Algebraic Number Theory
Classroom: CAB 657
Class Times: Mondays - Wednesdays - Fridays 12:00
No classes on October 13. Special arrangements for
the weeks November
2-7 and 20-25
Mondays 1:00-2:00, Wednesdays 10:00 -12:00, and by appointment.
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
Number Theory" by J. S. Milne
"A Brief Guide to Algebraic Number Theory", by H.P.F.
"Number Fields", by D. A. Marcus
"Algebraic Number Theory", by S. Lang.
- 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
- 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
- 12/3: extensions of archimedean valuations, places in a number
field, product formula, and whatever else that fits!!!!!
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
- 11/26: COURSE EVALUATIONS, completions of discrete archimedean
- 11/19: Weak approximation thm, Completions, the case of discrete
- 11/17: equivalent valuations, valuations in Q (Ostrowski),
- 11/14:Nonarchimedean valuations (aka p-adic numbers),
- 11/12: Fermat for regular primes. Definition of Valuations.
- 10/31: Scary cyclotomic fields
- 10/29: Pell's equation, cyclotomic fields
- 10/27: end of Continued fractions,
- 10/22: Real quadratic fields and continued fractions (a little
- 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
- 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
- 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
- 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
- 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)
In no particular order! I'll keep updating this, so let me know of any
- "Algebraic Number Theory", by Neukirch
Last update: November 12, 2008 (or