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 20-25Office:
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:**

Homework:

- due 12/3: Homework 6 Final version

- due 11/14: Homework 5

- due 10/29: Homework 4

- due 10/15: Homework 3

- due 10/1: Homework 2

- due 9/17: Homework 1

Special Announcements:

- From 9-12 and until further notice, classes will finish at 12:55.

- 10/3: A class has been recovered by using the 5 minute increments.

- 10/31: Another class has been recovered by using the 5 minute increments. We'll continue to do this.

- No class on the week of November 21 and 24.

- Course evaluations on November 26.

- Extra class on December 2.

- No classes on October 24, November 3,5,7,21,24

- Extra classes: Thursday September 18, Friday October 10, December 2

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

- 11/19: Weak approximation thm, Completions, the case of discrete archimedean valuations

- 11/17: equivalent valuations, valuations in Q (Ostrowski),

- 11/14:Nonarchimedean valuations (aka p-adic numbers), discrete valuations,

- 11/12: Fermat for regular primes. Definition of Valuations.

- 11/10: fall break...

- 11/3,5,7: No class

- 10/31: Scary cyclotomic fields

- 10/29: Pell's equation, cyclotomic fields

- 10/27: end of Continued fractions,

- 10/24: No class

- 10/22: Real quadratic fields and continued fractions (a little bit).

- 10/20: Computation of the rank Dirichlet's thm.

- 10/17: Units are finitely generated,

- 10/15: Units (statement of Dirichlet's theorem, definition of fundamental system of units, comments about the torsion part - roots of unity - (we proved they form a cyclic group), auxiliary statments to prove that units are finitely generated)

- 10/10: finiteness of the class number, quadratic forms

- 10/8: Mikowsky thm, volumes,

- 10/6: a discrete subgroup is a lattice, fundamental paralellogram

- 10/3: applications of Minkowsky bound, lattices, equivalences for discrete subgroups

- 10/1: (norm of ideals in number fields, intro to the statement of the finiteness of the class number in number fields

- 9/29:ramified primes divide de discriminant, norm of ideals in Dedekind domains

- 9/26: definitons of inertia group and fields, how to find prime decomposition in "nice" extensions

- 9/24:proof that sum e_i f_i = r, Galois extensions (definition of decomposition group and field)

- 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)
- "A Brief Introduction to Classical and Adelic Algebraic Number Theory", by William Stein
- "Algebraic Number Theory", by Neukirch
- Dario Alpern's website with Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

Other Resources:

In no particular order! I'll keep updating this, so let me know of any suggestions!

Last update: November 12, 2008 (or later)