Math 512

Algebraic Number Theory

Fall 2008

Instructor:    Matilde Lalín

Classroom:    CAB 657

Class Times:    Mondays - Wednesdays - Fridays 12:00 - 12:50 No classes on October 13.

Special arrangements for the weeks November 2-7 and 20-25

Office:    CAB 621

Office hours:   Mondays 1:00-2:00, Wednesdays 10:00 -12:00, and by appointment.

Phone:   (780) 492-3613

e-mail:    lalin at ualberta . ca or mlalin at math . ualberta . ca

Text:    I won't be following any specific book. Here are some possible books for reference

"Algebraic Number Theory" by J. S. Milne

"A Brief Guide to Algebraic Number Theory", by H.P.F. Swinnerton-Dyer

"Number Fields", by D. A. Marcus

"Algebraic Number Theory", by S. Lang.

Important links:


Special Announcements:

Important dates:

Topics covered in Class:

  • 12/3: extensions of archimedean valuations, places in a number field, product formula, and whatever else that fits!!!!!
  • 12/2: extra class: local-global principle for quadratic polynomials (you can read more about it in "Lectures on Elliptic Curves, by Cassels". I wanted to talk about Hilbert symbols but I didn't have time. To read about them: "A Course in Arithmetic, by Serre".
  • 12/1: extensions of non-archimedean discrete valuations in number fields and completions
  • 11/28: Hensel's lemma
  • 11/26: COURSE EVALUATIONS, completions of discrete archimedean valuations
9/22: localization of Dedekind domains are Dedekind (just the statement), Discrete Valuation rings, another version of CRT Factorization of ideals in extensions of Dedekind domains, defintion of ramification, residue class degree, examples in Z[i], statement of the theorem sum e_i f_i = r.
  • 9/19: The ideal class group, a Dedekind domain with finitely many prime ideals is principal. Background stuff: Chinese Remainder Theorem for ideals, Localization, ideals in localization
  • 9/18: Proof of unique factorization. (Lang's book)
  • 9/17: Review of Noetherian rings. Definition of Dedekind domains. Integral closures of Dedekind domains are Dedekind (we proved a weaker version). Statement of Unique factorization of ideals for Dedekind domains. Definition of fractional ideals. Every ideal in a Noetherian ring contains a product of prime ideals.
  • 9/15: existence of an integral basis for the integral closure of a PID, some results that help identifying integral bases: how to compute the sign of the discriminant, Stickelber's Thm, I've also distributed a copy of Proposition 2.11, pages 13-14 of Neukirch's book: Given a basis for L and a basis for L', how to get a basis for LL'
  • 9/12: the discriminant of a separable finite extension, the discriminant of a polynomial, the disc of x^2+ax+b,
  • 9/10: More traces and norms, comments about bilinear form, definition of discriminant, integral basis.
  • 9/8: More properties of the integral closure (its field of fractions, a UFD is integrally closed, minimal poly of an integral element has coefficients in the ring, integral closure is integrally closed). Review of Traces and Norms.
  • 9/5: Rings of integers, integral elements (from Milne, chapter 2). We proved that the integral elements in L over A form a ring that is called the integral closure of A in L.
  • 9/3: Intro to class. A resolution of Pythagorean triples (from Marcus, chapter 1), an example of a ring that is not UFD and introduction to ideal factorization (from Milne, introduction)

Other Resources:

In no particular order! I'll keep updating this, so let me know of any suggestions!
  • "Algebraic Number Theory", by Neukirch

Last update: November 12, 2008 (or later)