaurant, and they have proven that passive smoking has dropped dramatically. a case study on transition word smoking and it's effects. Tobacco was so popular that as part of the war effort, tobacco was made a protected crop in veterinary business, America and essay, cigarettes were included in soldiers rations.

Tobacco companies sent millions of veterinary free cigarettes to troops. With soldiers being conscripted and everyone having to do national service, a large majority of young men smoked. Another regard is having to go to school for another long minimum of 8 years. Hard To Find Analysis Essays. Most students get a 4-year bachelor's degree, and then go to a medical school for another 4 years to be a Medical Doctor. Doctor Green says, When you are young, your brain is veterinary clinic plan like a sponge, and you can soak up lots of information. When you get older, your brain gets harder and harder and less like a sponge, which means it is proud american much harder to learn and remember information. of student written work Annotated by. experienced teachers Ideas and feedback to. improve your own work. Marked by Teachers, The Student Room and Get Revising are all trading names of The Student Room Group Ltd.

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Veterinarian Veterinary Clinic Hospital Start Up Sample Business Plan!

Simple and Clean Resume Templates. Hiring managers look at very many of resumes. Their eyes get tired quickly, and business, having to transition essay look at elaborate resumes doesn’t help. In fact, fancy resume formats make the recruitment process even harder, since hiring managers have to look past the design elements in business plan, order to get to sigma bond the content. Many job seekers believe that an eye-catching resume alone will help them land a job. However, that is not usually the case. A clean, simple resume can help you stand out from the pool of applicants by being easy to veterinary business plan read and hard analysis essays, pleasant to veterinary look at. Professional resume formats will often use minimalist formatting to capture the attention of a hiring manager and showcase experience and skills. In this guide, we’ll talk about which types of employers prefer a simple resume format over a modern or creative one.

We’ll also walk you through how to sigma format and write a resume using one of our free resume templates. Let’s get you on the path to creating your resume today. A simple resume does not need to plan be a boring one. In fact, a simple resume format requires a lot of creativity because it should be both pleasing to the eye and tailored to your experience and essay, desired job. A simple resume can be more conservative or more trendy, depending on which template you use. All of our templates here have clean, simple layouts, but there is veterinary clinic plan a lot of variety within the essay, simple resume format, so you can choose one that suits your work history best. What Jobs Require a Simple Resume? If you’re applying to a conservative or old-fashioned place of clinic, employment, a simple resume is the way to go. The legal field is an example of a conservative workplace.

So are most professions in finance and many large corporate offices. If the word essay, job you want has a conservative dress code, you should probably use a simple resume format. This is like a dress code for veterinary your resume. Although a simple resume format is essential in those types workplaces, a well-crafted simple resume can also impress a potential employer in a creative, modern field. If you’re going after a job where design is important, then knowing how to use simple design elements without relying on fancy fonts or borders can actually make your resume stand out. Font, formatting, color scheme, and use of sigma, white space all make an veterinary plan, attractive simple resume. This requires solid design knowledge. Whether you’re applying to be a graphic designer or a fashion designer, a simple resume can be an excellent example of literature essays, your creative skills.

If you’re applying to a newer company, the veterinary plan, employer can also be impressed by a clean, traditional simple resume, even if the corporate culture is more laid-back. Many startups and essay, tech companies use a minimalist aesthetic for their offices, websites, and products. You can show you’re a good fit by reflecting that same clean aesthetic in your resume. You need to decide what you’ll be putting on your resume before you can write and format it. To help you decide what should go on your resume, you can use your existing resume or you can note down some specifics about your skills and experience. What you put on your resume will also depend on the details of the desired job and the job description.

If you’re applying in the medical field for example, it’s important to clinic have a section for definition essay friendship certifications and credentials. If you’re applying for a retail job, however, this section is probably not necessary. Here are the sections that should always go on your resume: Here are some ideas for optional sections: Career Objective, Objective Statement, or Professional Summary Interests and Activities Relevant Skills Achievements and clinic business plan, Awards References. Your resume should have the minimum number of sections necessary to showcase your abilities as a job candidate. Ruthlessly delete unnecessary information. Now that you’ve decided what sections to good essay include, it’s time to clinic business plan format those sections into definition essay on true friendship, a beautiful, simple resume.

You don’t need to do this work on your own. We’ve compiled a list of clinic business, simple resume templates you can use. However, you should know about good resume design so you’ll know how to best customize the template you choose. With all of transition essay, our resume templates, you can change the veterinary clinic plan, bullet points, fonts, colors, and metathesis, arrangement as needed. Avoid large blocks of text in your resume. Having white space between sections adds balance.

If a section on your resume involves a list (such as a “Skills” section), you will want to clinic plan choose a template with bullet points. These make your resume look organized and man is to find, easy to read. There’s nothing wrong with a classic font like Times New Roman, but a simple resume often looks best with a clean, modern, sans-serif font like Helvetica Neue. If you choose to use font colors to dress up your resume, use them sparingly. Pick just one or two colors to highlight aspects of your resume. For example, you might choose a conservative color, like gray or dark blue, for resume headers.

Make sure any font color you choose is easy to read. Avoid pink, yellow, and veterinary business, lighter shades, which look unprofessional and are hard to essay proud american notice. Always be consistent. Have your “Objective” header in the same color as your “Education” header. Don’t write your “Skills” section in Times New Roman and your “Work History” section in business, Helvetica. The following common resume mistakes make your resume look too complicated and poorly designed: Too many colors and fonts Hard-to-detect colors and fonts Unconventional fonts like Comic Sans Elaborate borders Colored or needlessly fancy paper Too many sections Unnecessary images Inconsistent formatting. If your resume is overly designed or messy, employers will be distracted from the good essay, content and more likely to veterinary put your resume in essay, the “No” pile. Veterinary Clinic Business. A really elaborate resume might even be automatically discarded before it gets read.

Once you’ve chosen the sigma metathesis, right resume template for veterinary business you, it’s time to write your unique resume content. Transition. If you’ve never written a resume before, use one of our detailed resume guides, or a free resume builder to help you get started. A simple resume is about more than just layout. Clinic Business. There should be simplicity in your phrasing too. This is the on true, case for all resumes, but especially for veterinary clinic business one with a simple design. Don’t hide your impressive experience under flowery language or long sentences. State things simply so employers can easily learn about on true friendship you. It may be tempting to add detail with more words, but wordiness will only make your resume look messy. Keep in mind that many job applications will also require a cover letter, where you can add more detail about your experience. It doesn’t all need to clinic business go on good essay your resume. Use active verb forms and veterinary, short sentences or bullet points when describing things like skills or achievements.

Find places to cut or reword. It’s okay to use sentence fragments in your bullet points. Not only practice literature essays do these points make your resume look good, they also make it more likely that a prospective employer will read all of it. We have a variety of business plan, free simple resume examples to suit your needs, including chronological, functional, and friendship, combination resumes. Using what you’ve just learned in this guide, look through these templates and clinic plan, choose one that suits your current job search best. Good Psychology. Then customize it to reflect what you have to veterinary clinic business offer as a unique job candidate. Good use of white space gives this professional resume template an open feel while subtle colours add design touch. Two main sections let you detail your experiences and education in a reverse chronological order. Size: 183 B Downloads: 103689 Filename: upfront.docx Uploaded: 2014-05-09 20:18:00 / 6327. Very compact and practice literature, neat professional layout.

Notice how the use of red text draws the eye. Business. Just the right amount in the right places can be really helpful to the reader. Size: 183 B Downloads: 103040 Filename: neat.docx Uploaded: 2014-05-06 11:38:00 / 4968. Traditional format with light orange headers and subtle gray titles. Great sample resume for metathesis graphic designers, corporate writers, or administrative professionals. Size: 183 B Downloads: 52699 Filename: subtle-creativity.docx Uploaded: 2014-05-09 19:42:00 / 6315. Margin column for titles and veterinary clinic business, headings provides lots of space for detailed, paragraph statements.

Perfect resume example for business, industrial, or vocational professionals. Size: 183 B Downloads: 34032 Filename: broad-appeal.docx Uploaded: 2014-05-09 20:48:00 / 6348. Labor, manufacturing, educational, and metathesis, service industry skills template. Veterinary. Good use of white space to catch the eye with blue section divider lines and large, bold section titles. Size: 183 B Downloads: 17061 Filename: commencing.docx Uploaded: 2014-05-09 22:42:00 / 6409. Just a bit of colour and creative use of indent make add a creative twist to practice literature essays this very traditional straight-forward design. Indented text helps the veterinary clinic, headings stand out. Size: 183 B Downloads: 16056 Filename: indent-line.docx Uploaded: 2014-05-06 17:22:00 / 5631. Excellent sample for those with lots of skills and training and maybe not a lot of experience. Bold headings with vertical divider lines and essay friendship, blue titles provide focus for the reader. Size: 183 B Downloads: 15305 Filename: fresh-take.docx Uploaded: 2014-05-08 16:54:00 / 6092.

Conventional, two-page layout with profile photo and red and blue fonts add design touch. Plus signs used for bullet points. Great for technology, engineering, or financial career seekers. Size: 183 B Downloads: 14463 Filename: creative-conventions.docx Uploaded: 2014-05-08 15:32:00 / 6023. Two-page format with a two-column layout offers much space for clinic business multiple sections. Light blue titles and dashed section lines add design touch. Essay. Give all of your details here. Size: 183 B Downloads: 13273 Filename: point-left.docx Uploaded: 2014-05-09 20:57:00 / 6354. Subtle red titles and block, sans serif fonts give this template a stately feel. Great use of white space complements paragraph descriptions of plan, your experiences, strengths, and a good to find analysis, objectives.

Size: 183 B Downloads: 13079 Filename: stately.docx Uploaded: 2014-05-08 16:19:00 / 6065. Touches of red shadings and gray background really draw in veterinary clinic business, the reader#039;s focus. Plus signs used for proud american bulleted lists with room for clinic multiple sections. Size: 183 B Downloads: 12904 Filename: bold-red-shading.docx Uploaded: 2014-05-09 20:34:00 / 6345. Check box bullets tick off all of your great career highlights in this two-page resume with blue section divider lines.

Excellent resume style for sigma bond metathesis very experienced and accomplished professionals. Size: 183 B Downloads: 12725 Filename: accomplished.docx Uploaded: 2014-05-08 17:47:00 / 6135. Perfect use of typography to veterinary clinic draw attention to to find your name and a short message. Horizontal and veterinary clinic, vertical lines help to divide up the information for an easy read. Size: 183 B Downloads: 10157 Filename: important-message.docx Uploaded: 2014-05-06 11:46:00 / 4977. Small profile picture and thin, blue-shaded headers add design to sigma metathesis this classic format with lots of veterinary plan, space for psychology essay summary texts and bulleted lists. Veterinary Clinic Plan. Business, health and IT professionals benefit here. Size: 183 B Downloads: 9065 Filename: packed-classic.docx Uploaded: 2014-05-08 11:53:00 / 5908.

Light brown headings and transition, vertical bright green dashed lines catch the eye and break content into veterinary, easy to read sections. Bold titles bring focus to key experiences and good psychology, skills. Great sample for marketing, public relations, or communications professionals. Size: 183 B Downloads: 8747 Filename: beaming.docx Uploaded: 2014-05-09 19:15:00 / 6278. Bold, open squares and thin black lines divide the veterinary clinic plan, two-column page layout. Essays. Dates and key words offset in clinic business plan, the margins act as section intros. Excellent template for psychology university and IT job seekers. Size: 183 B Downloads: 8619 Filename: checkmark-timeline.docx Uploaded: 2014-05-08 12:18:00 / 5929. Thin and thick blue lines define each squared section with headings in clinic plan, bold, white fonts to give this template pop.

Size: 183 B Downloads: 7126 Filename: blue-pop.docx Uploaded: 2014-05-07 20:24:00 / 5795. Offset one-column layout with red and orange colours and ap english, bold, large headers and titles with paragraph descriptions. Creative sample with a business feel. Size: 183 B Downloads: 7014 Filename: next-level.docx Uploaded: 2014-05-09 20:32:00 / 6342. Section titles offset in the margin with blue and veterinary business plan, light gray fonts.

Small vertical lines used to separate columns. Good Essay. Excellent template for those who have been in one industry for several years. Size: 183 B Downloads: 6472 Filename: industry-lifer.docx Uploaded: 2014-05-08 15:49:00 / 6041. Dark black section dividers complement light green and blue fonts and bullets. Veterinary Business. Large, three-columned Core Skills section for sigma many key attributes. Serif font type completes mature, professional feel. Size: 183 B Downloads: 5792 Filename: cosmopolitan.docx Uploaded: 2014-05-08 15:41:00 / 6032. A traditional format that portrays a simple elegance. An initialed ribbon provides a smart touch which can be personalized with you initials, your highest qualification or your most prestigious award.

Bullet points help you to squeeze in clinic business, a lot more information without making your resume too long. Size: 183 B Downloads: 5627 Filename: elegant-traditional.docx Uploaded: 2014-05-06 11:35:00 / 4958. Two-page, one-column layout with red and orange for headings and transition word essay, titles. Veterinary Clinic Business. Lots of space for detailed, paragraph descriptions. To Find Essays. Corporate and academia feel with this sample. Size: 183 B Downloads: 5626 Filename: tidy.docx Uploaded: 2014-05-09 19:23:00 / 6287. Traditional layout with bold and italic font types. Large red square with white font provides good space for objective. Great sample for corporate, IT, and nonprofit job seekers.

Size: 183 B Downloads: 5560 Filename: bold-red.docx Uploaded: 2014-05-08 13:37:00 / 5993. This simple block colour really makes a statement and is the perfect backdrop for the black text. Size: 183 B Downloads: 5118 Filename: pretty-pink.docx Uploaded: 2014-05-06 18:13:00 / 5665. One look and everyone will know who you are. The placement size and bold text really make the veterinary plan, name on this resume stand out. Size: 183 B Downloads: 5106 Filename: name.docx Uploaded: 2014-05-06 11:44:00 / 4974.

Large and small margins add white space while offering plenty of room for multiple sections. Bold orange and blues for titles focus the reader. Great template for entry level positions or new graduates. Size: 183 B Downloads: 4695 Filename: offset.docx Uploaded: 2014-05-08 16:08:00 / 6056. Bold section titles with arrow points and definition, light maroon titles give this sample a vivid touch.

Full page layout for multiple sections to let you list plenty of information. Size: 183 B Downloads: 4652 Filename: striking.docx Uploaded: 2014-05-08 16:38:00 / 6080. This block and plan, bulleted resume offers a professional-looking style. Hard. The enlarged font size for the objective section clearly defines your goals. Size: 183 B Downloads: 4501 Filename: professional-block.docx Uploaded: 2014-05-06 19:31:00 / 5711. The use of veterinary clinic, bold text, simple colours, underlines and transition word essay, spacing makes this a very easy read. Size: 183 B Downloads: 4376 Filename: simple-underline.docx Uploaded: 2014-05-06 17:46:00 / 5649. Plus signs and veterinary clinic business, lines outline each area with arrows providing the bulleted direction to your skills and experience. Scientific, data, and good essay, IT feel with this one. Size: 183 B Downloads: 4303 Filename: plus.docx Uploaded: 2014-05-06 19:46:00 / 5724.

Easy to scan, one column template with section and title headings in colour as well as all caps make your information quick to read. Large Skills Abilities section great for the career changer. Size: 183 B Downloads: 4123 Filename: academia.docx Uploaded: 2014-05-06 20:17:00 / 5748. Open and plan, airy one-column format centered on the page. Blue headings and bold titles with plus signs and outlined circle bullets provide subtle design appeal. Good Essay. Service and labour workers take a look here. Size: 183 B Downloads: 4083 Filename: breezy.docx Uploaded: 2014-05-08 15:35:00 / 6026. Colour contrast and different font types dictate the style and impact of this resume. The use of the clinic business, red always pulls you back to the name of the applicant. Size: 183 B Downloads: 3990 Filename: simple-red.docx Uploaded: 2014-05-06 18:02:00 / 5655.

Oversized captions for key words or job titles. Small profile photo with blue shaded section headers. Excellent use of practice essays, space for intern experience, new graduates, and veterinary, creative professionals. Size: 183 B Downloads: 3942 Filename: caption.docx Uploaded: 2014-05-08 18:00:00 / 6144. Want a different way to distinguish your sub-headings from proud american your headings? This layout shows how different shades of the same colour can do this. Size: 183 B Downloads: 3823 Filename: mono-shading.docx Uploaded: 2014-05-06 11:48:00 / 4986. Light blue titles, bold headings, and carat bullets with good use of white space in this basic resume template. Perfect example for service industry and labour staff as well as new graduates. Size: 183 B Downloads: 3335 Filename: burgeoning.docx Uploaded: 2014-05-08 16:52:00 / 6089. Simple layout with bold headings and veterinary clinic, thick lines to separate different sections.

Not the best format if you have a lot of essay american, information to include. Size: 183 B Downloads: 3183 Filename: simple-central.docx Uploaded: 2014-05-06 18:10:00 / 5662. Offset sections balance out this sample. Clinic Plan. Good use of white space helps focus the reader#039;s eye. Orang titles provide design touch.

Great format to take your few years of experience to definition essay on true friendship the next level. Size: 183 B Downloads: 1182 Filename: moving.docx Uploaded: 2014-05-08 18:05:00 / 6147. Thin margins with stacked sections provide lots of area for all of your achievements. Clinic Plan. Unique Projects area lets you describe your accomplishments. Transition Word. IT, project management, and senior-level sample.

Size: 183 B Downloads: 167238 Filename: substantial.docx Uploaded: 2014-05-08 16:36:00 / 6077. Thin page border and section divider lines with red and bold, black titles and veterinary business plan, key words provide an metathesis, executive feel. Plan. Project managers, directors, and definition essay on true friendship, senior VPs can do well with this example. Size: 183 B Downloads: 11561 Filename: top-level.docx Uploaded: 2014-05-08 16:42:00 / 6083. A compact resume format that draws the reader’s eye to the important information across different sections. Size: 183 B Downloads: 10825 Filename: in-a-nutshell.docx Uploaded: 2015-10-10 08:21:00 / 8305. Two well-defined columns give this sample structure. Veterinary Business Plan. Thick, orange-coloured bar graphs make your capabilities pop on good essay the page and the italicized, muted gray job titles add the professionalism. Size: 183 B Downloads: 10049 Filename: raise-bar.docx Uploaded: 2014-05-06 20:28:00 / 5758. What stands out in clinic business plan, this sample is the Skills section.

This format provides an instant snapshot of sigma bond metathesis, your skill levels that looks very professional and easy to read. Size: 183 B Downloads: 9660 Filename: self-assessment.docx Uploaded: 2014-05-06 11:47:00 / 4980. Entry-level workers ready for the next step here. Conventional sections with blue headers and light gray arrows and open circular bullets point all of business plan, your strengths and skills. Plenty of room for goals, key word lists, and good, more. Size: 183 B Downloads: 7397 Filename: space.docx Uploaded: 2014-05-08 15:38:00 / 6029. A very smart style that gets straight down to business. Can you summarize your abilities into veterinary business, a couple of sentences? Most prospective employers don#039;t have a lot of time to read through lengthy resumes.

Size: 183 B Downloads: 6467 Filename: classic-elegance.docx Uploaded: 2014-05-06 11:36:00 / 4961. Great use of contrasting text colour and line thickness to differentiate between sets of psychology essay, information. Size: 183 B Downloads: 5485 Filename: section-lines.docx Uploaded: 2014-05-06 17:49:00 / 5652. Small margins and thin gray shadowed headings with white fonts offer lots of veterinary plan, room in this example. Multiple sections for your training, certifications, skills, and other information. Size: 183 B Downloads: 4862 Filename: traditional-plus.docx Uploaded: 2014-05-07 21:02:00 / 5836. Just a bit of colour goes a long way in definition essay friendship, breaking up information. Size: 183 B Downloads: 4623 Filename: nonsense.docx Uploaded: 2014-05-06 19:10:00 / 5698. Bold headings and shading flush out this sample with plus signs and veterinary clinic, open circles for bullet points. Good skills resume for those with lots of essay on true, experience and training outside the veterinary plan, traditional job environment.

Size: 183 B Downloads: 3894 Filename: skilled.docx Uploaded: 2014-05-07 20:05:00 / 5783. Bold, black headings and section dividers with light-gray lines used for bullets and essay on true, blue font type for your name make this a non-traditional traditional template. Size: 183 B Downloads: 3881 Filename: non-traditional-traditional.docx Uploaded: 2014-05-08 12:13:00 / 5923. Complimentary colours for your personal information and the section headings catch the eye. Simple bullet list in veterinary clinic business, each section offer a straightforward, no nonsense approach. Size: 183 B Downloads: 3275 Filename: career-starter.docx Uploaded: 2014-05-06 19:34:00 / 5714.

Gray shading and colorful section headers give this contemporary resume a bold and confident air. Here, your skills, strengths, awards, achievements, and bond, career highlights take center stage. This resume lends a decisive and authoritative tone, while not being overbearing or fussy; it still allows your personality and background to shine. This is plan a perfect template for bond individuals seeking to change careers or industries, ex-military looking to enter the veterinary business, civilian workforce, and others who want to emphasise their talents and strengths, and not their previous job titles or employers. Size: 183 B Downloads: 40797 Filename: bulletin.docx Uploaded: 2014-05-08 18:20:00 / 6162. Standard Europass CV / Resume Template used in definition on true, Europe.

Size: 183 B Downloads: 30414 Filename: europass-cv-template.doc Uploaded: 2014-09-19 17:26:00 / 7451. Free Modern Resume templates that will get you noticed. Simple, elegant, discreetly modern, with a touch of color and veterinary plan, creativity. Size: 183 B Downloads: 30378 Filename: discreetly-modern.docx Uploaded: 2014-09-19 17:09:00 / 7448. Very classy and professional full-page two-column sample. Lots of sections included for key skills, awards, and attributes. Red and blue titles add touch of colour. Great example for those with substantial work experience. Size: 183 B Downloads: 12938 Filename: firm.docx Uploaded: 2014-05-08 16:48:00 / 6086. Two column layout for full page with sections styled into various columns.

Light gray divider lines and red and blue titles. Stylish with a corporate feel for marketing director, senior VPs, and principal managers. Size: 183 B Downloads: 9111 Filename: aristocratic.docx Uploaded: 2014-05-08 17:05:00 / 6104. Two-column layout with profile photo. Essay Proud. Large Awards section to list your sales, academic, or artistic achievements. Great sample for advertising, creative, sales, or communication professionals. Size: 183 B Downloads: 8426 Filename: award-winner.docx Uploaded: 2014-05-08 16:15:00 / 6062. Traditional classic layout with a high-end executive feel. Veterinary Business. Great use of bordered headings, indented bullets and tabbed text to psychology essay create an elegant professional look. Size: 183 B Downloads: 7021 Filename: executive-elegant.docx Uploaded: 2014-05-06 11:47:00 / 4983. A round portrait option makes this sample a bit different.

Then a strong opening statement in bold lettering is business plan great for impact, and makes the reader want to find out more about you. Proud. At the end of the veterinary business, resume a simple scale allows you to give a snapshot of your levels of competence. Size: 183 B Downloads: 6913 Filename: bold-simplicity.docx Uploaded: 2014-05-06 11:32:00 / 4946. Two-column format used for the whole page offers lots of good psychology, section areas. Light grays for headers and veterinary clinic plan, dashed section dividers as well as shaded background to highlight key work experience. Size: 183 B Downloads: 6672 Filename: split-page.docx Uploaded: 2014-05-08 15:47:00 / 6038. Bold headings shaded with olive and dark greens for a sophisticated template. Essay. Managers, administrators, directors, and accountants take advantage of the three-columned skills area and oversized sections for experience, certificates, and licenses.

Size: 183 B Downloads: 6551 Filename: polished.docx Uploaded: 2014-05-07 20:43:00 / 5812. Functional two-page sample with three-column core skills lists and clinic business, areas for professional experience, work history and word, more. Great resume style for those looking to clinic plan switch industries. Size: 183 B Downloads: 6162 Filename: new-agenda.docx Uploaded: 2014-05-08 17:00:00 / 6098. Checkmarks for bullets with blue, green, and proud american, gray fonts catch the eye. Veterinary Business. Underlines connect job titles to dates. Practice. Analysts, developers, academia, or health care template.

Size: 183 B Downloads: 4747 Filename: check.docx Uploaded: 2014-05-08 15:51:00 / 6044. Deep and veterinary clinic business plan, light blue headers with bold font titles add pop for this traditional template. Three-column key word section and two-column bulleted experience lists offer space for everything. Size: 183 B Downloads: 4370 Filename: airy-traditional.docx Uploaded: 2014-05-08 11:43:00 / 5899. Want the focus to be on your opening statement? This resume does that. Make your pitch for this job a real eye opener.

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Tax Examiners and Collectors, and Revenue Agents. Tax examiners and collectors, and plan, revenue agents ensure that federal, state, and local governments get their tax money from businesses and citizens. They review tax returns, conduct audits, identify taxes owed, and collect overdue tax payments. Tax examiners and essay american, collectors, and revenue agents work for federal, state, and local governments. Many work primarily in an office environment; others spend most of veterinary clinic their time doing field audits in taxpayers’ homes or places of business. How to Become a Tax Examiner or Collector, or Revenue Agent. Most tax examiners and collectors, and revenue agents need a bachelor’s degree in good essay, accounting or a related field.

However, the level of clinic business plan education and experience required varies by position and employer. The median annual wage for tax examiners and collectors, and practice ap english literature, revenue agents was $52,060 in May 2016. Employment of clinic tax examiners and collectors, and revenue agents is projected to decline 6 percent from 2014 to 2024. Employment change will depend primarily on future changes to federal, state, and local government budgets. Explore resources for employment and wages by ap english literature, state and veterinary clinic, area for tax examiners and collectors, and american, revenue agents. Compare the job duties, education, job growth, and pay of tax examiners and collectors, and revenue agents with similar occupations.

More Information, Including Links to O*NET. Learn more about tax examiners and collectors, and revenue agents by visiting additional resources, including O*NET, a source on key characteristics of business workers and occupations. What Tax Examiners and Collectors, and good psychology, Revenue Agents Do About this section. Tax examiners and collectors, and revenue agents ensure that federal, state, and local governments get their tax money from businesses and business, citizens. They review tax returns, conduct audits, identify taxes owed, and collect overdue tax payments. Tax examiners and collectors, and a good hard to find essays, revenue agents typically do the following: Review filed tax returns to determine whether credits and deductions claimed are allowed by law Contact taxpayers to address problems and to request supporting documentation Conduct field audits and investigations of business plan income tax returns to verify information or to update tax liabilities Evaluate financial information, using their familiarity with accounting procedures and knowledge of changes to tax laws and definition essay, regulations Keep records on each case they deal with, including contacts, telephone numbers, and actions taken Notify taxpayers of any overpayment or underpayment and either issue a refund or request additional payment. Tax examiners and veterinary clinic, collectors, and revenue agents are responsible for ensuring that individuals and a good hard to find, businesses pay the taxes they owe.

In addition to verifying that tax returns are filed properly, they follow up with taxpayers whose returns are questionable or who owe more money. Different levels of government collect different types of taxes. The federal government deals primarily with personal and business income taxes. State governments collect income and sales taxes. Local governments collect sales and property taxes. Because many states assess individual income taxes based on plan, the taxpayer’s reported federal income, tax examiners working for the federal government report to the states any adjustments or corrections they make. State tax examiners then determine whether the adjustments affect how much the to find taxpayer owes the state.

Tax examiners and collectors, and revenue agents have different duties and responsibilities: Tax examiners usually deal with the simplest tax returns—those filed by individual taxpayers who claim few deductions and business plan, those filed by small businesses. At the entry level, many tax examiners do clerical tasks, such as reviewing tax returns and word, entering them into a computer system for processing. Tax examiners also may contact individual taxpayers in order to resolve any outstanding problems with their returns. Much of a tax examiner’s job involves making sure that tax credits and deductions claimed by taxpayers are lawful. If a taxpayer owes additional taxes, tax examiners adjust the total amount by assessing fees, interest, and penalties and then notify the taxpayer of the veterinary plan total amount owed. Revenue agents specialize in tax-related accounting for the U.S. Internal Revenue Service (IRS) and for equivalent agencies in state and practice ap english literature essays, local governments. Like tax examiners, they review returns for veterinary clinic business plan accuracy. However, revenue agents handle complicated tax returns of large businesses and metathesis, corporations.

Many experienced revenue agents specialize in a particular area. For example, they may focus exclusively on clinic plan, multinational businesses. Regardless of sigma metathesis their specialty, revenue agents must keep up to date with changes in the lengthy and complex tax laws and regulations. Collectors , also called revenue officers in the IRS, deal with overdue accounts. Clinic Plan? The process of collecting an overdue payment starts with the revenue agent or tax examiner sending a report to the taxpayer. If the taxpayer makes no effort to pay, the case is assigned to a collector. When a collector takes a case, he or she first sends a notice to the taxpayer. The collector then works with the taxpayer to settle the debt. Settlement may involve setting up a plan in which the amount owed is paid back in small amounts over time. When delinquent taxpayers claim that they cannot pay their taxes, collectors investigate and verify these claims. Transition Essay? Collectors research information on taxpayer mortgages or financial statements and locate taxpayer-owned items of clinic value through third parties, such as neighbors or local departments of motor vehicles.

Ultimately, collectors must decide whether the essay proud american IRS should take a lien—a claim on an asset such as a bank account, real estate, or an automobile—to settle a debt. Collectors also have the authority to garnish wages—that is, take a portion of earned wages—to collect taxes owed. Work Environment About this section. Tax examiners and collectors, and revenue agents held about 67,900 jobs in 2014. The largest employers of veterinary business tax examiners and collectors, and revenue agents were as follows: Many tax examiners and man is hard to find analysis, collectors, and revenue agents work primarily in an office environment; others spend most of their time conducting field audits in business plan, taxpayers’ homes or places of proud american business. Most tax examiners and collectors, and revenue agents work full time. How to Become a Tax Examiner or Collector, or Revenue Agent About this section. Most tax examiners and collectors, and revenue agents need a bachelor’s degree in accounting or a related field.

However, the required level of education and experience varies by position and employer. Tax examiners need a bachelor’s degree in accounting or a related field, or a combination of relevant education and specialized experience in accounting, auditing, or tax compliance work. Candidates for clinic business plan tax examiner positions at the Internal Revenue Service (IRS) must have a bachelor’s degree or 1 year of full-time specialized experience. Revenue agents need a bachelor’s degree in accounting, business administration, economics, or a related discipline. A combination of relevant education and full-time experience in business administration, accounting, or auditing is also qualifying. Revenue agents with the IRS must have either a bachelor’s degree or 30 semester hours of accounting coursework, along with specialized experience. Specialized experience includes work in accounting, bookkeeping, or tax analysis. Collectors usually must have some combination of psychology essay relevant college education and specialized experience. Specialized experience may include previous work as a loan officer or credit manager , or a background in collections, management, customer service, or tax compliance.

A bachelor’s degree is needed for employment as a collector with the IRS; no additional experience is required, and experience may not be substituted for the degree. Employers desire degrees in veterinary, business, finance, accounting, and essay, criminal justice. At the state and local levels, a bachelor’s degree is not always required, although related work experience is veterinary clinic plan desired. Newly hired tax examiners get some formal training, which typically lasts between 1 month and 1 year. A Good Man Is? All tax examiners must keep current with changes in the tax code and enforcement procedures. Entry-level collectors get both formal training and on-the-job training under an veterinary plan, instructor’s guidance before working independently.

Collectors also are encouraged to essay, continue their professional education by veterinary business, attending meetings to exchange information about literature, how modifications to tax laws affect collection methods. Some state and local governments accept work experience as a substitute for education. In these cases, employers may hire tax examiners and revenue agents who have work experience in accounting, bookkeeping, or tax analysis. Employers may also hire collectors who have work experience in related areas, such as collections, customer service, or credit checking. Tax examiners, revenue agents, and collectors have different opportunities for career advancement. Tax examiners who review individual tax returns may advance to veterinary business, revenue agent positions, working on more complex business returns. Essay? Those with experience in supervisory or managerial roles may move to jobs that involve supervision of other examiners and clinic plan, revenue agents.

Collectors who demonstrate leadership skills and a thorough knowledge of tax collection activities may advance to supervisory or managerial collector positions. Analytical skills. Tax examiners and revenue agents must be able to identify questionable claims for credits and deductions. Ultimately, they must be able to hard to find, determine, on further review of veterinary clinic plan financial documentation, if the credits or deductions are lawful. Computer skills. Proud? Tax examiners and revenue agents must be comfortable using a variety of computer programs. These programs include tax preparation and bookkeeping software used by individuals and businesses. Detail oriented. Tax examiners and revenue agents verify the accuracy of clinic business each entry on the tax returns they review. Therefore, it is important that they pay attention to detail.

Interpersonal skills. Collectors must be comfortable dealing with people, including speaking with them during confrontational situations. When pursuing overdue accounts, collectors should be firm and hard essays, composed. Organizational skills. Tax examiners and business plan, revenue agents often work with multiple returns and a variety of transition essay financial documents.

Keeping the various pieces of information organized is essential. Tax Examiners and Collectors, and Revenue Agents. Median annual wages, May 2016. Note: All Occupations includes all occupations in clinic business plan, the U.S. Economy. Source: U.S. Practice Ap English Literature? Bureau of Labor Statistics, Occupational Employment Statistics. The median annual wage for tax examiners and veterinary business plan, collectors, and revenue agents was $52,060 in May 2016.

The median wage is the wage at which half the workers in an occupation earned more than that amount and half earned less. The lowest 10 percent earned less than $31,860, and the highest 10 percent earned more than $97,440. In May 2016, the median annual wages for american tax examiners and collectors, and revenue agents in the top industries in business, which they worked were as follows: Most tax examiners and collectors, and revenue agents work full time. Compared with workers in all occupations, tax examiners and collectors, and revenue agents had a higher percentage of workers who belonged to a union in 2014. Tax Examiners and man is hard analysis essays, Collectors, and Revenue Agents. Percent change in employment, projected 2014-24.

Note: All Occupations includes all occupations in the U.S. Economy. Source: U.S. Bureau of Labor Statistics, Employment Projections program. Employment of tax examiners and collectors, and revenue agents is veterinary business plan projected to decline 6 percent from essay, 2014 to 2024. Veterinary Clinic Business? Employment change will depend primarily on future changes to federal, state, and local government budgets. Budget reductions in recent years have resulted in decreased hiring for the agencies that employ these workers. Within the federal government, the primary employer of word essay these workers is the Internal Revenue Service (IRS), which has experienced more severe budget cuts than other federal agencies. Further employment declines for these workers in the federal government may occur if the IRS continues to operate with budget cuts.

Overall employment in federal government, excluding postal service, is projected to decline 19 percent. At the state and local level, funding for these workers’ departments has been more stable. Business? Therefore, employment of sigma bond these workers in state and local government is expected to grow in line with overall state and local government employment. SOURCE: U.S. Bureau of Labor Statistics, Employment Projections program. Tax examiners and collectors, and revenue agents.

State Area Data About this section. Occupational Employment Statistics (OES) The Occupational Employment Statistics (OES) program produces employment and wage estimates annually for over 800 occupations. Veterinary Plan? These estimates are available for good the nation as a whole, for individual states, and for metropolitan and nonmetropolitan areas. Veterinary Clinic Business? The link(s) below go to OES data maps for definition essay on true friendship employment and wages by state and area. Occupational employment projections are developed for all states by Labor Market Information (LMI) or individual state Employment Projections offices. All state projections data are available at www.projectionscentral.com. Information on veterinary clinic plan, this site allows projected employment growth for an occupation to be compared among states or to be compared within one state. In addition, states may produce projections for areas; there are links to each state’s websites where these data may be retrieved. America’s Career InfoNet includes hundreds of essay american occupational profiles with data available by state and veterinary, metro area. There are links in the left-hand side menu to compare occupational employment by proud, state and occupational wages by local area or metro area.

There is also a salary info tool to search for wages by zip code. Similar Occupations About this section. This table shows a list of clinic business occupations with job duties that are similar to those of tax examiners and collectors, and revenue agents. Accountants and auditors prepare and proud american, examine financial records. They ensure that financial records are accurate and that taxes are paid properly and on time. Accountants and auditors assess financial operations and work to help ensure that organizations run efficiently. Budget analysts help public and clinic, private institutions organize their finances. They prepare budget reports and monitor institutional spending. Cost estimators collect and analyze data in order to estimate the bond metathesis time, money, materials, and labor required to manufacture a product, construct a building, or provide a service. They generally specialize in a particular product or industry. Financial analysts provide guidance to businesses and individuals making investment decisions.

They assess the veterinary plan performance of stocks, bonds, and other types of investments. Financial managers are responsible for the financial health of an organization. Good Essay? They produce financial reports, direct investment activities, and develop strategies and plans for the long-term financial goals of their organization. Loan officers evaluate, authorize, or recommend approval of loan applications for people and businesses. Personal financial advisors provide advice on clinic business plan, investments, insurance, mortgages, college savings, estate planning, taxes, and retirement to help individuals manage their finances. Bookkeeping, Accounting, and transition word essay, Auditing Clerks. Bookkeeping, accounting, and veterinary business plan, auditing clerks produce financial records for organizations. They record financial transactions, update statements, and check financial records for accuracy. Contacts for More Information About this section.

For more information about tax examiner and collector, and a good analysis essays, revenue agent careers at the Internal Revenue Service (IRS), visit. For more information about requirements for federal government positions as an internal revenue officer, visit For more information about requirements for federal government positions as a tax specialist, visit For more information about requirements for federal government positions as an internal revenue agent, visit Bureau of Labor Statistics, U.S. Department of Labor, Occupational Outlook Handbook, 2016-17 Edition , Tax Examiners and Collectors, and Revenue Agents, Publish Date: Thursday, December 17, 2015. The What They Do tab describes the typical duties and responsibilities of workers in the occupation, including what tools and equipment they use and how closely they are supervised. This tab also covers different types of occupational specialties.

The Work Environment tab includes the number of jobs held in veterinary clinic business, the occupation and describes the practice ap english workplace, the business plan level of physical activity expected, and typical hours worked. It may also discuss the major industries that employed the occupation. This tab may also describe opportunities for part-time work, the amount and type of travel required, any safety equipment that is a good to find used, and plan, the risk of injury that workers may face. The How to Become One tab describes how to prepare for a job in the occupation. This tab can include information on education, training, work experience, licensing and certification, and important qualities that are required or helpful for entering or working in the occupation. The Pay tab describes typical earnings and how workers in the occupation are compensated—annual salaries, hourly wages, commissions, tips, or bonuses.

Within every occupation, earnings vary by experience, responsibility, performance, tenure, and geographic area. This tab may also provide information on earnings in the major industries employing the occupation. The State and Area Data tab provides links to state and area occupational data from the definition Occupational Employment Statistics (OES) program, state projections data from Projections Central, and business plan, occupational information from the Department of Labor's CareerOneStop. The Job Outlook tab describes the factors that affect employment growth or decline in the occupation, and in some instances, describes the relationship between the number of job seekers and the number of job openings. The Similar Occupations tab describes occupations that share similar duties, skills, interests, education, or training with the occupation covered in the profile.

The More Information tab provides the Internet addresses of associations, government agencies, unions, and definition essay on true friendship, other organizations that can provide additional information on the occupation. This tab also includes links to relevant occupational information from the Occupational Information Network (O*NET). The wage at which half of the workers in veterinary plan, the occupation earned more than that amount and definition essay on true, half earned less. Median wage data are from the BLS Occupational Employment Statistics survey. In May 2016, the veterinary median annual wage for all workers was $37,040. Additional training needed (postemployment) to attain competency in psychology, the skills needed in veterinary, this occupation. Typical level of education that most workers need to enter this occupation. Work experience in essay proud american, a related occupation. Work experience that is commonly considered necessary by employers, or is a commonly accepted substitute for more formal types of clinic business plan training or education.

The employment, or size, of this occupation in 2014, which is the base year of the 2014-24 employment projections. The projected percent change in employment from 2014 to 2024. The average growth rate for all occupations is bond metathesis 7 percent. The projected numeric change in clinic business plan, employment from 2014 to 2024. Typical level of education that most workers need to enter this occupation. Additional training needed (postemployment) to attain competency in the skills needed in this occupation. Employment Change, projected 2014-24.

The projected numeric change in employment from 2014 to 2024. The percent change of employment for each occupation from 2014 to 2024. The projected numeric change in employment from 2014 to 2024. The projected percent change in employment from 2014 to 2024. The wage at which half of the workers in the occupation earned more than that amount and half earned less.

Median wage data are from the BLS Occupational Employment Statistics survey. In May 2016, the sigma metathesis median annual wage for all workers was $37,040.

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Algebraic Number Theory - Essay - Mathematics. Plan? Algebraic Number Theory. Version 3.03 May 29, 2011. An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and proud american units in the ring of veterinary clinic business, integers, the extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the metathesis, field with abelian Galois group.

Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and the sequel with class field theory. v2.01 (August 14, 1996). First version on clinic business plan the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Good Psychology Essay? Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008). Fixed problem with hyperlinks; 163 pages. v3.02 (April 30, 2009). Fixed many minor errors; changed chapter and page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on clinic business plan my web page. Essay? The photograph is of the Fork Hut, Huxley Valley, New Zealand.

Copyright c 1996, 1998, 2008, 2009, 2011 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. . . . . . Clinic Plan? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Word? 5 Prerequisites . . . Business Plan? . . . . . . . . . . . . . . . Essay? . . . . . . . . Veterinary Plan? . . . . . . . . . . . 5 Acknowledgements . . . A Good Hard To Find? . . . . . . . . . . . . . . Plan? . . . . . . . . . . . . . On True? . . . 5 Introduction . . . . . . . . . . . . . . . . Veterinary Clinic Business Plan? . . . . . . . . . . . . . . . . . . . . . 1 Exercises . . . . Bond Metathesis? . . . . . . Veterinary Clinic Business Plan? . . . Definition? . . . . . . . . . . . . . . . . Clinic Business Plan? . . . . . . . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . Transition Essay? . . . . . . . . Veterinary Business Plan? . . . . . . . A Good Analysis Essays? . . . . . . . . . . . . . . . . . Veterinary Business? 7 Ideals in products of rings . . . Psychology? . . . . . . . . . . . . . Business Plan? . . . . . . . . . . . . . . 8 Noetherian rings . . . . . . . . . . . . . . . Essay Proud? . . Clinic Business Plan? . . . . . . . . . . . . . . . . . . 8 Noetherian modules . . . . . . . . . . . . . . . . . . . . . . . . Ap English Literature? . . . . . . . . . 10 Local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plan? . . . . . . . Essay? . 10 Rings of fractions . . . . . . . . . . . Business? . . . Ap English Literature Essays? . . . . . Clinic Business? . . . . . . . . . . Metathesis? . . . Business Plan? . . 11 The Chinese remainder theorem . . . . American? . . . . . . . . . Plan? . . . . . . . . . . . . . 12 Review of tensor products . . . . . . . . . . . . Man Is Analysis? . . Veterinary Business Plan? . . Essay On True Friendship? . . . . . . . . . . . . . . Veterinary Plan? 14 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sigma? 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . . . . . . . Business Plan? . . . . Transition Essay? . . . . Veterinary Plan? . 19 Dedekind’s proof that the integral elements form a ring . . . Proud American? . . Clinic Business Plan? . . . . . . . . . 20 Integral elements . . . . . . . . . . . . Metathesis? . . . . . . . . . . . Veterinary Clinic Plan? . . . . Sigma Bond Metathesis? . . . . . . . 22 Review of bases of A-modules . . . . Veterinary Business? . . . . . . Transition Word? . . . . . . . . . . . . . . . . . 25 Review of norms and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Veterinary Business? . Essay? 25 Review of bilinear forms . . . . . . . . . . . . Plan? . . . . . . . . . . . . Literature Essays? . . . . . . 26 Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Rings of integers are finitely generated . . . . . . . . . . . . . . Clinic Plan? . . Good Psychology? . . . . . . . 29 Finding the ring of veterinary business plan, integers . . . . . . . . . Proud American? . . . . . . . . . . . . . . . . . . . . 31 Algorithms for finding the clinic business plan, ring of integers . . . . . . . . Bond Metathesis? . . . . . . . . . . . . . Business Plan? 34 Exercises . . . . Transition Essay? . Veterinary Business Plan? . . . . . . . . . . . . . . . . . Word? . . . . Veterinary Clinic Business Plan? . . . . . . . . . . . . 38.

3 Dedekind Domains; Factorization 40 Discrete valuation rings . . Ap English? . . . . . . . . . . . . . Clinic Business Plan? . . . . . . . . . Man Is Hard Analysis Essays? . . . . Clinic? . . . 40 Dedekind domains . . . . . . . . Practice Ap English? . . . . Clinic? . . . . . . . . . Essay Friendship? . . . . . . . . . . . . 42 Unique factorization of ideals . . . . . . . . . . . . Veterinary Clinic? . . . . . . . . . . . . . Transition Word Essay? . . . 43 The ideal class group . . . . . . . Veterinary Clinic Business Plan? . . . . . . . . . . . . . . . Sigma Bond Metathesis? . . Plan? . . . . . . . . 46 Discrete valuations . . . . . . . . . . . . . . Sigma Bond? . . . . . . . . . . . . . . . . . . . 49 Integral closures of Dedekind domains . . . . . . Veterinary? . . . . . . . . . . . . . . . . . 51 Modules over Dedekind domains (sketch). . . . . Word Essay? . . Clinic Plan? . . . . . . . . . . . . . . . 52. Factorization in extensions . . Essay Proud American? . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 The primes that ramify . . . . . . . . . . . . . . . Clinic Business? . . . . . . . . . . . . . . . . 54 Finding factorizations . . . . . . . . . . . . . . . . . . . . . . Essay? . . . . . . . . . . 56 Examples of factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Eisenstein extensions . . . . . . . . . . . Business Plan? . . . . . . . . Practice Ap English Literature? . . . . . . . Clinic Plan? . . . . . . 60 Exercises . . . . . . . . . . . . . . Sigma Bond Metathesis? . . . . . . . . . . . . . . . . . . . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . . . . Veterinary? . . . Psychology Essay? . . . . . . Clinic Business? . . . . . . . . . Psychology Essay? . . . . . . . 63 Statement of the main theorem and its consequences . . . Business? . A Good Analysis Essays? . . . . . Clinic? . . Ap English Literature Essays? . . . . . Veterinary Business? 65 Lattices . . . . . Essay American? . . . . . . . Clinic Business Plan? . . . . . Sigma Bond Metathesis? . . . . . . Veterinary Clinic? . . . . . . . . . . . . . . . . A Good To Find Essays? 68 Some calculus . . . Veterinary Business? . . . . . . . . . . A Good Analysis? . . . Clinic Business? . . . . . . . . . . . . . . . . . . . . 73 Finiteness of the class number . . . . . . . . . . . . . . . . . . . A Good To Find Essays? . . . . . Clinic? . . . 75 Binary quadratic forms . . . . . . . . . . Essays? . . . . . . . . . . Veterinary Business? . . . . . . . Good Essay? . . . . 76 Exercises . . . . . . Veterinary Business Plan? . . . . . . . . . . . . . . . . . . Proud? . . . . . . . . . . Veterinary Clinic Plan? . . . . 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . . . . . . . . . . Essay American? . . . . . . Business? . . . . . . . . 80 Proof that UK is finitely generated . . . . . . . . . . . . . Ap English? . . . . . . . . Veterinary Business Plan? . . . . 82 Computation of the on true friendship, rank . . . . . . Veterinary Business? . . . . . . . . Word? . . . . Veterinary Clinic Business? . . . . . . . . . . . . 83 S -units . . . . . . . . . . . . . . . . . . . . . . . . Psychology Essay? . . . . . . . . . Veterinary Clinic Business Plan? . . . . . . . 85 Example: CM fields . . Sigma Bond Metathesis? . . Veterinary Clinic? . . . . Good Psychology Essay? . . . . . . . . . . . . Clinic Business? . . . . . . . . . . . Word? . . 86 Example: real quadratic fields . . . . . . . . . . . . . . . . . . . Clinic Business Plan? . . . . . . . . 86 Example: cubic fields with negative discriminant . Man Is Hard To Find Analysis Essays? . . . . . . . . Veterinary Business? . . . . . . . . 87 Finding .K/ . . . Essay? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Finding a system of fundamental units . . . . . . . . . . . . . . . . . Clinic Business Plan? . . . . . . 89 Regulators . . . . . . . . . . . . . . . . . . . . . . . . . Essay Proud American? . . . . . . . . . . . . . 89 Exercises . . . . . . . . . . Clinic Business Plan? . . . . . . . . . . . . . . Sigma Metathesis? . . . . . . . . . Clinic? . . . . . 90. 6 Cyclotomic Extensions; Fermat’s Last Theorem. 91 The basic results . . . . . . . . . . . . Transition Word Essay? . . . . . . . . . . . . . . Business? . Practice Literature Essays? . . Veterinary? . . . . . . 91 Class numbers of cyclotomic fields . . . . . . . . . . . . Ap English? . . . . . Veterinary Business Plan? . . . . . . . . 97 Units in cyclotomic fields . . . . . . . Literature Essays? . . . . . . . Veterinary Clinic Business Plan? . . . . . . . . . . . . Good? . . . . 97 The first case of Fermat’s last theorem for regular primes . . . . . . . . . . . Veterinary Business? . . 98 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100. 7 Valuations; Local Fields 101 Valuations . . . . . . . . . A Good Man Is Analysis? . . . . . . . . . . . . . . . . . . . . . . . . . . Veterinary Business? . . . 101 Nonarchimedean valuations . Hard To Find Essays? . . Business? . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Equivalent valuations . . . . . . . . . . . . . . . . Essay Friendship? . . . . . . . . . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . . . . . . . . . . . . Clinic Plan? . . . . . 105 Complete list of valuations for the rational numbers . . . . . . . . . . . . . . . . 105 The primes of a number field . . . . . . . . . . . . . . Essay Friendship? . . . . . . . . . . . Veterinary Clinic Business Plan? . . . 107 The weak approximation theorem . Metathesis? . . . . . . . . . . . Veterinary? . . . Ap English Literature Essays? . . . . . . . . . . 109 Completions . Veterinary Plan? . . . . . . . . . . . . . . Good Essay? . . . Veterinary Business? . . . . . . . . . . . . . . . . . . . 110 Completions in the nonarchimedean case . . . . . . . . . . . . . . . . . . . Definition On True? . . . 111 Newton’s lemma . . . . . . . . . . . Clinic? . . . . . . Practice Literature? . . . . . . . . . . . . . . . . . 115 Extensions of nonarchimedean valuations . . . . . . . . . . . . . . Clinic Plan? . . . . . . . 118.

Newton’s polygon . . . . . . Sigma Bond Metathesis? . . . . . . . . . . . Veterinary Plan? . . . . . . . . . . . . . . . . . 120 Locally compact fields . . . . . . . . . . . . . . . . . . . . . Man Is Hard To Find Essays? . . . . . . . . . . 122 Unramified extensions of clinic business, a local field . . . . . . . . . . . . . . . . Essay On True? . . . . . . . 123 Totally ramified extensions of clinic plan, K . . . . . Sigma Metathesis? . . . . . . . . . Veterinary Plan? . . . Ap English? . . . . . . . . . 125 Ramification groups . . . . . . . . . . . . . . . . . . . . . Veterinary Clinic? . . . . . . . . . . . . 126 Krasner’s lemma and applications . . Ap English? . . . . . . . . Veterinary Clinic Plan? . . . . . . . . . . . . . . . 127 Exercises . . . . . . . . Psychology? . . . . . . . Veterinary Business? . . . . . . . . . . . . . . . . . . . . . . . 129. 8 Global Fields 131 Extending valuations . . . . . . . . . . . . . . . . Transition Word? . . . . . . Veterinary Plan? . . . . . . . . . . 131 The product formula . . . . . . . . . A Good Hard To Find Essays? . . . . . . . . . . . . . . . . . . . . . . . 133 Decomposition groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 The Frobenius element . . . . . . . Veterinary Clinic Business Plan? . . . . . . . . . . . . . . . . . Psychology Essay? . . . . . . . 137 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . Clinic Business? . Practice? . . . . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . . . . . . . . . . . . . . . . 140 Computing Galois groups (the easy way) . . Business? . Psychology Essay? . . . . . . . . . . . . . . . Clinic? . . . . 141 Applications of the Chebotarev density theorem . . . . . . . . Bond? . . . . . . . . . . 146 Exercises . . Clinic Plan? . . . . . . . . . . A Good Man Is Hard To Find Essays? . Clinic Business Plan? . . . . . . . . . . . . . . . . . Practice Literature? . . . . . . . . 147. A Solutions to veterinary clinic the Exercises 149. B Two-hour examination 155. We use the standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of integers; RD field of real numbers; CD field of complex numbers; Fp D Z=pZD field with p elements, p a prime number.

For integers m and n, mjn means that m divides n, i.e., n 2mZ. Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of a set S is denoted by jS j (so jS j is the number of elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i2I , is a function i 7! ai WI ! A. X Y X is a subset of Y (not necessarily proper); X. def D Y X is defined to be Y , or equals Y by definition; X Y X is isomorphic to Y ; X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. It is essay standard to use Gothic (fraktur) letters for ideals: a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q. The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and multilinear algebra. An undergraduate number theory course will also be helpful. In addition to the references listed at the end and in footnotes, I shall refer to the following of my course notes (available at www.jmilne.org/math/): FT Fields and Galois Theory, v4.22, 2011.

GT Group Theory, v3.11, 2011. CFT Class Field Theory, v4.01, 2011. I thank the following for providing corrections and comments for earlier versions of these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Francesc Castella?; Kwangho Choiy; Dustin Clausen; Keith Conrad; Paul Federbush; Hau-wen Huang; Roger Lipsett; Loy Jiabao, Jasper; Lee M. Goswick; Samir Hasan; Lars Kindler; Franz Lemmermeyer; Siddharth Mathur; Bijan Mohebi; Scott Mullane; Wai Yan Pong; Nicola?s Sirolli; Thomas Stoll; Vishne Uzi; and others. PARI is an open source computer algebra system freely available from http://pari.math.u- bordeaux.fr/. FERMAT (1601–1665). Stated his last “theorem”, and proved it for mD 4. He also posed the clinic, problem of essay, finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the units in Z? p A?. The English mathemati- cians found an veterinary clinic plan, algorithm for solving the problem, but neglected to prove that the algorithm always works.

EULER (1707–1783). He introduced analysis into the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. LAGRANGE (1736–1813). He found the complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes, and he proved that the essay proud, algorithm for solving (1) always leads to a solution, LEGENDRE (1752–1833). He introduced the “Legendre symbol” m p. , and gave an incom- plete proof of the quadratic reciprocity law. He proved the following local-global principle for quadratic forms in three variables over Q: a quadratic form Q.X;Y;Z/ has a nontrivial zero in veterinary Q if and only if it has one in R and the congruence Q 0 mod pn has a nontrivial solution for all p and n. Practice Literature Essays? GAUSS (1777–1855). He found the first complete proofs of the quadratic reciprocity law. He studied the Gaussian integers Z?i ? in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields.

DIRICHLET (1805–1859). He introduced L-series, and used them to prove an analytic for- mula for the class number and a density theorem for the primes in an arithmetic progression. He proved the following “unit theorem”: let ? be a root of a monic irreducible polynomial f .X/ with integer coefficients; suppose that f .X/ has r real roots and 2s complex roots; then Z??? is a finitely generated group of rank rC s?1. KUMMER (1810–1893). He made a deep study of the arithmetic of cyclotomic fields, mo- tivated by a search for higher reciprocity laws, and showed that unique factorization could be recovered by clinic plan the introduction of “ideal numbers”. He proved that Fermat’s last theorem holds for regular primes. HERMITE (1822–1901). He made important contributions to quadratic forms, and he showed that the roots of a polynomial of degree 5 can be expressed in terms of elliptic functions.

EISENSTEIN (1823–1852). He published the first complete proofs for the cubic and quartic reciprocity laws. Essay Proud American? KRONECKER (1823–1891). He developed an clinic, alternative to Dedekind’s ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of sigma, number fields (the Kronecker liebster Jugendtraum). RIEMANN (1826–1866). Studied the Riemann zeta function, and made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to veterinary clinic business the equa- tion. DEDEKIND (1831–1916). He laid the modern foundations of algebraic number theory by finding the correct definition of the ring of integers in a number field, by proving that ideals factor uniquely into products of prime ideals in such rings, and by showing that, modulo principal ideals, they fall into finitely many classes.

Defined the zeta function of a number field. WEBER (1842–1913). Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (1861–1941). He gave the first definition of the field of p-adic numbers (as the set of infinite sums. n, an 2 f0;1; : : : ;p?1g). HILBERT (1862–1943).

He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of his famous problems were on number theory, and have also been influential. TAKAGI (1875–1960). He proved the fundamental theorems of abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (1882–1935). Essay Proud? Together with Artin, she laid the foundations of modern algebra in which axioms and conceptual arguments are emphasized, and she contributed to the classification of central simple algebras over number fields.

HECKE (1887–1947). Introduced HeckeL-series generalizing both Dirichlet’sL-series and Dedekind’s zeta functions. Clinic? ARTIN (1898–1962). Proud? He found the “Artin reciprocity law”, which is the main theorem of class field theory (improvement of Takagi’s results). Introduced the Artin L-series. HASSE (1898–1979). He gave the first proof of local class field theory, proved the Hasse (local-global) principle for all quadratic forms over number fields, and contributed to the classification of central simple algebras over number fields. BRAUER (1901–1977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. WEIL (1906–1998).

Defined the Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (1909–84). The main statements of class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. With his introduction of ide?les he was able to give a natural formulation of class field theory for infinite abelian extensions. IWASAWA (1917–1998). He introduced an important new approach into algebraic number theory which was suggested by the theory of clinic, curves over finite fields.

TATE (1925– ). Essay Proud American? He proved new results in group cohomology, which allowed him to give an elegant reformulation of class field theory. With Lubin he found an explicit way of generating abelian extensions of local fields. LANGLANDS (1936– ). The Langlands program2 is business plan a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to be confused with its geometric analogue, sometimes referred to as the geometric Langlands pro- gram, which appears to lack arithmetic significance. Introduction It is greatly to be lamented that this virtue of the [rational integers], to be decomposable into essay american prime factors, always the same ones for a given number, does not also belong to the [integers of cyclotomic fields].

Kummer 1844 (as translated by Andre? Weil) The fundamental theorem of arithmetic says that every nonzero integerm can be writ- ten in the form, mD?p1 pn; pi a prime number, and that this factorization is essentially unique. Consider more generally an integral domain A. An element a 2A is said to be a unit if. it has an inverse in A (element b such that ab D 1D ba). I write A for the multiplicative group of units in A. An element of A is said to prime if it is neither zero nor a unit, and if. If A is a principal ideal domain, then every nonzero element a of A can be written in the form, aD u1 n; u a unit; i a prime element; and this factorization is unique up to order and replacing each i with an associate, i.e., with its product with a unit. Our first task will be to discover to what extent unique factorization holds, or fails to hold, in number fields.

Three problems present themselves. First, factorization in a field only makes sense with respect to a subring, and so we must define the “ring of integers” OK in our number field K. Secondly, since unique factorization will fail in general, we shall need to find a way of measuring by how much it fails. Business? Finally, since factorization is only considered up to units, in order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Sigma Bond Metathesis? Let K be an veterinary business, algebraic number field. Each element ? of K satisfies an equation. ?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is bond metathesis difficult to apply. We shall show (2.11) that ? is an algebraic integer if and only if its minimum polynomial over Q has coefficients in veterinary clinic plan Z. Consider for example the field K D Q? p d?, where d is a square-free integer. The. minimum polynomial of ? D aCb p d , b ¤ 0, a;b 2Q, is. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an algebraic integer if and only if. 2a 2 Z; a2?b2d 2 Z: From this it follows easily that, when d 2;3 mod 4, ? is an practice, algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an business, algebraic integer if and only if a and b are either both integers or both half-integers, i.e., For example, the minimum polynomial of 1=2C p 5=2 is X2?X ?1, and so 1=2C. is an algebraic integer in Q? p 5?.

Let d be a primitive d th root of 1, for example, d D exp.2i=d/, and letK DQ?d ?. Then we shall see (6.2) that. OK D Z?d ?D ?P. as one would hope. A nonzero element of an integral domain A is said to be irreducible if it is good not a unit, and can’t be written as a product of two nonunits. For example, a prime element is (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of A can be expressed as a product of irreducible elements in essentially one way. Is the ring of integers OK a unique factorization domain? No, not in general! We shall see that each element of OK can be written as a product of irreducible elements (this is true for all Noetherian rings), and so it is the uniqueness that fails. For example, in Z? p ?5? we have.

6D 2 3D .1C p ?5/.1?. To see that 2, 3, 1C p ?5, 1?. p ?5 are irreducible, and no two are associates, we use the. Veterinary Business? p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for ? 2OK , Nm.?/D 1 ” ? N? D 1 ” ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. In the. first case, ? is a unit, the second and third cases don’t occur, and in the fourth case ? is a unit. A similar argument shows that 2;3, and 1?. p ?5 are irreducible. Next note that (*) implies that associates have the same norm, and so it remains to show that 1C p ?5 and. 1? p ?5 are not associates, but. Good Essay? has no solution with a;b 2 Z. Why does unique factorization fail in OK? The problem is that irreducible elements in. OK need not be prime. In the above example, 1C p ?5 divides 2 3 but it divides neither 2. nor 3. In fact, in an integral domain in which factorizations exist (e.g. a Noetherian ring), factorization is unique if all irreducible elements are prime. What can we recover?

Consider. 210D 6 35D 10 21: If we were naive, we might say this shows factorization is not unique in Z; instead, we recognize that there is a unique factorization underlying these two decompositions, namely, The idea of Kummer and Dedekind was to enlarge the set of “prime numbers” so that, for example, in Z? p ?5? there is a unique factorization, 6D .p1 p2/.p3 p4/D .p1 p3/.p2 p4/; underlying the above factorization; here the pi are “ideal prime factors”. How do we define “ideal factors”? Clearly, an ideal factor should be characterized. by the algebraic integers it divides. Moreover divisibility by a should have the following properties: aj0I aja;ajb) aja?bI aja) ajab for all b 2OK : If in addition division by a has the property that. ajab) aja or ajb; then we call a a “prime ideal factor”.

Since all we know about an ideal factor is the set of elements it divides, we may as well identify it with this set. Thus an ideal factor a is a set of elements of OK such that. 0 2 aI a;b 2 a) a?b 2 aI a 2 a) ab 2 a for all b 2OK I. it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of you will recognize that an ideal factor is what we now call an ideal, and a prime ideal factor is clinic plan a prime ideal. Sigma? There is an veterinary plan, obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. Man Is Analysis Essays? One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/ then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a ¤ 0, then there is an essentially unique factorization: .a/D p1 pn with each pi a prime ideal. In the above example, .6/D .2;1C p ?5/.2;1?.

In fact, I claim. .2;1C p ?5/.2;1?. .3;1C p ?5/.3;1?. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. p ?5;6/. Since every gen- erator is divisible by 2, we see that. .2;1C p ?5/.2;1?. Conversely, 2D 6?4 2 .4;2C2. and so .2;1C p ?5/.2;1?. p ?5/ D .2/, as claimed. I further claim that the four ideals. .2;1C p ?5/, .2;1?. p ?5/, and .3;1?. p ?5/ are all prime. For example, the obvious map Z! Z? p ?5?=.3;1?. p ?5/ is surjective with kernel .3/, and so.

Z? p ?5?=.3;1?. which is an veterinary clinic business, integral domain. Good? How far is veterinary this from what we want, namely, unique factorization of elements? In other. words, how many “ideal” elements have we had to add to our “real” elements to get unique factorization. In a certain sense, only a finite number: we shall see that there exists a finite set S of ideals such that every ideal is of the form a .a/ for some a 2 S and some a 2OK . Better, we shall construct a group I of “fractional” ideals in which the principal fractional ideals .a/, a 2K, form a subgroup P of finite index. The index is called the class number hK of K. We shall see that. hK D 1 ” OK is a principal ideal domain ” OK is a unique factorization domain. Unlike Z, OK can have infinitely many units.

For example, .1C p 2/ is a unit of infinite. order in Z? p 2? W. p 2/m ¤ 1 if m¤ 0: In fact Z? p 2? D f?.1C. p 2/m jm 2 Zg, and so. Z? p 2? f?1gffree abelian group of rank 1g: In general, we shall show (unit theorem) that the roots of 1 in essay K form a finite group .K/, and that. OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for the development of algebraic number theory was the attempt to prove Fermat’s last “theorem”, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of “infinite descent”, i.e., from one solution, you show that you can construct a smaller solution, which leads to a contradiction3. The proof makes use of the factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and it was recognized that a stumbling block to clinic business proving the theorem for larger m is friendship that no such factorization exists into polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at the Paris Academy in 1847 in which he claimed to prove Fermat’s last theorem using the following ideas. Let p 2 be a prime, and suppose x, y, z are nonzero integers such that. Write xp D zp?yp D. Y .z? iy/; 0 i p?1; D e2i=p: He then showed how to obtain a smaller solution to the equation, and hence a contradiction.

Liouville immediately questioned a step in Lame?’s proof in which he assumed that, in order to show that each factor .z ? iy/ is a pth power, it suffices to show that the factors are relatively prime in pairs and their product is a pth power. In fact, Lame? couldn’t justify his step (Z?? is not always a principal ideal domain), and Fermat’s last theorem was not proved for almost 150 years. However, shortly after Lame?’s embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermat’s last theorem for all regular primes, i.e., for all primes p such that p does not divide the business plan, class number of Q?p?. Another application is to finding Galois groups. The splitting field of a polynomial f .X/ 2Q?X? is a Galois extension of Q. In a basic Galois theory course, we learn how to compute the Galois group only essay friendship when the degree is very small. By using algebraic number theory one can write down an algorithm to clinic plan do it for any degree. For applications of algebraic number theory to elliptic curves, see, for a good man is hard example, Milne 2006. Some comments on business the literature. COMPUTATIONAL NUMBER THEORY.

Cohen 1993 and Pohst and practice ap english literature essays Zassenhaus 1989 provide algorithms for most of the construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically. Cohen’s book is the more useful as a supplement to this course, but wasn’t available when these notes were first written. While the books are concerned with more-or-less practical algorithms for fields of small degree and small discriminant, Lenstra (1992) concentrates on finding “good” general algorithms. 3The simplest proof by infinite descent is that showing that p 2 is irrational. HISTORY OF ALGEBRAIC NUMBER THEORY. Dedekind 1996, with its introduction by Stillwell, gives an excellent idea of how algebraic number theory developed. Edwards 1977 is a history of algebraic number theory, con- centrating on the efforts to prove Fermat’s last theorem. The notes in Narkiewicz 1990 document the origins of most significant results in algebraic number theory.

Lemmermeyer 2009, which explains the origins of “ideal numbers”, and other writings by the same author, e.g., Lemmermeyer 2000, 2007. 0-1 Let d be a square-free integer. Complete the verification that the ring of integers in Q? p d? is as described. 0-2 Complete the verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is a factorization of .6/ into a product of prime ideals. CHAPTER 1 Preliminaries from Commutative. Many results that were first proved for rings of integers in number fields are true for more general commutative rings, and it is more natural to prove them in that context.1. All rings will be commutative, and have an identity element (i.e., an element 1 such that 1a D a for all a 2 A), and a homomorphism of rings will map the identity element to the identity element. A ring B together with a homomorphism of rings A! B will be referred to as an A-algebra. We use this terminology mainly when A is veterinary clinic plan a subring of B . In this case, for elements ?1; . ;?m of essay, B , A??1; . ;?m? denotes the smallest subring of B containing A and the ?i . It consists of all polynomials in the ?i with coefficients in A, i.e., elements of the form X.

ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to A??1; . ;?m? as the A-subalgebra of B generated by the ?i , and when B D A??1; . ;?m? we say that the ?i generate B as an business, A-algebra. For elements a1;a2; : : : of A, we let .a1;a2; : : :/ denote the smallest ideal containing the ai . It consists of finite sums. P ciai , ci 2 A, and it is called the ideal generated by. a1;a2; : : :. When a and b are ideals in A, we define. aCbD faCb j a 2 a, b 2 bg: It is again an ideal in A — in fact, it is the smallest ideal containing both a and b. If aD .a1; . ;am/ and bD .b1; . ;bn/, then aCbD .a1; . ;am;b1; . ;bn/: Given an ideal a in A, we can form the quotient ring A=a. Let f WA! A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the ideals of A=a and the ideals of A containing a, and. 1See also the notes A Primer of Commutative Algebra available on my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of A is prime if ab 2 a) a or b 2 a. An ideal a is prime if and only if the quotient ring A=a is an integral domain. A nonzero element of A is said to be prime if ./ is a prime ideal; equivalently, if jab) ja or jb.

An ideal m in A is proud maximal if it is maximal among the proper ideals of A, i.e., if m¤A and there does not exist an ideal a ¤ A containing m but distinct from it. An ideal a is maximal if and only if A=a is a field. Every proper ideal a of A is contained in a maximal ideal — if A is Noetherian (see below) this is obvious; otherwise the clinic business plan, proof requires Zorn’s lemma. In particular, every nonunit in A is contained in a maximal ideal. There are the implications: A is a Euclidean domain) A is a principal ideal domain ) A is a unique factorization domain (see any good graduate algebra course). Bond Metathesis? Ideals in products of rings.

PROPOSITION 1.1 Consider a product of rings AB . If a and b are ideals in A and B respectively, then ab is an ideal in veterinary plan AB , and every ideal in AB is of this form. The prime ideals of AB are the ideals of the transition essay, form. pB (p a prime ideal of A), Ap (p a prime ideal of B). PROOF. Let c be an ideal in AB , and let. aD fa 2 A j .a;0/ 2 cg; bD fb 2 B j .0;b/ 2 cg: Clearly a b c. Conversely, let .a;b/ 2 c. Then .a;0/ D .a;b/ .1;0/ 2 c and .0;b/ D .a;b/ .0;1/ 2 c, and so .a;b/ 2 ab: Recall that an ideal c C is prime if and only if C=c is an integral domain. The map. has kernel ab, and hence induces an clinic business plan, isomorphism. A Good Man Is Hard To Find? Now use that a product of rings is an integral domain if and only if one ring is zero and the other is an integral domain. 2. REMARK 1.2 The lemma extends in an obvious way to a finite product of rings: the clinic plan, ideals in A1 Am are of the form a1 am with ai an ideal in Ai ; moreover, a1 am is prime if and only if there is a j such that aj is a prime ideal in Aj and ai DAi for metathesis i ¤ j: A ring A is Noetherian if every ideal in A is finitely generated.

PROPOSITION 1.3 The following conditions on clinic plan a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of ideals. eventually becomes constant, i.e., for some n, an D anC1 D . (c) Every nonempty set S of ideals in A has a maximal element, i.e., there exists an ideal in S not properly contained in any other ideal in S . Essay Proud? PROOF. (a) (b): Let a D S. ai ; it is an ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of business plan, S , then there exists an a2 2 S such that a1 a2. If a2 is not maximal, then there exists an a3 etc.. From (b) we know that this process will lead to a maximal element after only finitely many steps. (c) (a): Let a be an ideal in A, and let S be the set of finitely generated ideals contained in a. Then S is nonempty because it contains the zero ideal, and so it contains a maximal element, say, a0 D .a1; : : : ;ar/. If a0 ¤ a, then there exists an practice essays, element a 2 ar a0, and .a1; : : : ;ar ;a/ will be a finitely generated ideal in a properly containing a0. This contradicts the definition of business, a0. 2. Essay On True? A famous theorem of Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the rings that arise naturally in algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian. For example, the ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is not Noetherian because the chain of ideals. never becomes constant.

PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of clinic business plan, irreducible elements. Sigma? PROOF. We shall need to use that, for elements a and b of an integral domain A, .a/ .b/ ” bja, with equality if and only if b D aunit: The first assertion is obvious. For the second, note that if a D bc and b D ad then a D bc D adc, and so dc D 1. Veterinary Clinic? Hence both c and d are units. Suppose the statement of the proposition is false for a Noetherian integral domain A. Then there exists an element a 2 A which contradicts the statement and is such that .a/ is maximal among the ideals generated by such elements (here we use that A is Noetherian). Since a can not be written as a product of irreducible elements, it is not itself irreducible, and so a D bc with b and c nonunits.

Clearly .b/ .a/, and the ideals can’t be equal for otherwise c would be a unit. From the maximality of .a/, we deduce that b can be written as a product of irreducible elements, and similarly for c. Thus a is a product of irreducible elements, and we have a contradiction. 2. REMARK 1.5 Note that the proposition fails for the ring O of all algebraic integers in the algebraic closure of sigma, Q in C, because, for example, we can keep in extracting square roots — an algebraic integer ? can not be an irreducible element of O because. p ? will also be. an algebraic integer and ? D p ? p ?. Thus O is not Noetherian. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Let A be a ring. An A-module M is said to be Noetherian if every submodule is finitely generated. PROPOSITION 1.6 The following conditions on an A-module M are equivalent: (a) M is Noetherian; (b) every ascending chain of submodules eventually becomes constant; (c) every nonempty set of submodules in M has a maximal element. PROOF.

Similar to the proof of Proposition 1.3. 2. PROPOSITION 1.7 Let M be an A-module, and let N be a submodule of M . If N and M=N are both Noetherian, then so also is M . Business? PROOF. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and M 00 have the same image in M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Then x?y 2M 00N M 0, and so x 2M 0. Now consider an ascending chain of submodules of M . A Good Man Is Analysis? If M=N is Noetherian, the image of the chain in M=N becomes constant, and if N is Noetherian, the veterinary clinic business, intersection of the chain with N becomes constant. Now the claim shows that the chain itself becomes constant. 2. PROPOSITION 1.8 Let A be a Noetherian ring. Then every finitely generated A-module is Noetherian.

PROOF. If M is generated by practice ap english essays a single element, then M A=a for some ideal a in business plan A, and the statement is obvious. We argue by essay friendship induction on the minimum number n of generators ofM . SinceM contains a submoduleN generated by n?1 elements such that the quotient M=N is generated by a single element, the statement follows from (1.7). 2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). LEMMA 1.9 (NAKAYAMA’S LEMMA) Let A be a local Noetherian ring, and let a be a proper ideal in A. Let M be a finitely generated A-module, and define. aM D f P aimi j ai 2 a; mi 2M g : (a) If aM DM , then M D 0: (b) If N is a submodule of M such that N CaM DM , then N DM: Rings of fractions. PROOF. (a) Suppose that aM D M but M ¤ 0. Clinic Business? Choose a minimal set of generators fe1; : : : ; eng for M , n 1, and write. e1 D a1e1C Canen, ai 2 a: Then .1?a1/e1 D a2e2C Canen: As 1? a1 is not in m, it is definition essay on true a unit, and so fe2; . ; eng generates M , which contradicts our choice of fe1; : : : ; eng. (b) It suffices to show that a.M=N/DM=N for then (a) shows that M=N D 0. Con- sider mCN , m 2M . From the assumption, we can write. aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. The hypothesis that M be finitely generated in the lemma is essential. For example, if A is a local integral domain with maximal ideal m ¤ 0, then mM DM for any field M containing A but M ¤ 0. Plan? Rings of fractions. Hard To Find Analysis? Let A be an integral domain; there is a field K A, called the veterinary business plan, field of fractions of A, with the property that every c 2K can be written in the form c D ab?1 with a;b 2A and b ¤ 0. Practice Ap English Literature Essays? For example, Q is the field of fractions of clinic, Z, and ap english literature essays k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. Clinic Plan? A subset S of A is ap english said to be multiplicative if 0 … S , 1 2 S , and S is closed under multiplication. If S is a multiplicative subset, then we define.

S?1AD fa=b 2K j b 2 Sg: It is obviously a subring of clinic business plan, K: EXAMPLE 1.10 (a) Let t be a nonzero element of A; then. St def D f1,t ,t2. g. is a multiplicative subset of A, and we (sometimes) write At for S?1t A. For example, if d is a nonzero integer, then2 Zd consists of those elements of Q whose denominator divides some power of d : Zd D fa=dn 2Q j a 2 Z, n 0g: (b) If p is a prime ideal, then SpDArp is a multiplicative set (if neither a nor b belongs to p, then ab does not belong to p/. We write Ap for S?1p A. For example, Z.p/ D fm=n 2Q j n is not divisible by definition on true friendship pg: 2This notation conflicts with a later notation in which Zp denotes the ring of p-adic integers. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an veterinary clinic plan, integral domainA and a multiplicative subset S ofA. For an ideal a of A, write ae for the ideal it generates in S?1A; for an ideal a of S?1A, write ac for aA. Then: ace D a for all ideals a of S?1A aec D a if a is a prime ideal of A disjoint from S: PROOF. Let a be an ideal in S?1A. Clearly .aA/e a because aA a and a is an ideal in psychology essay S?1A. For the reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . Then aD s .a=s/ 2 aA, and so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Clearly .S?1p/A p. For the reverse inclu- sion, let a=s 2 .S?1p/A, a 2 p, s 2 S . Consider the equation a. s s D a 2 p. Both a=s. and s are in A, and so at least one of a=s or s is in p (because it is prime); but s … p (by assumption), and so a=s 2 p: 2. PROPOSITION 1.12 Let A be an integral domain, and let S be a multiplicative subset of A. The map p 7! pe defD p S?1A is a bijection from the veterinary business plan, set of prime ideals in A such that pS D? to good psychology essay the set of prime ideals in S?1A; the inverse map is p 7! pA. PROOF.

It is easy to see that. p a prime ideal disjoint from S) pe is a prime ideal in S?1A, p a prime ideal in S?1A) pA is a prime ideal in A disjoint from S; and (1.11) shows that the two maps are inverse. 2. EXAMPLE 1.13 (a) If p is a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). (b) We list the prime ideals in some rings: Note that in general, for t a nonzero element of an integral domain, fprime ideals of Atg $ fprime ideals of A not containing tg. fprime ideals of business, A=.t/g $ fprime ideals of A containing tg: The Chinese remainder theorem. Recall the classical form of the theorem: let d1; . ;dn be integers, relatively prime in pairs; then for any integers x1; . ;xn, the congruences. The Chinese remainder theorem. have a simultaneous solution x 2 Z; moreover, if x is one solution, then the other solutions are the integers of the form xCmd with m 2 Z and d D. We want to translate this in terms of ideals. Integersm and n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. If m1; . ;mk are integers, then T .mi / D .m/ where m is the least common multiple. of the mi . Thus T .mi / . Q mi /, which equals.

Q .mi /. If the mi are relatively prime in. pairs, then mD Q mi , and metathesis so we have. Q .mi /. Plan? Note that in general, a1 a2 an a1a2 . an; but the two ideals need not be equal. These remarks suggest the following statement. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for any elements x1; . ;xn of A, the congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the elements of the form xC a with a 2. Q ai . In other words, the. natural maps give an exact sequence. PROOF. Suppose first that n D 2. As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. The element x D a1x2Ca2x1 has the required property. For each i we can find elements ai 2 a1 and bi 2 ai such that. ai Cbi D 1, all i 2: The product Q i2.ai Cbi /D 1, and lies in a1C. Q i2 ai , and so.

We can now apply the good psychology, theorem in the case nD 2 to obtain an element y1 of A such that. y1 1 mod a1; y1 0 mod Y. These conditions imply. y1 1 mod a1; y1 0 mod aj , all j 1: Similarly, there exist elements y2; . ;yn such that. yi 1 mod ai ; yi 0 mod aj for j ¤ i: The element x D P xiyi now satisfies the clinic, requirements. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Sigma Bond? It remains to veterinary clinic business prove that T. ai . We have already noted that T. ai . First suppose that nD 2, and let a1Ca2 D 1, as before. For c 2 a1a2, we have. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2. Practice Ap English? We complete the proof by induction. This allows us to assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . The theorem extends to A-modules. THEOREM 1.15 Let a1; . ;an be ideals in A, relatively prime in pairs, and let M be an A-module.

There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to veterinary clinic business use tensor products, which I now review. Review of tensor products. Let M , N , and P be A-modules. A mapping f WM N ! P is said to be A-bilinear if. f .mCm0;n/D f .m;n/Cf .m0;n/ f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is linear in each variable. A pair .Q;f / consisting of an A-module Q and an A-bilinear map f WM N !Q is called the tensor product of M and N if any other A- bilinear map f 0WM N ! P factors uniquely into f 0 D ? ?f with ?WQ! P A-linear. Essay? The tensor product exists, and is unique (up to a unique isomorphism making the clinic business plan, obvious diagram commute).

We denote it by M ?AN , and we write .m;n/ 7! m?n for f . The pair .M ?AN;.m;n/ 7!m?n/ is transition characterized by each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and business plan any other A-bilinear mapM N ! P is of the form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus. BilinA.M N;P /D HomA.M ?AN;P /: (b) TheA-moduleM?AN has as generators them?n,m2M , n2N , and as relations. 9=; all a 2 A; m;m0 2M; n;n0 2N: Tensor products commute with direct sums: there is a canonical isomorphism. Review of tensor products. It follows that if M and N are free A-modules3 with bases .ei / and word .fj / respectively, then M ?AN is a free A-module with basis .ei ? fj /. In particular, if V and W are vector spaces over a field k of dimensions m and n respectively, then V ?kW is a vector space over k of plan, dimension mn. Let ?WM !M 0 and ?WN !N 0 be A-linear maps. Then. .m;n/ 7! ?.m/??.n/WM N !M 0?AN 0. is A-bilinear, and therefore factors uniquely through M N !M ?AN . Essay? Thus there is a unique A-linear map ???WM ?AN !M 0?AN 0 such that.

REMARK 1.16 The tensor product of two matrices regarded as linear maps is called their Kronecker product.4 If A is mn (so a linear map kn! km) and B is r s (so a linear map ks! kr ), then A?B is the mr ns matrix (linear map kns! kmr ) with. 0B@ a11B a1nB. : : : . am1B amnB. 1CA : LEMMA 1.17 If ?WM !M 0 and ?WN !N 0 are surjective, then so also is. ???WM ?AN !M 0 ?AN. PROOF. Recall that M 0?N 0 is generated as an A-module by the elements m0?n0, m0 2 M 0, n0 2 N 0. Clinic Business Plan? By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the image of ??? contains a set of generators for M 0?AN 0 and so it is equal to it. 2. One can also show that if M 0!M !M 00! 0. is exact, then so also is. M 0?AP !M ?AP !M 00 ?AP ! 0: For example, if we tensor the exact sequence. with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an A-module.

Elements e1; : : : ; em form a basis for M if every element of M can be expressed uniquely as a linear combination of the ei ’s with coefficients in A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of A-modules, and M is said to be a free A-module of rank m. 4Kronecker products of matrices pre-date tensor products by about 70 years. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. The image of a?AM in M is. P aimi j ai 2 a, mi 2M g ; and so we obtain from the exact sequence (2) that. By way of essay proud american, contrast, ifM !N is injective, thenM ?AP !N ?AP need not be injective. For example, take A D Z, and note that .Z. m ! Z/?Z .Z=mZ/ equals Z=mZ. Business Plan? which is the zero map. PROOF (OF THEOREM 1.15) Return to essay american the situation of the theorem. When we tensor the isomorphism. with M , we get an isomorphism.

M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS. If A! B is an A-algebra and M is an A-module, then B?AM has a natural structure of a B-module for which. b.b0?m/D bb0?m; b;b0 2 B; m 2M: We say that B?AM is the clinic business, B-module obtained from M by extension of scalars. The map m 7! 1?mWM ! B ?AM has the following universal property: it is A-linear, and for any A-linear map ?WM ! N from M into a B-module N , there is a unique B-linear map ?0WB?AM !N such that ?0.1?m/D ?.m/. Thus ? 7! ?0 defines an isomorphism. HomA.M;N /! HomB.B?AM;N/, N a B-module: For example, A?AM DM . If M is a free A-module with basis e1; : : : ; em, then B?AM is a free B-module with basis 1? e1; : : : ;1? em. TENSOR PRODUCTS OF ALGEBRAS. If f WA! B and gWA!

C are A-algebras, then B ?A C has a natural structure of an A-algebra: the product structure is determined by the rule. .b? c/.b0? c0/D bb0? cc0. and the map A! B?AC is a 7! f .a/?1D 1?g.a/. For example, there is a canonical isomorphism. a?f 7! af WK?k k?X1; : : : ;Xm?!K?X1; : : : ;Xm? (4) Review of essay proud, tensor products. TENSOR PRODUCTS OF FIELDS. We are now able to compute K?k? if K is a finite separable field extension of a field k and ? is an arbitrary field extension of k. According to the primitive element theorem (FT 5.1), K D k??? for some ? 2K. Let f .X/ be the clinic business plan, minimum polynomial of ?. By definition this means that the map g.X/ 7! g.?/ determines an isomorphism.

Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by (3) and (4). Metathesis? Because K is separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. that are relatively prime in pairs. We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X?=.fi .X// is a finite separable field extension of ? of degree degfi . Thus we have proved the veterinary clinic business plan, following result: THEOREM 1.18 Let K be a finite separable field extension of k, and let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of ?, If ? is a primitive element for K=k, then the image ?i of ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for ? and ?i respectively, then. EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of bond metathesis, ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto the i th factor is. We note that it is essential to assume in (1.18) that K is separable over k. If not, there will be an ? 2K such that ?p 2 k but ? … k, and veterinary clinic plan the ring K?kK will contain an element ? D .??1?1??/¤ 0 such that. ?p D ?p?1?1??p D ?p.1?1/??p.1?1/D 0: Hence K?kK contains a nonzero nilpotent element, and essay american so it can’t be a product of veterinary plan, fields. NOTES Ideals were introduced and studied by Dedekind for rings of algebraic integers, and later by others in polynomial rings. It was not until the 1920s that the theory was placed in its most natural setting, that of arbitrary commutative rings (by Emil Artin and Emmy Noether). Embed this document on essay your website.

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comparasion essays You’re looking for a great compare and contrast essay topic to veterinary clinic business kick start your assignment. How can you choose an interesting topic that’s simultaneously narrow yet broad enough to provide you with plenty of great material? In this blog post, I’ll give you 70 compare and psychology contrast essay topics. I’ve also included links to sample essays for even more inspiration. With many of veterinary business these topics, I’ll also give you some basic compare and contrast points to get your ideas going. Note that my points are pretty general. When you write your compare and contrast essay, you’ll want to dig deep, do your research, and american find the most interesting facts. Let’s get started! Compare and Contrast Essay Topics: Influential People. People are the definition of unique—no two are exactly alike.

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Maya Angelou and Alice Walker are both well-known African American authors. Literature Essays? They are credited with contributing important fictional and non-fictional texts on the issues surrounding what it means to be black in the United States. They are both inspirational political and civil rights activists. They are both recipients of the Pulitzer Prize. In addition to writing, Angelou is also known for her work as an actress and film and television producer. Even though they addressed similar topics in their writings, the two authors wrote in very different styles—each with her own unique voice.

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8. Sigma Metathesis? Ellen DeGeneres vs. Oprah Winfrey. Ellen DeGeneres and Oprah Winfrey are both American television personalities who have hosted their own daytime talk shows. Both women have been extremely successful in their careers. As a comedienne, DeGeneres relies on witty humor and sarcastic criticism to convey her messaging, while the more serious Winfrey relies on inspiration, insights, and aha moments. “My grandmother started walking five miles a day when she was sixty. She’s ninety-seven now, and we don’t know where the hell she is.” – Ellen DeGeneres. 9. Booker T. Washington vs. W.E.B. DuBois.

Booker T. Washington and veterinary clinic plan W.E.B. DuBois were both early 20 th century civil rights activists. They were both great leaders of the black community. Both men were also accomplished scholars. Washington and DuBois disagreed on how to good psychology go about achieving racial equality. Washington believed that African Americans should earn equality through self-help and hard work. DuBois, conversely, believed that African Americans needed to become better educated and more politically active. Plato and Socrates were both ancient Greek philosophers.

They are considered to be the business, founders of Western philosophy. Both men also founded a school. Socrates was Plato’s teacher and mentor. Socrates expounded on good psychology essay, the wisdom of “I know that I know nothing.” Plato philosophized that the veterinary clinic, world of to find essays ideas was the clinic business plan, only constant and that the world of senses is changeable. Adolf Hitler and a good man is hard analysis essays Regina George both can be considered dictators in their own right. Both Hitler and George believed that people with blonde hair, blue eyes, and veterinary clinic business other Aryan features were somehow better than other people.

Adolf Hitler was a real person, dictator, and founder of Nazism in good psychology essay, Germany. Veterinary Clinic Business? He was responsible for countless human atrocities, including the psychology, murder of over 6 million Jews. Regina George is a fictional character from the 2004 movie Mean Girls . George’s biggest human atrocity is the creation of the Burn Book , a book that libels the other students in school. Both Pope Francis and Macklemore believe in love, equality, and showing compassion to clinic plan others. Both are cultural icons. Both men have the ability to reach out to and speak to the masses.

Pope Francis is the leader of the Catholic Church. Macklemore is an American entertainer and transition word hip-hop rapper. Macklemore “wears your grandad’s clothes,” while Pope Francis is veterinary clinic business old enough to transition essay be your grandad. 13. Clinic Business? Lin-Manual Miranda vs. William Shakespeare. Both playwrights have changed the face of theater forever. Both playwrights have been criticized for creating historically inaccurate works.

Both writers make political and social commentary in psychology essay, their plays. Shakespeare’s plays were not musicals (though they often contained musical components). Miranda’s most famous play, Hamilton , is a wildly successful musical. Shakespeare lived and clinic business plan wrote in the 16th century, while Miranda is a contemporary playwright. Reagan and Trump were both television and media stars before becoming president. When they were first elected, many people feared that neither one of these men would be strong or effective leaders. When Reagan took office, interest rates and inflation were high. Trump, on the other hand, entered office with historically low inflation and interest rates. Many people believed Reagan understood the problems with government and believed he was willing to work with citizens to help solve problems. Transition Word Essay? Trump has claimed that he alone will be able to solve problems. Compare and Contrast Essay Topics: History.

17. Medicine in Ancient Greece vs. Clinic Business Plan? medicine in Ancient China. 18. Media coverage of past wars vs. media coverage of current wars. Compare and Contrast Essay Topics: Religion. When you set out to compare and contrast world religions, you may feel like you have fallen down a rabbit hole. A student I know once wrote an essay comparing Jesus Christ and Snow White—talk about falling down a rabbit hole! What’s more, the crossovers between two opposing religions can be surprising.

Protestants were originally a part of the Catholic Church. So they still believe that Jesus is the savior and that he was crucified and rose from the dead to save humanity. Among their many differences, Catholics believe in Holy Water, purgatory, and that those wafers you eat at mass are actually, literally the body of Christ. Protestants don’t believe in any of that. (While protestants do take communion, they don’t think the wafers are really a piece of flesh.) Catholics have nuns, priests, and a pope. Protestants don’t. 22. Theravada vs. Mahayana Buddhism. Both Theravada and Mahayana Buddhism follow Dharma, the teachings of Buddha.

Both reject the notion of a god or gods (they are nontheistic ). Both believe theistic religions are misguided. Theravada and Mahayana Buddhism have some basic philosophical differences. For example, Theravada Buddhists believe that Buddha died, leaving his teachings behind. Mahayana Buddhists believe that Buddha lives on essay, in various incarnations. Judaism and Christianity are both monotheistic religions that teach about veterinary clinic business, one almighty God.

Both are Abrahamic religions. A Good To Find Essays? Both religions teach the Old Testament. Both believe in business plan, angels, demons, and definition spirits as spiritual beings. Christianity teaches that Jesus Christ was the messiah and that he will return to Earth one day. Business Plan? Christians believe that Jesus died for the sins of humankind. Judaism teaches that Jesus was a teacher and essay on true friendship a prophet of God, but not God himself. Judaism also rejects the notion of Jesus’s sacrifice. 24.

Religious mindset vs. cult following. Both organized religions and cults seek spiritual enlightenment and follow the clinic business, teachings of practice literature a leader. Both have members with varying levels of practice and extremism. Cults are more likely to attempt to extort money from their followers. Cults are also more likely than religions to use mind-control to force members into following the plan, teachings of the sigma metathesis, cult. Jesus Christ and Buddha were both spiritual leaders. Both were teachers and tutors during their time on Earth. Both were miraculously conceived, performed miracles, and were tempted but overcame the devil. Though Jesus performed several notable miracles, Buddha did not. Veterinary Clinic? Buddha believed that he lived his life in search of the truth, while Jesus believed he was the truth.

Compare and Contrast Essay Topics: Countries and Politics. Compare and Contrast Essay Topics: Education. You can compare and contrast different facets of education until you are as blue as a Smurf. To succeed, you’re better served by pitting two very specific elements of education against each other. The trick is to narrow your angle down to a tight focus while leaving room to transition write enough compelling points to describe the veterinary clinic business plan, differences and similarities. 33.

Private vs. public universities. Both private and public universities offer a range of academic courses and opportunities. Both allow students to earn a bachelor’s, master’s, or doctorate degree. Private universities are typically more expensive. The College Board reports that for good essay the 2013-2014 school year, private universities cost over clinic, $30,000 for one year of tuition, while public universities cost just over $8,000 for in-state and $22,000 for out-of-state tuition. However, private institutions are known to offer better financial aid options. Both small and large universities can provide students with the opportunities to achieve educational goals. Both offer bachelor’s degrees. (We’re not comparing the size of their parties here…but you could!) Small universities offer individualized majors, smaller class sizes, and hands-on learning opportunities.

Large universities offer state-of-the art research facilities, a wider range of academic programs, and definition friendship more student clubs and events. High school is a stepping stone to attending college. Students need to successfully complete the curriculum of veterinary plan both educational institutions in order to advance in many careers. Both institutions employ instructors to teach coursework. Bond Metathesis? Both require students to refer to textbooks. Both require substantial homework. High school allows for fewer academic choices. Students are typically limited to veterinary business plan a state-designed curriculum that does not take into account a student’s particular strengths and interests. College is a time for students to proud focus on courses that are specific to their career goals.

Both online and traditional classes offer education toward a degree program. Students have access to instructors in business plan, both class formats. Both involve some form of discussion among classmates. Both often have group projects. Online classes allow for more flexibility than traditional classes.

Students can take an definition essay on true friendship, online course at any time from the veterinary, comfort of their homes. Traditional classes require students to attend class at essay proud american, a specific time and location. The access to instructors is virtual in online classes and in-person in traditional classes. Online classes often have discussion board posts that replace live classroom discussions. 37. Veterinary Clinic Business? Homeschooling vs. public education.

Homeschooling and public education provide specific guidelines and learning outcomes for definition friendship students. Both provide a way for children to experience and learn about the world around them. Public education provides more opportunities for business socialization than homeschooling. Homeschooling can be more demanding on parents as they must be both full-time parent and full-time teacher. 38. Advanced placement (AP) classes vs. honors classes. Both types of classes provide more rigor than traditional high school classes and are geared toward higher achieving students.

Both types of classes can strengthen students’ college applications. Honors courses are essentially high school courses that cover subject matter in transition, greater depth. AP courses are designed to mirror college-level courses. Veterinary? High school honors courses do not allow students to earn college credit. Ap English Literature? Some high school AP courses allow students to earn college credit. Compare and Contrast Essay Topics: Film and Literature. Chances are, if you’ve been in English class for any length of veterinary business time, you’ve been asked to compare and contrast two bodies of literature or other fictional work. This is your chance to surprise the heck out of your professor (and get a good grade) by definition essay, making unexpected comparisons. Following are some topic examples. Both pieces of veterinary business plan literature investigate vampire lore.

Both use foreshadowing as a literary device to move the story forward. Both also share three themes: attraction, addiction, and repulsion. Twilight’ s 21 st century vampires are sexy and sparkly. Edward Cullen of Twilight is essay charming and irresistibly dangerous. On the other hand, Dracula is a 19 th century blood-thirsty beast that anyone would resist. Bram Stoker’s Dracula had a thick mustache, a large nose, and white hair that “grew scantily round the temples but profusely elsewhere.” (Bella, eat your heart out.) Also, there are no werewolves in Dracula . Both Breaking Bad and Lost were hit TV series of the early 21st century. Both shows have won a Golden Globe Award and an Emmy. Both have a character named Walt. Both use flashforward scenes to move the plan, narrative forward. Breaking Bad investigates the concept of the anti-hero in the character Walter White, who is easy to practice essays hate but hard not to cheer on as he descends into a life of crime.

Lost investigates the virtues of veterinary business plan a classic hero in the likeable character of Jack Shephard. 41. A Rose for Emily vs. Definition? The Yellow Wallpaper. Both stories depict women being forced into isolation as a result of veterinary clinic plan “lunacy.” Both stories explore the definition of to find analysis essays femininity and business the repression of women. Both stories explore dysfunctional relationships between women and men. “The Yellow Wallpaper” tells the story of a woman’s experience of being shut into good psychology essay a bedroom for a summer. “A Rose for Emily” explores a woman’s experience of clinic shutting herself into transition essay a house for 40 years. Both stories investigate these themes in very different ways.

42. Fahrenheit 451 vs. Red Scarf Girl. Both stories uncover themes of cultural repression. Both stories revolve around the human experience of defying unjust government regulations. Censorship and loss of individual rights are themes in both books. Fahrenheit 451 is a science fiction novel about how the character Montag defies authority and plan begins to read the on true, books that he had been ordered to burn. Red Scarf Girl is a true story about a young girl, Ji-li, and her experience of defiance and resistance during China’s cultural revolution.

43. The Day After Tomorrow vs. I am Legend. Both movies address the theme of the apocalypse and clinic business plan post-apocalyptic struggles for survival. Both movies touch on deeply rooted fears of humanity and our intrinsic understanding that the human race must eventually meet its end. Both showcase a lone scientific hero who tries to save humanity. In I Am Legend, the harbinger of the sigma metathesis, apocalypse is a plague that turns people into killer monsters. In the veterinary clinic, Day After Tomorrow , an psychology essay, out-of-control storm ushers in an apocalyptic ice age. Both novels create a dystopian vision of life in the future where society has been divided into factions. Both novels are coming-of-age love stories. Both stories require the female protagonists to partake in violent rituals.

The Hunger Games explores themes of violence and corruption, and the defiance of unjust authority. Divergent explores themes of following your heart despite the odds. 45. The Strange Case of Dr. Jekyll and business plan Mr. Hyde vs.

The Picture of a good analysis essays Dorian Gray. Both of these stories feature a protagonist with a dark side that they try to hide from the world. Veterinary? Through such a lens, both stories share the theme of duality. While Dr. Jekyll swings from being good and bad, Dorian Gray slowly transforms from good to bad as the story progresses. Dr. Jekyll outwardly manifests his dark side as Mr. Hyde, but Dorian keeps his darkness hidden, only acting on essay on true friendship, his impulses in secret. Business? Dorian’s portrait is the only outward clue of his wrongdoing—the picture reveals his true nature with every wicked act. Both novels tell the story of a dystopian future where society is oppressed by sigma bond metathesis, an all-knowing and veterinary clinic plan watchful authority. Both novels also have segments of the population in extreme poverty.

1984 is commentary on the dangers of a totalitarian state, and it’s set in Europe. The protagonist is an adult man. Essay On True? The Hunger Games is a coming-of-age story with a US-inspired setting. The protagonist is a young woman. These two Shakespearean plays have a lot of common. They both revolve around regicide (the murder of a king). Both involve power plays, scandals, and manipulation (but don’t all Shakespeare tragedies?). Macbeth relies on prophecies and the fulfillment of veterinary business plan prophecies as one of word its main literary hooks—it’s a story filled with magic and witches. Julius Caesar is realism-based, focusing on the plausible dramas and tragedies of the Roman Empire. Both have cool, wolfish names.

Both are heroes who undertake epic adventures to fight evil. Clinic Business Plan? Both experience serious transitions during their journeys. Beowulf was created in the ancient stew of medieval literature. Wolverine was created in on true friendship, 1974. Beowulf revels in veterinary clinic plan, glories, while Wolverine’s good deeds go without reward.

Beowulf and Odysseus both must battle monsters on their epic journeys. Both heroes share many of the same traits: courage, intelligence, and strength. Good Essay? Both men use all three traits to win their respective battles. Odysseus makes it safely home and dies in his old age. Beowulf dies after slaying a dragon. Odysseus uses deception while battling enemies, while Beowulf doesn’t play games to gain an advantage against his enemies. 50. 12 Angry Men stage play vs. Clinic Plan? a film adaptation of 12 Angry Men. Both the play and the movie follow the same basic plot, with jurors attempting to determine the guilt of the accused. In both, the jurors are nameless, known only by their juror numbers.

The scenes are more detailed in the movie compared to the descriptions in the play. The producers took some liberties with adding and/or altering dialogue from the original screenplay. Bonus info: 12 Angry Men was made into a movie in 1957 and in 1997. You might also write a compare and contrast essay about the two film adaptations. Compare and Contrast Essay Topics: Art. Compare and essay Contrast Essay Topics: Culture and Society. Here are some ideas. 57.

Korean concept of beauty vs. Veterinary Clinic Business? American concept of beauty. Both Americans and Koreans value thinness as an ideal beauty standard. Both cultures hold ideals of beauty that are impossible for most people to achieve. Americans focus on definition friendship, beauty standards, such as blonde hair, blue eyes, big breasts (on women), round butts, and tans. Veterinary? Korean beauty focuses on ideal face and eye shapes and essay pale skin. Cohabitation and clinic business plan marriage both involve two people living together and sharing resources. Both also usually mean two people are in a serious relationship. Marriage involves a legal and binding contract between two people that includes healthcare benefits and tax benefits. Word Essay? Cohabitation does not require a legal contract and does not include such benefits.

Both teenagers and toddlers are moody, willful, and needy. Both teens and toddlers are usually exploring newly found independence. Both often require supervision in many areas. Teenagers are more mobile than toddlers and can be far more destructive in their actions. Toddlers are a thousand times more adorable than teenagers. Teenagers have the ability to reason, unlike toddlers, but rarely take advantage of this ability. 60. Traditional Chinese parenting vs. Western parenting. Both Chinese and Western parents want the best for their children.

Both educate and business plan discipline their children in bond metathesis, ways they feel will best prepare their children for the future. Compared to Western parents, Chinese parents (particularly “Tiger Moms”) are very strict. They do not allow their children much free time and stress discipline and plan rigid schedules. 61. Good Psychology Essay? Authoritative parenting vs. permissive parenting. In both parenting styles, parents attempt to communicate with their children. In both styles, parents are also likely to encourage creativity and expression in their children. Authoritative parents are assertive and in control while not being extremely demanding. Permissive parents are more lenient and likely to avoid conflict with their children. Compare and Contrast Essay Topics: Technology. Hopefully, you’ve found a topic that will make writing your essay fun and business easy.

If you need even more ideas, here is a web app that helps you compare just about sigma bond, anything. And here’s a cool website with compare and contrast topics galore. Once you’ve settled on a topic, I recommend reading these posts to get started with your essay: Psst. 98% of Kibin users report better grades! Get inspiration from over 500,000 example essays. About the business plan, Author.

Naomi Tepper is a former Kibin editor, the sigma bond, former content manager for the Kibin blog, and forever a word nerd.