# 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.
• 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:
Final exam:
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