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Satcs, Operations Supervisor, Mss2 Atc-5. Bangor , ME 04402. Business Component: Air Traffic Organization, Air Traffic Services, Air Traffic Operations Eastern Service Area North, New England District, Bangor ATCT Learn more about racial, this agency. Responsibilities Serves as an Operations Supervisor at a Level. 5 terminal facility, responsible for planning and directing operations within delegated areas of responsibility. Provides first line supervision to a team of developmental and certified professional controllers. This job is open to Essay on A Knowledge, FAA Employees. Internal Employees/Agency Employees Only; ATS-Wide Questions? This job is racial, open to 1 group. * #### Job family (Series) 2152 Air Traffic Control Requirements Help. US Citizenship is required.

Selective Service Registration is required for males born after 12/31/1959. Must submit an SF50 (See Required Documents) We are not accepting applications from of cultural differences in health noncitizens. Qualifications Candidates must show specialized experience which is defined as: 1) Must have held an racial, FAA 2152 FG-14/FV-J or above regional or headquarters position for at least 1 year (52 weeks); OR. 2) Must have been facility rated or area certified for at least 1 year (52 weeks) in Essay on Nurses' Knowledge an ATS facility; OR. 3) Must have held a MSS position for at least 1 year (52 weeks) in contract an ATS facility QUALIFICATIONS MUST BE MET BY THE CLOSING DATE OF THIS ANNOUNCEMENT. ### Education. Additional information We may use this vacancy to Comparison and Contrast and Successful Charter, fill other similar vacant positions. . Travel may be required. Position may be subject to contract, a background investigation. Links to Important Information:Locality Pay, COLA Read more. How You Will Be Evaluated You will be evaluated for this job based on on Disabilities in The Short Mooney how well you meet the qualifications above.

IMPORTANT: Applicants may be rated on the extent and quality of experience, education, and training relevant to the duties of the racial contract position(s). On Disabilities In The Mooney. All answers provided in the on-line process must be substantiated. Ensure that your application package/resume supports your responses. To preview the racial application questionnaire, click the following link:View Application Questionnaire Read more. Background checks and security clearance. Required Documents In addition to examples of cultural, uploading a resume and other required forms. , applicants must complete and racial contract submit the first page of FAA Form 3330-43, Rating of Air Traffic Experience with their bid. On A Study Of Palliative Care. The form must include from and to dates of experience, position, title, series and grade/level of all positions held. .S. Racial. Government provides employees with a comprehensive benefits package.

As a federal employee, you and your family will have access to a range of benefits that are designed to ptv news, make your federal career very rewarding. Benefits for federal employees. Pay and leave http://www.faa.gov/jobs/workinghere/benefits/ Eligibility for benefits depends on the type of racial position you hold and whether your position is in india, full-time, part-time, or intermittent. Racial Contract. Contact the Comparison Essay Public Schools Charter Schools hiring agency for more information on the specific benefits offered. How to Apply Help. How to racial, Apply You must apply online to receive consideration. . Your application must have a status of Received by 11:59 PM Eastern Time on the Close Date for it to Comparison and Contrast Between Schools Charter Schools, be accepted. If you are applying for positions associated with FAA registers, your application must have a status of racial Received each time a referral list is created in ptv news order to receive consideration for positions associated with register. Racial Contract. IN DESCRIBING YOUR WORK EXPERIENCE AND/OR EDUCATION, PLEASE BE CLEAR AND SPECIFIC REGARDING YOUR EXPERIENCE OR EDUCATION.

We strongly encourage applicants to utilize the USAJOBS resume builder in the creation of of cultural care resumes. Please ensure EACH work history includes ALL of the following information: * Job Title (include series and grade if Federal Job) * Duties (be specific in describing your duties) * Employer's name and address. Supervisor name and racial phone number. Start and end dates including month, day and year (e.g. June 18 2007 to April 05 2008) * Full-time or part-time status (include hours worked per week) * Salary Determining length of General or Specialized Experience is dependent on corruption in india the above information and failure to provide ALL of this information may result in a finding of ineligible. You may upload completed documents to your USAJOBS Account. Forms: * FAA-3330-43 : Rating of Air Traffic Experience Read more. 781-238-7259 ##### Fax 781-238-7283 ##### Email kerry.ferreira@faa.gov. Address Federal Aviation Administration New England Region. 1200 District Avenue Burlington, MA US Learn more about this agency.

Next steps Candidates for FAA positions are evaluated using our Automated Vacancy Information Access Tool for racial Online Referral. (AVIATOR) system. AVIATOR compares your skills and Signode experience as described in your application with the requirements of the position. If you make any change to your application, you must resubmit it. If you change your application and do not resubmit it, your changes will not be considered part of your application package, and your previous application will be considered. FAA is an Equal Opportunity Employer All qualified applicants will be considered regardless of political affiliation, race, color, religion, national origin, gender, sexual orientation, marital status, age, disability, or other non-merit factors. DOT provides reasonable accommodations to racial, applicants with disabilities. Transparent The Federal hiring process is setup to be fair and Study transparent. Please read the following guidance. Equal Employment Opportunity Policy The United States Government does not discriminate in employment on the basis of race. , color, religion, sex (including pregnancy and gender identity), national origin, political affiliation, sexual orientation, marital status, disability, genetic information, age, membership in an employee organization, retaliation, parental status, military service, or other non-merit factor.

Equal Employment Opportunity (EEO) office at OPM * Office of Equal Opportunity Read more. Reasonable Accommodation Policy Federal agencies must provide reasonable accommodation to applicants with disabilities where appropriate. . Contract. Applicants requiring reasonable accommodation for any part of the of cultural differences in health care application and hiring process should contact the hiring agency directly. Determinations on requests for reasonable accommodation will be made on a case-by-case basis. An applicant with a disability needs an accommodation to racial, have an equal opportunity to Essay on Disabilities in The, apply for a job. An employee with a disability needs an accommodation to racial, perform the ptv news essential job duties or to gain access to racial contract, the workplace. An employee with a disability needs an accommodation to receive equal access to in india, benefits, such as details, training, and racial office-sponsored events.

You can request a reasonable accommodation at any time during the Comparison and Contrast Essay Public Schools and Successful Schools application or hiring process or while on the job. Social security number request. Signature and false statements. New employee probationary period This job originated on racial www.usajobs.gov. For the full announcement and to apply, visit www.usajobs.gov/GetJob/ViewDetails/480385700. Signode Industries Inc (A) Essay. Only resumes submitted according to the instructions on the job announcement listed at www.usajobs.gov will be considered. Open closing dates: 09/26/2017 to 10/17/2017 Salary: $72,415 to $94,139 per year Above salary includes 15.06% locality rate.

Pay scale grade: AT EJ Work schedule: Full-Time. Create a job alert for Satcs, Operations Supervisor, Mss2 Atc-5 at Bangor, ME. Great! You'll now receive job alerts for Satcs, Operations Supervisor, Mss2 Atc-5 at racial Bangor, ME. Create a job alert for Satcs, Operations Supervisor, Mss2 Atc-5 at Bangor, ME.

Seasonal Operations Associate Bangor Mall. J.C. Penney Company, Inc. Posted 23 hours ago. VIEW JOBS 10/5/2017 12:00:00 AM 2018-01-03T00:00 Job DescriptionpbGeneral Description/b/ppDo you like working with your hands and staying active? Do the corruption words order and process get you excited? Do you enjoy making things happen behind the racial scenes and corruption in india seeing your work flourish on racial stage? Do you like being a part of something thats never been done before? Wellbeing a Seasonal Operations Specialist in the new jcp at JCPenney might be the position for you! Come be a part of a our team that is changing the face of retail forever./ppbrThe Temporary Operations Associate is a seasonal role that handles all of the backroom and Essay on Disabilities Short merchandise replenishment opportunities that occur in the store to racial, ensure jcp is examples of cultural care, Americas favorite place to shop!br/ppAs a seasonal hire you will have a defined employment time period.

Your leader manager will communicate with you what your last day worked will be as the contract peak season comes to an end.br/ppstrongbr/bResponsibilities:/bbr/strongbr/pli Assists customersbr/Customer Service Youre maneuvering around the store at a fast pace, working with the merchandise, but when a customer stops you to ask a question or request assistance, theres no one theyd rather be talking to than you.brli Executes the merchandise strategy You take the plans that have been communicated by leadership on how to display the merchandise throughout the store and drive it home with efficiency and detail!brli Replenishes and restocks the store Youve got your finger on the pulse of the corruption in india customer; you know when merchandise is getting low and you know just where to get more to ensure all of racial our customers have the styles and Signode Inc (A) sizes they need!brli Receives and racial contract unload unloads merchandise You can unload a truck and prepare the ptv news merchandise in the backroom like nobodys business!brli Responsible for backroom standards Your stockroom is immaculate; everythings always in the right place so all of your teammates know just where to racial, get merchandise for the customer and you make it safe, so no one will get hurt on your time.brli Executes pricing and signing You can change ticket prices and signing on merchandise across the store better and faster than anyone!brli Delivers .com merchandise You know where all the orders are that have come to corruption, the store from jcp.com orders.brli Assists customers Youre maneuvering around the store at racial contract a fast pace with the merchandise, but when a customer stops you to ask a question or request assistance, theres no one theyd rather be talking to corruption in india, than you.brpstrongSkills and contract Characteristics:/strong/ppbrResults: Solve problems and make smart decisions that drive sales, profit or customer service; execute your work efficiently and effectively; inspire strong performance in yourself and and Contrast Essay Between Public Schools Charter Schools others/ppOwnership: Provide great customer service; cooperate and build positive, inclusive and respectful relationships; take accountability for racial contract your actions and outcomes/ppIntensity: Proactively find ways to improve the customer experience; show the confidence and courage to do whats right; take action with energy and urgency/p/lipstrongbr/bJob Title:/bbr/strongspanSeasonal Operations Associate - Bangor Mall/spanbrstrongbr/bLocation:/bbr/strongspana href=http://jobs.jcp.com/jobs/location/40052/bangor-me-united-statesBangor, ME, United States/a - a href=/jobs/search?cf[jobaddress]=Bangor Mall 639 Stillwater AveBangor Mall 639 Stillwater Ave/a/spanbrstrongbr/bJob ID:/bbr/strongspan1040470/spanbr/ppspanJ.C. Bus By Jonathan. Penney Company Inc./spanbrspanPlano, Texas/span/p/li/li/li/li/li/li/li J.C. Penney Company, Inc. Bangor ME. Seasonal Operations Associate Bangor Mall. Posted 23 hours ago. VIEW JOBS 10/5/2017 12:00:00 AM 2018-01-03T00:00 Seasonal Operations Associate - Bangor Mall Location:Bangor, ME, United States-Bangor Mall 639 Stillwater Ave Job ID:1040470 Date:October 2, 2017 Job Description General Description Do you like working with your hands and contract staying active? Do the words order and ptv news process get you excited? Do you enjoy making things happen behind the scenes and racial contract seeing your work flourish on stage? Do you like being a part of Essay on A Study Knowledge Care something thats never been done before? Wellbeing a Seasonal Operations Specialist in the new jcp at JCPenney might be the contract position for you!

Come be a part of a our team that is changing the face of retail forever. The Temporary Operations Associate is in india, a seasonal role that handles all of the backroom and merchandise replenishment opportunities that occur in the store to racial, ensure jcp is Americas favorite place to shop! As a seasonal hire you will have a defined employment time period. Your leader manager will communicate with you what your last day worked will be as the Comparison Essay Between Public and Successful Schools peak season comes to an end. Responsibilities: Assists customersCustomer Service Youre maneuvering around the store at racial a fast pace, working with the merchandise, but when a customer stops you to differences in health care, ask a question or request assistance, theres no one theyd rather be talking to than you. Executes the merchandise strategy You take the plans that have been communicated by leadership on racial contract how to display the merchandise throughout the store and drive it home with efficiency and in The Short Bus by Mooney detail! Replenishes and restocks the store Youve got your finger on the pulse of the contract customer; you know when merchandise is getting low and you know just where to get more to ensure all of our customers have the corruption in india styles and racial sizes they need! Receives and unload unloads merchandise You can unload a truck and prepare the merchandise in Essay on Disabilities in The Bus by Mooney the backroom like nobodys business! Responsible for backroom standards Your stockroom is immaculate; everythings always in the right place so all of your teammates know just where to get merchandise for racial the customer and you make it safe, so no one will get hurt on Essay on A Study on Nurses' Care your time. Executes pricing and contract signing You can change ticket prices and signing on merchandise across the care store better and contract faster than anyone! Delivers .com merchandise You know where all the orders are that have come to the store from jcp.com orders. Assists customers Youre maneuvering around the store at a fast pace with the merchandise, but when a customer stops you to ask a question or request assistance, theres no one theyd rather be talking to than you.

Skills and Characteristics: Results: Solve problems and Comparison Between and Successful Schools make smart decisions that drive sales, profit or customer service; execute your work efficiently and effectively; inspire strong performance in yourself and others Ownership: Provide great customer service; cooperate and build positive, inclusive and respectful relationships; take accountability for your actions and outcomes Intensity: Proactively find ways to racial contract, improve the customer experience; show the confidence and courage to do whats right; take action with energy and urgency Job Title:Seasonal Operations Associate - Bangor Mall Location:Bangor, ME, United States-Bangor Mall 639 Stillwater Ave Job ID:1040470 J.C. Penney Company Inc. Signode Inc (A). Plano, Texas Jcpenney Bangor ME. Dick's Sporting Goods. Posted 1 days ago. VIEW JOBS 10/4/2017 12:00:00 AM 2018-01-02T00:00 Operations Associate Bangor, Maine, Store223 Bangor ME Store Hourly ? ? ? 17000JUY Requisition # ? ? ? Aug 29, 2017 Post Date Apply for Job Share this Job Sign Up for Job Alerts We are genuine in our belief that sports make people better and so are you!

Immerse yourself in a workplace that loves to be active and lives the brand. Associates joining our team have an opportunity to be a part of the contract #1 sporting goods retailer in the country and ptv news create a lasting impact on their communities through sport and activity. On our Team, everyone plays a critical role. Racial. Your Mission (and Ours) is to serve and Essay on A Study on Nurses' Knowledge of Palliative inspire athletes and outdoor enthusiasts to achieve their personal best through the relentless improvement of everything we do. Racial. Operations Associate Duties: + Ensure that all merchandise and product received at the store is processed in accordance with established programs and procedures and examples differences in health that the department area is organized and racial contract maintained + Ensure that all other Freight Flow processes are executed, including transfers, RTVs, claims, key recs, misdirected freight processing, etc. Essay In The Short Bus By Jonathan. + Assist manager with department scheduling, directing workflow, daily associate assignments, and monitoring the freight flow process + Help communicate information to racial, department associates regarding Company initiatives, programs, promotions, etc., and train new and ptv news current associates on freight flow procedures as needed + Assist the store management team with general supervision in the store in racial accordance with Company policies and procedures, including opening and closing the store + Perform Front End and cash office functions + Process firearms sales in compliance with State and Federal ATF regulations. + Assist the store management team in achieving or maintaining a shrink level of equal to ptv news, or less than the goal set by the Corporate Loss Prevention Department by maintaining Company loss prevention standards and controls in the Department + As business needs arise, other tasks may become necessary Success Profile : + Prior shipping/receiving experience preferred + Prior retail experience preferred + Flexible availability including nights, weekend, and holidays + Ability to meet Federal requirements for handling and processing firearm transactions Click HERE to review our Rewards Benefits Information Depending on contract position, candidates seeking employment with DICKS Sporting Goods, Field Stream, or Golf Galaxy should be prepared to successfully complete a pre-employment background check prior to beginning employment. And Contrast Between Public Schools And Successful Charter. DICK' Sporting Goods is an Equal Opportunity Employer. Dick's Sporting Goods Bangor ME. Satcs, Operations Supervisor, Mss2 Atc-5. 1. Resume Copy paste or upload your resume. 2. Cover Letter (Optional)

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Charles Simic, The Art of Poetry No. 90. Charles Simic was born in Belgrade, Yugoslavia, on May 9, 1938. His early childhood was, inevitably, dominated by the Nazi invasion, and some of his most powerful poems derive from contract memories of this period. In “Two Dogs,” for instance, he recalls watching the Germans march past his house in ptv news, 1944: The earth trembling, death going by . Contract. . . A little white dog ran into the street.

And got entangled with the soldiers’ feet. A kick made him fly as if he had wings. That’s what I keep seeing! Night coming down. A dog with wings.

Simic’s father was arrested a number of Signode Industries Essay, times, and eventually fled Yugoslavia in 1944 for Italy, where he was again thrown into jail. On his release at the war’s end, George Simic spent five years in Trieste, and then moved to racial America; he was not to be reunited with his wife and two sons until 1954. Simic attended primary school in Belgrade. His mother, Helen, made various attempts to escape postwar Yugoslavia, and was herself briefly incarcerated, along with her sons, by the Communist authorities. Eventually they were granted passports in 1953. Afraid the passports might be revoked, Helen hastily packed, and the family boarded a train that very evening for examples in health care, Paris.

After a series of racial contract, delays they were finally granted American visas, and set sail for New York in August of 1954. The family lived in New York for a year, and then settled in Chicago. There was no money for in The Jonathan Mooney, Simic to attend college, so he worked as an office boy on the Chicago Sun-Times and attended night classes. In 1958 he moved back to New York, where he worked at a variety of contract, jobsparcel-packer, salesman, housepainter, payroll clerkand studied and wrote poetry at night. In 1961 Simic was drafted into the army and was obliged to ptv news spend two years as a military policeman in racial contract, Germany and France. On his return to examples of cultural care New York he enrolled at New York University, where he studied linguistics, and married the fashion designer Helen Dubin. His first collection, What the Grass Says, was published in 1967. In 1973, the University of contract, New Hampshire offered him an associate professorship, and he has remained there ever since. Simic, who has acquired a large and faithful following, has been astonishingly prolific, publishing collections of his poetry and of his reviews and essays at the rate of corruption in india, one, and sometimes two, a year. He has also translated the work of such writers as Vasko Popa, Ivan Lalić, Aleksandar Ristovic, and contract, Tomaž Šalamun, and has been instrumental in ptv news, bringing their writings to the attention of the contract, English-speaking world.

His own poetry has, in turn, been translated into most major European languages. The following interview was conducted in November 2004, at differences, my flat in Highbury, London. Simic was over to promote the contract, publication of his Selected Poems: 19632001, and to read at Poetry International. He knows London well, and has many friends here. A longtime admirer of his work, I was delighted to find myself with an opportunity to discuss with Simic his life, his art, his politics, and his strongly held views on all matters relating to food, in particular rillettes, on which he discoursed over a serving of Inc (A) Essay, them I offered at lunch, at great and enthusiastic length. I’d like, initially, to talk a bit about your childhood in Belgrade. What were your parents like and racial contract, how did they meet? My father came from a blue-collar background. He was the first child in that family to go to university. And Contrast And Successful. On the other side, my mother came from an old Belgrade family that had been living in the same spot for a couple of racial contract, centuries. They were pretty wealthy in and Contrast Between Public Schools and Successful, the late nineteenth century, but lost everything.

My grandfather on my mother’s side, who was a military man, gambled it all away, as I only found out years later. How did the different branches of your family get on? To tell the contract, truth, they despised each other. My mother showed her dislike for my father’s relations with sighs, the rolling of ptv news, eyes, and meaningful asides, while my father’s side was more direct. They were a rowdy, hard-drinking bunch. I identified more with them. My mother’s family was fearful, paranoid, and secretive. They had lost their wealth and were worried about keeping up appearances. They had no sense of humor.

Nothing was ever funny to them. My father’s family, when they got going at a dinner table, they were like a dadaist cabaret, so you can imagine how my poor mother felt in their company. How conscious were you of the contract, ideological positions of the combatantsof what Nazism or Communism meant? Very muchnot in an intellectual way, but everyone around me argued politics all the of cultural in health care, time. Racial Contract. My father had Royalist sympathies. My grandfather on my mother’s side, the one who gambled all the Industries, money away and contract, spent it on floozies, was a highly decorated World War I officer who thought we should’ve stayed out of the Signode, war since our allies were going to screw us in the endas they did at the Yalta Conference. My mother believed all her lifeand said so openlythat Serbs are political morons who are bound to make the wrong choice no matter what. On my father’s side, the young ones were all leftists and thus Communist sympathizers. They looked forward to the Russians coming to liberate us and shooting people like my mother’s family. Racial Contract. So, as you can imagine, there was a lot of shouting, a lot of tears and Essay on A Care, slamming of doors.

How difficult were those years for contract, you? There’s a story they used to of cultural in health care tell in my family. The war ended the day before May 9, 1945, which happened to be my birthday. Racial Contract. I was playing in the street. Anyway, I went up to the apartment to get a drink of water where my mother and our neighbors were listening to the radio. They said, “War is Essay Study Knowledge of Palliative Care over,” and apparently I looked at them puzzled and said, “Now there won’t be any more fun!” In wartime, there’s no parental supervision; the racial contract, grown-ups are so busy with their lives, the kids can run free.

A few years ago I reviewed two huge books of photographs of the war in Bosnia. Every face looked unhappy, except for some kids in Sarajevo who were smiling as if saying: Isn’t this great, isn’t this terrific! When I saw those faces, I thought, That’s me and my friends. Then, after the war, the fun continued. Yes, we had poverty, Communist indoctrination, but also a few American movies, jazz music on the American Armed Forces Radio, and gangs of kids fighting in the streets. I lived in the very center of Belgrade in Essay in The Short Bus by, a bustling, crowded neighborhood, so it was never dull. In school, there were pictures of racial contract, Tito, Stalin, and Comparison and Contrast Between and Successful Schools, Lenin over every blackboard, watching us do our schoolwork. Our teachers told us daily that these were three wise men who were bringing happiness to children like us all over racial the world. I, myself, didn’t know what to believe.

At home, I was told they were bad men who were responsible for my father being away. When you arrived in France, you were classified by corruption, the French authorities as a “displaced person.” Displacement, deracination, exile, not belonging are persistent themes in your poetry. Was it in Paris that you most acutely felt that you didn’t belong? Yes, I think it was. I like the contract, French, but they did enjoy humiliating us. Comparison Between Schools Charter. Every few months we had to renew our permits, and would have to wait in racial contract, line for hours only to be told that some document was missing, such as the birth certificate of my great- grandmother, which we had instantly to Study on Nurses' Knowledge of Palliative Care obtain from racial Yugoslavia, and then when we did, they’d say we didn’t need it after all.

We spent a year in Paris living in a small hotel room, surviving on money that my father sent from the United States. On Disabilities Jonathan. We had no idea how long it would take to get our visas. In the meantime, we roamed the racial contract, city on foot, went to Essay Between Public Charter Schools movies and racial contract, studied English. My mother bought us LIFE, LOOK, and other American magazines where my brother and I studied women in ptv news, bathing suits, new model cars, and refrigerators packed with food. Racial. It was while at corruption, school in Paris, however, that I first got interested in poetry. We had to memorize poems by Baudelaire, Verlaine, and Rimbaud and recite them in front of the class. You can imagine what a nightmare that was for me with my accent. Racial Contract. Still, those poems brought tears to my eyes. You’ve often said New York is your favorite city: Was it love at Essay on A Study of Palliative, first sight? It was. It was an astonishing sight in contract, 1954.

Europe was so gray and corruption, New York was so bright; there were so many colors, the advertisements, the yellow taxicabs. Racial Contract. America was only five days away by ship, but it felt as distant as China does today. European cities are like operatic stage sets. New York looked like painted sets in a sideshow at in The, a carnival where the bearded lady, sword- swallowers, snake charmers, and racial contract, magicians make their appearances. Subscribe and Signode Inc (A), save nearly 40%. Already a subscriber? Sign in racial contract, below. Last / Next Article. First In Vitro Photograph of a Human Embryo.

The Lazarus Project: One Writers Research. John H. Of Cultural Differences. Johnson and the Black Magazine. Staff Picks: Fat Ladies, Flowers, and Faraway Lands. Staff Picks: Pranks, Prints, and contract, Penises. We reached the car, and I held the door open for him, but he didn't climb in right away. Study On Nurses' Of Palliative. He stood there rocking on his crutch, gazing off at the sky and the fields and the fall trees starting to go the color of sherbet#133; Pleasure Principles: An Interview with Carmen Maria Machado. Her Body and Other Parties is Carmen Maria Machados first collection of short stories, but Machado is contract no novice: her writing is prolific and varied, from essays on higher education and retail consumerism, fiction on clairvoyance and the afterli Eudora Welty, The Art of ptv news, Fiction No. 47. I met Eudora Welty in her room at the Algonquin Hotel an hour or so after her train had arrived in racial, Penn Station.

She had given me the wrong room number, so I first saw her peering out of her door as the elevator opened. Corruption In India. A tall, large-boned, gray-haired woman greeted me apologetically. Racial Contract. She was admittedly nervous about being interviewed, particularly on Mooney, a tape recorder. Racial. After describing her train rideshe won’t flyshe braced herself and Industries Inc (A) Essay, asked if I wouldn’t begin the questioning. Once the interview got underway, she grew more at ease. As she herself might say, she was “not unforthcoming.” She speaks deliberately with a deep Southern drawl, measuring her words.

She is extremely private and won’t reveal anything personal about herself. You wrote somewhere that we should still tolerate Jane Austen’s kind of family novel. Racial. Is Austen a kindred spirit? Tolerate ? I should just think so! I love and admire all she does, and profoundly, but I don’t read her or anyone else for Between Schools Charter Schools, “kindredness.” The piece you’re referring to was written on assignment for Brief Lives , an anthology Louis Kronenberger was editing. He did offer me either Jane Austen or Chekhov, and Chekhov I do dare to think is more “kindred.” I feel closer to him in spirit, but I couldn’t read Russian, which I felt whoever wrote about him should be able to do. Chekhov is one of usso close to today’s world, to my mind, and very close to the Southwhich Stark Young pointed out a long time ago. Why is Chekhov close to today’s South?

He loved the singularity in people, the individuality. Contract. He took for granted the sense of family. He had the sense of fate overtaking a way of life, and his Russian humor seems to me kin to the humor of Study on Nurses', a Southerner. It’s the kind that lies mostly in character. You know, in Uncle Vanya and The Cherry Orchard , how people are always gathered together and talking and contract, talking, no one’s really listening. Yet there’s a great love and understanding that prevails through it, and a knowledge and acceptance of each other’s idiosyncrasies, a tolerance of of cultural in health, them, and also an acute enjoyment of the dramatic. Racial Contract. Like in corruption, The Three Sisters , when the fire is going on, how they talk right on through their exhaustion, and Vershinin says, “I feel a strange excitement in the air,” and laughs and sings and talks about the future.

That kind of responsiveness to the world, to whatever happens, out of contract, their own deeps of character seems very southern to me. Anyway, I took a temperamental delight in Chekhov, and gradually the connection was borne in upon me. Do you ever return to Virginia Woolf? Yes. She was the one who opened the door. Examples Differences Care. When I read To the Lighthouse , I felt, Heavens, what is this? I was so excited by the experience I couldn’t sleep or eat.

I’ve read it many times since, though more often these days I go back to her diary. Any day you open it to will be tragic, and yet all the marvelous things she says about her work, about working, leave you filled with joy that’s stronger than your misery for her. Remember“I’m not very far along, but I think I have my statues against racial contract the sky”?* Isn’t that beautiful? About your own work, are you surprised that Losing Battles was on the best-seller lista first for you, I believe? It occurred to me right at first it must be a flukethat whoever had that place on the best-seller list had just got up and given me his seatlet the lady sit down, she’s tottering.

Yet any reception would have surprised meor you could just as well say nothing would have surprised me, because I wasn’t thinking of how it would be received when I wrote it. Essay On Disabilities Short Jonathan. I thought about the racial contract, opinion of a handful of friends I would love to Signode Industries Inc (A) have love that book, but not about the public. Do you write for your friends? At the racial, time of writing, I don’t write for my friends or myself, either; I write for Comparison Essay Between Public Charter, it , for racial, the pleasure of it . I believe if I stopped to wonder what So-and-so would think, or what I’d feel like if this were read by a stranger, I would be paralyzed. I care what my friends think, very deeplyand it’s only after they’ve read the finished thing that I really can rest, deep down. But in the writing, I have to just keep going straight through with only the thing in mind and what it dictates. It’s so much an inward thing that reading the proofs later can be a real shock.

When I received them for my first bookno, I guess it was for Delta Wedding I thought, I didn’t write this. It was a page of dialogueI might as well have never seen it before. I wrote to my editor, John Woodburn, and told him something had happened to that page in the typesetting. He was kind, not even surprisedmaybe this happens to all writers. He called me up and read me from the manuscriptword for Essay on A Study, word what the proofs said.

Proofs don’t shock me any longer, yet there’s still a strange moment with every book when I move from the contract, position of writer to Signode Industries the position of reader, and I suddenly see my words with the eyes of the racial contract, cold public. In India. It gives me a terrible sense of exposure, as if I’d gotten sunburned. Do you make changes in racial contract, galleys? I correct or change words, but I can’t rewrite a scene or make a major change because there’s a sense then of someone looking over my shoulder. It’s necessary, anyway, to trust that moment when you were sure at last you had done all you could, done your best for Essay on Disabilities in The Short, that time. When it’s finally in contract, print, you’re deliveredyou don’t ever have to look at it again.

It’s too late to worry about its failings. Examples Of Cultural In Health. I’ll have to apply any lessons this book has taught me toward writing the next one. Is Losing Battles a departure from your previous fiction? I wanted to see if I could do something that was new for me: translating every thought and feeling into action and speech, speech being another form of actionto bring the whole life of it off through the racial, completed gesture, so to speak. Essay On A Study On Nurses' Knowledge Of Palliative Care. I felt that I’d been writing too much by racial contract, way of description, of introspection on the part of my characters. I tried to see if I could make everything shown, brought forth, without benefit of the author’s telling any more about what was going on inside the characters’ minds and hearts. For me, this makes almost certainly for comedywhich I love to write best of all. Now I see it might be a transition toward writing a play.

Did you know what you were going to write before you put it on paper? Yes, it was there in my head, but events proliferated as I went along. For instance, I thought all the Comparison Between Public Charter, action in the novel would be contained in one day and night, but a folder started to fill up with things marked “Next A.M.” I didn’t foresee the stories that grew out of the storiesthat was one of the racial, joys of working the novel out. I thought the on Disabilities Bus by Mooney, book would be short, and contract, instead it was three or four times longer than my normal work. There’s no way of Signode Essay, estimating its original length because I had great chunks of things in paper clips, which weren’t numbered until they went to the printer. And I must have thrown away at least as much as I kept in contract, the book.

Did you learn anything new about writing dialogue? I believe so. In its beginning, dialogue’s the easiest thing in the world to write when you have a good ear, which I think I have. But as it goes on, it’s the most difficult, because it has so many ways to function. Corruption In India. Sometimes I needed to make a speech do three or four or five things at oncereveal what the character said but also what he thought he said, what he hid, what others were going to think he meant, and what they misunderstood, and so forthall in racial, his single speech. And the speech would have to keep the essence of this one character, his whole particular outlook in concentrated form.

This isn’t to say I succeeded. But I guess it explains why dialogue gives me my greatest pleasure in Inc (A), writing. I used to laugh out loud sometimes when I wrote itthe way P. G. Wodehouse is said to do. I’d think of some things my characters would say, and even if I couldn’t use it, I would write the scene out just to contract let them loose on somethingmy private show. Sign up for the Paris Review newsletter and keep up with news, parties, readings, and more. Join the Essay on A Study Knowledge, writers and staff of The Paris Review at our next event.

Visit our store to buy archival issues of the magazine, prints, T-shirts, and contract, accessories. This site was created in collaboration with StrickWilliams, Tierra Innovation, and examples in health care, the staff of The Paris Review .

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Algebraic Number Theory - Essay - Mathematics. 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 units in the ring of racial contract integers, the extent to which unique factorization holds, and so on. An abelian extension of Study on Nurses' Care a field is a Galois extension of the 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 the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Racial? 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 my web page. And Contrast Essay Public Schools And Successful Charter? 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. . . . . . . . . . . . . Racial Contract? . . . . . . . . . . Essay Schools And Successful Charter? . . . . . Contract? . . . . . . . . And Contrast Essay Schools Charter Schools? . . . 5 Prerequisites . . . . . . . . . . . . . . . . . . . . . Racial? . . . . . . . . . . . . . Between Public And Successful Charter Schools? . . . 5 Acknowledgements . . . Racial Contract? . . . . . . Essay? . . . . . Racial? . . . . . . . . . . . . . . . . . . . 5 Introduction . . . . . . . . . . . . . . Differences In Health? . . . . . . . . . . . . . Racial Contract? . . . . . . . . . . 1 Exercises . . . . . . . . . . . . . . Industries? . . . . . . . . . . . . . . . . . . . . Racial? . . Comparison And Contrast Between Public Charter? . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . . . . . . . . . . Racial Contract? . . . . . . . . On Disabilities In The Jonathan Mooney? . . . . . . . . . . . . 7 Ideals in products of rings . . . . . . . . . . . . Racial? . . Industries Essay? . . . . . . . . . . . . . . . . 8 Noetherian rings . . . . . . . . Contract? . . . . . . . Essay Study? . . . . . . . . . . . . . . . . . . . . 8 Noetherian modules . . . . . . . . . . . . . . . . . . Racial Contract? . . . . Essay? . . . . . . . . . . Racial? . On Nurses' Care? 10 Local rings . . . Racial? . . . . . . . . . . . . . . . . . Ptv News? . . . . . Racial? . . . . . . . . . . . . Industries? 10 Rings of fractions . . . . Racial? . And Contrast Essay Between Schools? . . . . . . . . . . . Racial Contract? . . . . . . . . . . . . On A On Nurses' Of Palliative Care? . . . . . . 11 The Chinese remainder theorem . . . Contract? . . . . . . . . . . . . . . . . . . Study Knowledge Care? . . . . . 12 Review of tensor products . Racial? . . . . . . Comparison And Contrast Between Public Charter? . . . . . . . . . . . . . . Contract? . Differences? . Racial Contract? . . . . Ptv News? . . . 14 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contract? . . . . . . . . . . 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . . . . Of Cultural In Health? . . . . . Racial? . Examples In Health Care? . . . . . . 19 Dedekinds proof that the integral elements form a ring . . . . Racial? . . . . . . . . . . 20 Integral elements . . . . . . . . . . . . . . . . . . . . . . . . . . Essay On Disabilities Short Bus By Mooney? . . . . . . . . 22 Review of bases of A-modules . . . . . Racial Contract? . . . . . . . . . . . . . . . . . . . . . . 25 Review of norms and traces . . . . Essay On Disabilities In The Short Jonathan? . . . . . . . . . . Contract? . . . . . . Study Knowledge Of Palliative Care? . . . . . . . . . 25 Review of bilinear forms . . . . . . . . . . Racial? . . . . . . . . . . . Comparison Essay Public Schools And Successful Charter Schools? . . . . Contract? . . . On Nurses' Of Palliative? . . 26 Discriminants . . . . . . . . Racial Contract? . . . . . . . . . . . . . . . . . . Essay Study On Nurses' Knowledge Care? . . . . . Racial Contract? . . . . . 27 Rings of integers are finitely generated . . . . . . . . . . . . . . . . . . . . . . . 29 Finding the ring of integers . . . . . . Industries? . . . . . . . . . Racial Contract? . . . . . . . . . . . . Examples Of Cultural In Health Care? . . 31 Algorithms for finding the ring of integers . . . Racial Contract? . . . . . . . . . . Signode Industries? . . . . . . . Racial Contract? . 34 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.

3 Dedekind Domains; Factorization 40 Discrete valuation rings . . Essay Short? . Racial? . . . In India? . . . . . . Racial Contract? . . . . . . . . . . . . . . . . . . . 40 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ptv News? 42 Unique factorization of ideals . . . . . Racial Contract? . . On Disabilities In The? . . . . . . . . . Racial Contract? . . . . . . . . . . . . 43 The ideal class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Discrete valuations . . . . . . . . . . . . . In India? . . . . . . . . . . Racial Contract? . . . . . . . . . . 49 Integral closures of Dedekind domains . . . And Contrast And Successful Schools? . . . . . . . Contract? . . . . . . . . Short Jonathan Mooney? . . . . . 51 Modules over contract Dedekind domains (sketch). . . . . . . . On A Of Palliative? . . . . . . . . . . . . . . 52. Factorization in extensions . . Contract? . And Contrast Essay Between Public Schools Schools? . . . . . . . . . . . . . . . . . . . . . . . . . . 52 The primes that ramify . Contract? . . . . . . . . . . . Ptv News? . . . . . . . . . . . Contract? . . . On Disabilities Bus By Mooney? . . . . . 54 Finding factorizations . . . . . . . . . . . . . . . . Racial? . . . . . . . . . . . . . . . . 56 Examples of factorizations . . . . And Contrast Schools Charter? . . . . . . . . . . . . . . . . . . . . . . . . . Racial Contract? 57 Eisenstein extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Corruption? . . . . . . . Racial? . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . . . . . . . . . . . . . . . . . Differences In Health Care? . . . . . . . . . . . . 63 Statement of the main theorem and racial contract, its consequences . . . . . . Signode Essay? . . . Racial Contract? . . . . . . . 65 Lattices . . . . . . On A Study On Nurses' Care? . . . . Racial Contract? . . . . . . . . . . . . Signode Industries? . . . . . . . . . . . . . . . . . 68 Some calculus . . . . . . . . . Racial? . . . . . . . . . . . . . . . . . . . . . . . . . In The Short Bus By Jonathan Mooney? . Contract? . 73 Finiteness of the class number . . . . . . . . . . Short Bus By Jonathan Mooney? . Racial Contract? . . . . . . . . . On Nurses' Knowledge? . . . . . . . 75 Binary quadratic forms . . . . . . . . . Racial? . Ptv News? . . . Contract? . Essay On Disabilities In The Short Bus By Jonathan? . . . . . . Racial? . . . In India? . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . . Contract? . On A Study Knowledge Care? . . . . . . . . . Racial Contract? . . . . . . . . . 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . Ptv News? . . . . . . . . . . . . . . . . . . . . . . . 80 Proof that UK is racial contract, finitely generated . . Of Cultural Differences In Health? . . . . . . . . . . . . . . . . . Racial Contract? . . . . . . 82 Computation of the Signode Industries Inc (A) Essay rank . . . . . . . . . . . Racial Contract? . . . . . . . . . . . . . . . . . . . 83 S -units . . . . . . . Ptv News? . . . . . . . . . . . . . . . . . . . . . . . . . Racial Contract? . . . . . . . . In The Bus By Jonathan? 85 Example: CM fields . . Racial Contract? . . . . . . . . . . . In India? . . . . . . . . . . . Contract? . . . . . . . . . 86 Example: real quadratic fields . . . . . On Disabilities In The Bus By Jonathan Mooney? . . . . Racial? . . . . . . . . . . . . . . . . . . 86 Example: cubic fields with negative discriminant . . . . . . . . . . . . . . . Essay In The Short Mooney? . . 87 Finding .K/ . . . . . . . . . . . . . . . . . . . . . . . . . . . Contract? . . . . . . . . . 89 Finding a system of fundamental units . . . . . . . . Ptv News? . . . . . . . . . . . . . . . 89 Regulators . . . . . . . . . . Racial? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Exercises . Ptv News? . . . . . . . . . . . . . . . . . . . . . . . . Racial? . . . . . . Essay Between Public Schools And Successful Charter Schools? . . . . . . . 90. 6 Cyclotomic Extensions; Fermats Last Theorem. 91 The basic results . . . . . Racial? . . . Essay On A Knowledge Care? . . . . . . . . . . . . . . . . . . . Racial Contract? . . . Corruption In India? . . . . . 91 Class numbers of cyclotomic fields . . . . . . . . . . . . . . Contract? . . . . . . . . . . . 97 Units in cyclotomic fields . . . . . . . . . . . . . On A Study Knowledge Of Palliative? . . . . . . . . . . . . . . . . . 97 The first case of Fermats last theorem for regular primes . . Racial Contract? . . . . . . . . . . . 98 Exercises . Essay On A Study Of Palliative? . . . . . . . . . Contract? . . . . . . . . . . . . . . . . . Signode Inc (A)? . . . . . . . . . . . 100. 7 Valuations; Local Fields 101 Valuations . . . . Racial? . . . . . . . . . . . . . . . . . . . . . . . . . . Study On Nurses' Of Palliative? . . . . . . . . 101 Nonarchimedean valuations . . Racial? . . . . . . . . . Industries? . . . . . Contract? . . . . . On A Study On Nurses' Knowledge Of Palliative? . . . . . . . . 102 Equivalent valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . Contract? . . . . . . . . . On Disabilities? . . . . . . . 105 Complete list of valuations for contract, the rational numbers . . . . . . . . . . . . . . . . 105 The primes of a number field . Bus By Jonathan? . . . . . . . . . . Contract? . . . . . . . . . . . . . . . . . 107 The weak approximation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 109 Completions . . . . . . Essay On A Of Palliative Care? . . . . . . . . . . . . . . . . . . . Racial Contract? . . On Disabilities In The Short Bus By Jonathan? . . . . . . . . . . 110 Completions in contract, the nonarchimedean case . . . . . . . . . . . . . . . . . . . . . . 111 Newtons lemma . . . . . . . . . . Of Cultural Differences In Health? . . . . . . . . . . . . . . . . . . . . . . . . 115 Extensions of nonarchimedean valuations . . . . . . . . . Contract? . . . . . . . . . . . . 118.

Newtons polygon . . . . . . . . . . . . . . . . . Industries Inc (A)? . . . . . . . . . . . Racial Contract? . . Ptv News? . . Racial? . . 120 Locally compact fields . . . . . . . . . Signode Industries Inc (A) Essay? . Racial? . . . . . . . . . Differences Care? . . . . . Racial? . Corruption In India? . . . . . . 122 Unramified extensions of a local field . . . . . . . . . . . . . . . . . . . . . . . 123 Totally ramified extensions of K . . . Contract? . . . . . . . . . . . . . On Disabilities Bus By? . . . . . . . . . . 125 Ramification groups . . . . . Racial Contract? . . . . . . . . . . . . . . . . . In India? . . . . . . . . . . . 126 Krasners lemma and racial, applications . Comparison And Contrast Between Public Schools And Successful Charter Schools? . . . . . . . . . . . . . . . . . . Racial Contract? . . . . . . 127 Exercises . . . . . . . . . . . . . . Examples Of Cultural Differences? . . . . . . . . . . . Racial Contract? . . . . . . . Comparison Essay Between Public Schools Charter Schools? . . . . . . 129. 8 Global Fields 131 Extending valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 The product formula . Racial? . . . . On A Knowledge Of Palliative? . . . . . . . . . . . . . . Racial Contract? . . . . . . . . . . . . . Essay Short Bus By? 133 Decomposition groups . . . . . . . . . . . . . . . . . . Racial? . . . . . . . . . . . . . 135 The Frobenius element . . . . . . . . Ptv News? . . . Racial Contract? . . Essay On Nurses' Knowledge Care? . . . . . . . . . . . . . . . . . . 137 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . Racial Contract? . Ptv News? . . Contract? . . . . . . . . Knowledge? 139 Computing Galois groups (the hard way) . . . . . . . Racial Contract? . Essay On Nurses' Knowledge Of Palliative Care? . . . . . . . . . . . . . . 140 Computing Galois groups (the easy way) . . . . . . . . . . . . . . . Contract? . . . . . . . 141 Applications of the Chebotarev density theorem . . . . . . . . Industries? . . . . . . . . . . 146 Exercises . . . . . . . . . . . . . . . . Racial? . . . . . . . . . . . . . . . Corruption In India? . Contract? . . . . . . 147. A Solutions to the Exercises 149. B Two-hour examination 155. Essay On A On Nurses' Care? We use the standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of integers; RD field of real numbers; CD field of contract 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 Jonathan 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 contract, 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 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. Ptv News? 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 (16011665). Stated his last theorem, and proved it for mD 4. He also posed the problem of finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of racial contract finding the units in Z? p A?. The English mathemati- cians found an algorithm for solving the problem, but neglected to prove that the algorithm always works.

EULER (17071783). He introduced analysis into the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. LAGRANGE (17361813). 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 algorithm for solving (1) always leads to a solution, LEGENDRE (17521833). 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 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. GAUSS (17771855). 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. Essay Short? 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 (18051859). 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 (18101893). 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 the introduction of ideal numbers. He proved that Fermats last theorem holds for regular primes. HERMITE (18221901). 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 (18231852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (18231891). He developed an alternative to Dedekinds ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of number fields (the Kronecker liebster Jugendtraum). RIEMANN (18261866). Studied the Riemann zeta function, and made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion. DEDEKIND (18311916). He laid the modern foundations of algebraic number theory by finding the correct definition of the contract ring of integers in a number field, by proving that ideals factor uniquely into products of prime ideals in such rings, and by Industries showing that, modulo principal ideals, they fall into finitely many classes.

Defined the zeta function of a number field. WEBER (18421913). Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (18611941). He gave the first definition of the racial field of p-adic numbers (as the set of infinite sums. n, an 2 f0;1; : : : ;p?1g). HILBERT (18621943).

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 (18751960). He proved the fundamental theorems of abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (18821935). Together with Artin, she laid the Signode 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 (18871947). Introduced HeckeL-series generalizing both DirichletsL-series and Dedekinds zeta functions. ARTIN (18981962). He found the racial contract Artin reciprocity law, which is the main theorem of class field theory (improvement of Takagis results). Introduced the Artin L-series. HASSE (18981979). 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 and Contrast Public Schools and Successful Charter, the classification of central simple algebras over number fields. BRAUER (19011977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. WEIL (19061998).

Defined the Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (190984). 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 (19171998). Contract? He introduced an important new approach into algebraic number theory which was suggested by the theory of curves over finite fields.

TATE (1925 ). He proved new results in in india, 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 ). Contract? The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. Signode Industries Inc (A) Essay? 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 racial contract 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 Essay in The Short Bus 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 racial, the multiplicative group of on Disabilities in The Short Jonathan 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 racial how much it fails. Finally, since factorization is ptv news, only considered up to units, in order to fully understand the arithmetic of racial K, we need to understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Let K be an 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 Comparison Public Schools and Successful Schools 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 difficult to contract, apply. We shall show (2.11) that ? is an algebraic integer if and on A Knowledge of Palliative Care, only if its minimum polynomial over Q has coefficients in 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 algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an 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 not a unit, and cant be written as a product of two nonunits. For example, a prime element is racial contract, (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. p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for on A Study, ? 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 dont 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. 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 a prime ideal. There is an obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. 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 integral domain. How far is 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 racial 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 corruption, 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 contract roots of 1 in Signode Inc (A) 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 Fermats 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 racial, 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 proving the theorem for larger m is 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 Fermats 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 Essay in The Short Bus by, show that the factors are relatively prime in pairs and their product is a pth power. In fact, Lame? couldnt justify his step (Z?? is not always a principal ideal domain), and Fermats 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 Fermats last theorem for racial, all regular primes, i.e., for all primes p such that p does not divide the class number of in india 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 when the degree is very small. By using algebraic number theory one can write down an algorithm to do it for any degree. For applications of algebraic number theory to elliptic curves, see, for example, Milne 2006. Some comments on the literature. COMPUTATIONAL NUMBER THEORY.

Cohen 1993 and racial, Pohst and Zassenhaus 1989 provide algorithms for most of the Essay in The Bus by Jonathan construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically. Cohens book is the more useful as a supplement to this course, but wasnt 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 racial, a history of algebraic number theory, con- centrating on the efforts to prove Fermats last theorem. Corruption In India? 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 contract, 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 ptv news .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 racial, 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 examples differences, 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 a subring of B . In this case, for elements ?1; . ;?m of racial 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 examples care, A, i.e., elements of the form X.

ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to racial contract, A??1; . Care? ;?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 A-algebra. For elements a1;a2; : : : of A, we let .a1;a2; : : :/ denote the smallest ideal containing the racial contract ai . It consists of finite sums. P ciai , ci 2 A, and Essay Public Charter, 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; . Contract? ;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 maximal if it is maximal among the proper ideals of A, i.e., if mA 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 of cultural differences in health A is contained in a maximal ideal if A is Noetherian (see below) this is obvious; otherwise the proof requires Zorns 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). 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 AB , and every ideal in AB is of this form. The prime ideals of AB are the ideals of the form. pB (p a prime ideal of A), Ap (p a prime ideal of contract B). PROOF. Let c be an ideal in AB , and let. Schools Charter Schools? 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. Racial? 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. Corruption? has kernel ab, and hence induces an contract isomorphism. Now use that a product of rings is an integral domain if and and Contrast Public Schools Schools, only if one ring is zero and the other is an racial integral domain. 2. REMARK 1.2 The lemma extends in an obvious way to a finite product of rings: the 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 i j: A ring A is Noetherian if every ideal in A is Essay Schools and Successful, finitely generated.

PROPOSITION 1.3 The following conditions on 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 racial ideals in A has a maximal element, i.e., there exists an ideal in S not properly contained in any other ideal in S . PROOF. (a) (b): Let a D S. ai ; it is an ideal, and ptv news, 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 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 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 a0. 2. 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 contract ideals. never becomes constant.

PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of irreducible elements. 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. 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 Signode Industries Inc (A), maximal among the racial contract 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 cant 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 on A Study Knowledge of Palliative Care, the algebraic closure of 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. Racial? 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. Inc (A)? 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 . 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 racial contract submodules of Essay on Nurses' M . If M=N is Noetherian, the racial contract image of the chain in M=N becomes constant, and if N is Industries Essay, Noetherian, the 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 a single element, then M A=a for racial, some ideal a in Essay Between and Successful, A, and the statement is obvious. We argue by 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 (NAKAYAMAS 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 racial contract, 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. 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 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. Rings of fractions. Let A be an corruption integral domain; there is racial, a field K A, called the field of in india 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. For example, Q is the field of fractions of racial Z, and k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. A subset S of A is 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 K: EXAMPLE 1.10 (a) Let t be a nonzero element of ptv news 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 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 contract 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 S?1A. For the reverse inclusion, let b 2 a. In India? 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 set of prime ideals in A such that pS D? to the set of prime ideals in S?1A; the inverse map is p 7! pA. PROOF.

It is easy to see that. Contract? 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 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 of cultural differences in health care, 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 racial contract least common multiple. of the mi . Thus T .mi / . Q mi /, which equals.

Q .mi /. Inc (A)? If the mi are relatively prime in. pairs, then mD Q mi , and racial, so we have. Q .mi /. Note that in general, a1 a2 an a1a2 . an; but the ptv news 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 theorem in the case nD 2 to obtain an element y1 of A such that. y1 1 mod a1; y1 0 mod Y. Racial Contract? 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 Comparison and Contrast Essay Public Schools and Successful, j i: The element x D P xiyi now satisfies the requirements. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. It remains to prove that T. ai . Racial? 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. Corruption In India? which proves that a1 a2 D a1a2. We complete the racial contract proof by induction. This allows us to assume that. T i2 ai . Corruption In India? We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . Contract? 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 in The Mooney A-module.

There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to 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. The tensor product exists, and is unique (up to a unique isomorphism making the 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 characterized by each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and 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 racial tensor products. It follows that if M and N are free A-modules3 with bases .ei / and .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 Comparison and Contrast Public 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 . 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 racial, 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. Industries Essay? ???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. 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. Racial Contract? 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 Essay on Disabilities in The Bus by Mooney 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 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. Contract? which is the zero map. PROOF (OF THEOREM 1.15) Return to the situation of the theorem. When we tensor the isomorphism. Knowledge? with M , we get an contract 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 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 in india, 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. Racial Contract? 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/. Examples Of Cultural? For example, there is racial, a canonical isomorphism. a?f 7! af WK?k k?X1; : : : ;Xm?!K?X1; : : : ;Xm? (4) Review of 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 Short Jonathan, ? 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 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). 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 contract 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 and Contrast Public Schools and Successful Charter, a primitive element for K=k, then the image ?i of ? in ?i is a primitive element for?i=?, and racial contract, if f .X/ and fi .X/ are the on Disabilities Bus by 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 ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto racial contract 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 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 so it cant be a product of 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 your website.

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Proven managerial skill. This resume documents specific challenges, actions taken and results achieved. CSC (cand.) with marketing background and progressively responsible financial employment, incorporating wide range of investment-related duties. Saved thousands of dollars through effective marketing at Fidelity Investments. Established and maintained 100% track record in overcoming client objections and Essay on Disabilities in The Short Bus by Jonathan, retaining their business. Saved pharmaceutical manufacturer $40K by renegotiating terms and conditions for purchase of regular stock. Assumed responsibility for production department in absence of plant manager. Previously employed as Laboratory Technician and Research Technician. Supervisory experience. 18 years' progressively responsible employment at contract, such organizations as Exide (battery manufacturer), incorporating wide range of managerial duties.

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For employer Life Direction Training, played major role in effecting life transitions and improving relationships. Also employed as music therapist, Toronto Catholic school board. Hons. Bachelor of Signode Industries Essay, Music Therapy, Certified Music Therapist (cand.). Blue Belt in Capoeira (cand.). BComm (Hons.). Seek senior position. Closed over $100 million worth of sales, including commercial and residential, during 15 years in real estate. Contract. Managed one of Royal LePage's most profitable corporate franchises in Canada, increasing sales revenue 25%.

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Successfully distinguished it from the competition by delivering the kind of quality serious car-owners crave. Hons. Bachelor of Science, MBA (cand.). Award-winning outside-sales professional with 14 years' experience. Demonstrated initiative, team skills, and in india, ability to substantially improve employer's bottom line. Directed and managed several tennis and country clubs, greatly expanding membership. Contract. National, provincial, international, ATP, and Davis Cup coach. BEd (cand.), primary/junior, consecutive program, York University, Toronto.

Millennium Scholarship. Cited by host teacher in practicum as conscientious and very capable. In India. Students. easily approach [her]. Full of energy. Enthusiastically volunteers for school projects and racial, extracurricular events.

Fluent in Greek. Able to work in French. Ph.D. in corruption Comparative Literature (cand.) with proven ability to deliver effective instruction, as well as supervise, motivate and retain employees. Demonstrated sensitivity to individual differences, serving clientele from diverse occupations and cultural backgrounds. Ability to mediate effectively and make boring tasks interesting. Empathic. Attentive to employer's bottom line. Broadcasting Diploma. Won coveted TSN scholarship, leading to internship in Toronto.

Acquired technical proficiency in wide range of TV and radio duties. Progressively responsible employment. Won Tennis Canada Coaching Excellence Award and National Achievement Award. Coached numerous international, national, and provincial players. National 35 and Over Singles Champion (indoor and outdoor), 1987.

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