### Math 676: Introduction to AlgebraicNumber Theory: Syllabus

Text (primary): J. W. S. Cassels Local Fields, Cambridge University Press, 1994.

This syllabus will be superseded by later versions as the course evolves. (Nov. 20, 2008)

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 # date topics References 1 9/3 1.1 Intoduction, Local and Global Fields ; analogies 2 9/5 1.2 Integral elements over a ring, finitely generated modules 3 9/8 1.3 Intrinsic characterization of integality; ring of integers of number field; cyclotomic fields 4 9/10 1.4 Trace, norm, of field elements; discriminant of free K-module in L/K. 5 9/12 1.5 Nondegeneracy of discriminant; power bases; discriminant of integral basis; finite generation of integer rings 6 9/15 1.6 Free modules over PID; Dedekind domains 7 9/17 1.7 Unique ideal factorization in Dedekind domains HW#1 due 8 9/19 1.8 Unique ideal factorization in Dedekind domain: proof completed. Complements: Every ideal divides principal ideal; PID=UFD in Dedekind domain; all fractional ideals require at most 2 generators. 9 9/22 1.9 Z-basis for ring of integers O_K; Hermite normal form for integral matrices; Example: Quadratic Fields 10 9/24 1.10 Example: Cyclotomic fields, ring of integers, discriminant for prime power case; Discriminant of compositum of fields (rel.prime case) 11 9/26 1.11 Splitting of prime ideals in extension fields, sum of e_i*f_i =n; Localization 12 9/29 1.12 Localization preserves Dedekind domain property; complete proof \sum e_if_i=n; e_i, f_i for Galois extension are constants 13 10/1 1.13 Ramified primes; iff p divides discriminant; factorization of irreducible f(x) (mod p) shows how (p) splits in extension field by root of f(x) HW#2 due 14 10/3 1.14 Proof of factorization f(x) (mod p); Example: Cyclotomic field prime ideal factorization; Frobenius automorphism 15 10/6 1.15 Completion of cyclotomic field prime ideal factorization; 2.1 Ideal class group and units. Fractional ideals are group; ideal class group; Statement of finiteness of class number. 16 10/8 2.2. Norms of ideals: statement of Minkowski ideal norm bound; r real embeddings, 2s complex embeddings of K. 17 10/10 2.3 Consequences of Minkowski norm bound; no unramified extensions of Q, small discriminant fields have small class number; Geometry of numbers, lattices and convex bodies 18 10/13 2.4 Centrally symmetric bodies and norms; characterizing lattices by discreteness; lattice parameters: determinant, successive minima 19 10/15 2.5 Minkowski's first fundamental theorem (convex body theorem) proved; second fundamental theorem stated, Lattice in R^n attached to integers O_K number field [K:Q]=n, determinant of this lattice. 20 10/17 2.6 Ideals in number field as sublattice in R^n, determinant of this. Completion proof of Minkowski ideal norm bound. Ideal sublattice has norm form bounded away from 0; norm form is non-convex unbounded star body, invariant under diagonal torus in SL(n, R). (McMullen handout.) HW#3 due 10/20 Fall Study Break 21 10/22 2.7 Unit theorem stated: unit rank is r+s-1. Logarithmic lattice in R^{r+s} introduced, contained in hyperplane H. 22 10/24 2.8 Completion of proof of unit theorem; logarithmic lattice has rank r+s-1. Lemma on special units big in one embedding, small in all other embeddings. 23 10/27 3.1 Valuations: examples and properties [Cassels Chap. I, II.1] 24 10/29 3.2 Ostrowski's theorem; Weak Approximation theorem [Cassels II.2, II.3] 25 10/31 3.3 Complete valuation rings; Weak approximation thm (restated). [Cassels II.4] 26 11/3 3.4 Nonarchimedean discrete valuation rings, residue class field, Characterization of Local fields [Cassels IV.1] 27 11/5 3.5 Complete discrete valuation fields, NASC for Local compactness of valued field [Cassels IV.1] HW#4 due 28 11/7 3.6 Hensel's Lemma I, Applications of Hensel's lemma, polynomial roots [Cassels IV. 3] 29 11/10 3.7 Hensel Lemma II, [Cassels,VI.4] basic p-adic analysis [Cassels IV.4] 30 11/12 3.8 Strassmann's theorem, Embedding theorem, three lemmas [Cassels Chap V.3] 31 11/14 3.9 Proof of Embedding theorem, linear recurrences, Skolem-Mahler-Lech thm (statement) [Cassels Chap V.5] 32 11/17 3.10 Skolem-Mahler-Lech Theorem (proof) [Cassels V.5], Ramanujan-Nagell equation [Cassels IV.6] 33 11/19 3.11 Transcendental extensions, valuations ||.||_c, Newton polygons, Eisenstein irreducibility criterion [Cassels VI.1, V1.2] HW#5 due 34 11/21 3.12 Eisenstein irreducibility over nonarchimedean fields, Newton factorization theorem over complete nonarchimedean fields. [Cassels VI.3] 35 11/24 3.13 Proof of Newton factorization, Hensel Lemma III. [Cassels VI.3] Algebraic extensions, unique extension of valuation (statement). Equal absolute values of roots of pure polynomials. [Cassels VII.1] 36 11/26 3.14 Unique extension of valuation to algebraic extension. Krasner's lemma and applications. [Cassels VII.2- VII.3] 11/28 Thanksgiving Break 37 12/1 3.15 Algebraic extensions of complete fields: e and f. Unramified extensions. Completely ramified extensions. [Cassels VII.4-VII.5] 38 12/3 3.16 Sect. HW#6 due 39 12/5 3.17 40 12/8 3.18 Local Kronecker-Weber Theorem (Cassels VIII.4), 4.1 Kronecker-Weber Theorem (Cassels X.12)