UCSB Math 225AB Algebraic Number Theory - Math 225AB Algebraic Number Theory Simon RubinsteinSalzedo Winter and Spring 2006 0.1 Introduction Professor

UCSB Math 225AB Algebraic Number Theory - Math 225AB...

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Math 225AB: Algebraic Number Theory Simon Rubinstein–Salzedo Winter and Spring 2006
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0.1 Introduction Professor: Adebisi Agboola. Office Hours: Tuesday: 11:15-12:30, Thursday: 11:15-12:30 (225A), Wednesday 10:00- 12:00 (225B). Textbooks: Algebraic Number Theory by Fr¨ ohlich and Taylor, Algebraic Number The- ory by Lang, Number Fields by Marcus, Introduction to Cyclotomic Fields by Wash- ington. Course Outline: We shall aim to cover the following topics. Additional topics will be covered if time permits. Basic commutative algebra: Noetherian properties, integrality, ring of integers. More commutative algebra: Dedekind domains, unique factorization of ideals, local- ization. Norms, traces, and discriminants. Decomposition of prime ideals in an extension field. Class numbers and units. Finiteness of the class number: Minkowski bounds. Dirich- let’s unit theorem. Explicit calculation of units. Decomposition of prime ideals revisited: the decomposition group and the inertia group associated to a prime ideal. A nice proof of quadratic reciprocity. Basic Theory of completions and local fields. The Dedekind zeta function and the analytic class number formula. Dirichlet characters; Dirichlet L -functions; primes in arithmetic progressions; the ex- plicit class number formula for cyclotomic fields. Artin L -functions: definitions and basic properties. Miscellaneous topics, e.g. Stickelberger’s theorem, p -adic L -functions, Stark’s conjec- tures. Additional books that may be of use are: Galois Theory — Lang’s Algebra Number Theory — Hecke’s Theory of Algebraic Numbers , Borevich and Shafarevich’s Number Theory , and Serre’s A Course in Arithmetic Commutative Algebra — Atiyah and MacDonald’s Introduction to Commutative Alge- bra , Zariski and Samuel’s Commutative Algebra , and Eisenbud’s Commutative Algebra with a View Toward Algebraic Geometry . 1
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Chapter 1 Basic Commutative Algebra Example. In Z [ - 6], we do not have unique factorization of elements, e.g. 6 = - - 6 - 6 = 2 · 3. We shall later establish unique factorization into prime ideals. We will then have 6 Z [ - 6] = ( - 6 , 2) 2 ( - 6 , 3) 2 . Definition 1.1 Let M be an R -module, where R is a commutative ring with a 1. We say that M is a noetherian R -module if every R -submodule is finitely generated over R . Example. 1. M is finite. 2. R is a field and M is a finite-dimensional vector space. Definition 1.2 We say that the ring R is noetherian iff R is a noetherian R -module, i.e. iff all ideals are finitely generated over R . Example. A PID is a noetherian ring. Proposition 1.3 (See 220ABC) The following are equivalent: 1. R is a noetherian ring. 2
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2. Every ascending chain of R -ideals stabilizes. 3. Every nonempty set of R -ideals has a maximal element. Proposition 1.4 Suppose that the following sequence of R -modules is exact: 0 M N P 0 .
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  • Fall '13
  • SimonRubinstein-Salzedo
  • Algebra, Number Theory, Algebraic number theory, Principal ideal domain, prime ideal, L/K

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