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  1. math.berkeley.edu › ~apaulin › NumberTheoryNumber Theory - University of California, Berkeley

    Number Theory Alexander Paulin August 31, 2009 Lecture 2 Number Fields Throughout this section all rings will be commutative with unit. Dedekind Domains A number field is a finite field extension E/Q. The simplest example is Q. By construction we have the inclusion Z ⊂ Q. Q is the field of fractions of Z. It is elementary to show that

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  2. people.math.harvard.edu › ~landesman › assetsMATH 154. ALGEBRAIC NUMBER THEORY - Harvard University

    MATH 154. ALGEBRAIC NUMBER THEORY 7 and 4 = (1 + p 3)(1 p 3). We claim: Lemma 1.16. The element 2 is irreducible in Z[p 3]. Proof. Say 2 = ab, so by conjugating both sides we have 2 = ab. Multiply-ing both relations gives 4 = (aa)(bb). with aa and bb each a non-negative integer since for a = u + v p 3 with u,v 2Z we have aa = u2 +3v2. But u2 ...

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  4. people.maths.ox.ac.uk › greenbj › papersAlgebraic Number Theory Ben Green - University of Oxford

    Algebraic Number Theory. Ben Green. Contents. Preface. 0.1. A brief introduction. 0.2. These notes. 0.3. Prerequisites. Chapter 1. Algebraic numbers. 1.1. Algebraic numbers. Minimal polynomials. 1.2. The algebraic numbers are a eld. 1.3. Number elds. The primitive element theorem. 1.4. More examples. 1.5. Conjugates and embeddings. 1.6. Norms. 1.7.

  5. wstein.org › 129-05 › notesIntroduction to Algebraic Number Theory - wstein

    Algebraic number theory involves using techniques from (mostly commutative) algebra and finite group theory to gain a deeper understanding of number fields. The main objects that we study in algebraic number theory are number fields, rings of integers of number fields, unit groups, ideal class groups,norms, traces,

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  7. assets.cambridge.org › 97805218 › 32502INTRODUCTORY ALGEBRAIC NUMBER THEORY

    Algebraic number theory is a subject that came into being through the attempts of mathe-maticians to try to prove Fermat’s last theorem and that now has a wealth of applications to Diophantine equations, cryptography, factoring, primality testing, and public-key cryp-tosystems. This book provides an introduction to the subject suitable for ...

  8. pi.math.cornell.edu › cornell › 18spMath 6370: Algebraic Number Theory - Cornell University

    The main objects of algebraic number theory are number fields. Definition 1.1. A number field is an extension field of Q of finite degree, i.e. K Qwith [K: Q] = dim Q K<1. Example 1.2. Q, Q(p 2), Q(p-3), Q(3 p 5). Theorem 1.3 (Primitive Element). For any number field K, K= Q( ) for some . In number theory, we study the integers Z Q. The ...

  9. math.jhu.edu › ~iyengar › ANTALGEBRAIC NUMBER THEORY - Mathematics

    2. Number fields and algebra7 2.1. Galois theory 7 2.2. Different kinds of rings8 2.3. Number fields 10 2.4. Integral bases 12 3. Dedekind domains 14 3.1. Failure of unique factorization14 3.2. The definition 15 3.3. Unique factorization in Dedekind domains16 4. Finiteness of the class group19 4.1. Norms of ideals 19 4.2. General strategy ...

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