MAT 1200/415: Algebraic Number Theory, Fall 2018

This course is an introduction to algebraic number theory.

Instructor: Florian Herzig (my last name at math dot toronto dot edu)
Office Hours: Tuesdays 3-5pm at BA 6186
Lectures: Mon 4-5, Tue 10-12 in BA B026

Outline:

review of some commutative algebra
discriminant
Dedekind domains
ideal class group
splitting of prime ideals
decomposition and inertia groups, Frobenius elements
finiteness of class number
unit theorem
local fields (Ostrowski's theorem, p-adic numbers, Hensel's lemma, Newton polygon, unramified and totally ramified extensions)
comparison between ideal theory and valuation theory

Prerequisites:

Solid knowledge of abstract algebra is essential (e.g. Dummit and Foote, MAT347, MAT1100-1101).

References:

The main reference will be Milne's course notes.
Other useful references include Marcus' Number fields and Cassels' Local fields. Both are available electronically at U of T: Marcus and Cassels.

Grade:

The course grade will be based on roughly 5 homework sets (50%) and a final exam (50%).
The lowest homework grade will be dropped.

Homework:

Assignment 1, due Fri Sep 28.
Assignment 2, due Fri Oct 12.
Assignment 3, due Fri Oct 26.
Assignment 4, due Fri Nov 16.
Assignment 5, due Fri Dec 14.

Final exam:

Thu Dec 20, 7-10pm (sorry!), location: BA 2185.

Weekly plan:

Week of Sept 10: review of rings, algebras, modules
Week of Sept 17: review of modules over a PID, integral ring extensions, definition of O_K
Week of Sept 24: trace/norm, discriminants, computing O_K
Week of Oct 1: discriminants, Dedekind domains, localisation of rings, DVRs
Week of Oct 8: Dedekind domains (unique factorisation into prime ideals), fractional ideals, class group
Week of Oct 15: factorisation of prime ideals in extensions, ramification
Week of Oct 22: Kummer-Dedekind, ideal norm and numerical norm
Week of Oct 29: finiteness of class number, unit theorem
Week of Nov 12: cyclotomic fields, decomposition and inertia groups
Week of Nov 19: Frobenius elements, quadratic reciprocity; absolute values
Week of Nov 26: more on absolute values, Ostrowski's theorem, weak approximation, completions
Week of Dec 3: Hensel's lemma, Newton's method, extensions of valuations, Newton polygons, comparison with number fields

Links:

A relative extension of number rings that is not free. Section 2 should be understandable by now. (For later: the extension is still "locally free". See Thm. 3.1 in the note and compare it with the classification of finitely generated modules over Dedekind domains in Milne's notes.)
This paper mentions a slightly improved version of the Kummer-Dedekind theorem, due to Dedekind, see Theorem 1.1 on page 2. Namely it makes it easier to check the hypothesis that p does not divide the index. (Please don't cite this in your homework/exam solutions though.)
Wikipedia on Gauss' class number problem and Number fields with class number 1
Article of Cohen-Lenstra on their heuristics (see e.g. Section 9).