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This syllabus will be superseded by later versions as the course evolves.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)
| # | 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 | [TBA]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) |