- 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

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

- 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.

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

- 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.

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

- 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

- 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).