# Math 254A: Algebraic Number Theory

This is the course description and syllabus for Math 254A, an
introductory graduate course in
algebraic number theory, to be offered at UC Berkeley in fall 2001.
(Last updated: 23 Aug 2001.)

## About the professor

Kiran S. Kedlaya

Email: `kedlaya@math.berkeley.edu`

Web: `http://www.math.berkeley.edu/~kedlaya/math254.html`

Office: 757 Evans

Office Phone: 642-6923

Office Hours: Monday 1-2 PM, or by appointment, or just drop by!

## Course description

Algebraic number theory is the theory of *algebraic numbers*, those
numbers which are roots of polynomials with integer coefficients. Such
numbers occur often in number theory (e.g., when solving Diophantine
equations), but also elsewhere in mathematics. For example, the eigenvalues
of a matrix with integer entries are algebraic numbers; sequences satisfying
a recurrence relation are often expressible in terms of algebraic numbers,
as are solutions of some differential equations.
In this course, we will explore the "ground floor" of this theory,
as developed by the masters of the 19th century and refined using modern
perspectives. These perspectives include the algebraic (the methods
of commutative algebra), the geometric (the analogy between the ring of
integers and the ring of polynomials over a field) and the algorithmic
(both methods of computing within the subject, and exports from the subject
to other areas).
This course will be largely driven by the exercises; if you don't keep
up with the exercises, lectures will be correspondingly less useful.
Some exercises build up or reinforce core results from the course.
Some are applications requiring additional insights; these will largely
be optional, and are directed at those of you who like Putnam-type
problems. Some require nontrivial computations; for these, the
computer algebra package Magma is available on the department
network and can be installed on personal machines if needed.
## Prerequisites

Required: two semesters of
undergraduate algebra (Math 113 or equivalent).
Specifically, we will use basic facts about groups, rings, fields
(especially finite fields), and Galois theory.
Even better: one semester of graduate algebra (Math 250 or equivalent).
Also possibly helpful: some familiarity with classical
number theory.
If you're interested in the course but not sure how well your background
matches these prerequisites, let me know your precise situation and
I'll let you know whether this course is right for you.

## Textbook

Recommended: Jürgen
Neukirch's Algebraic Number Theory, published by Springer-Verlag.
I would have listed this as required if it were not so expensive.
I will put one copy on reserve in the math library (1st floor of Evans).
Some homework will be drawn from the text.
## Homework, exams

There will be weekly homework sets drawn from the text and the lecture.
Homework will be due each Monday at the end of class.
Some problems will require (minimal) use of a computer algebra system
such as *Magma*, available on the math department computers.
(Type \verb+magma+ at the prompt on any of the Unix machines. For
documentation, go to the Magma home page.)
Each set will contain more problems than are required. Students are
encouraged to work together, but should write up their own solutions;
students working together are especially encouraged to work on all of the
problems, and have each student write up a different subset of the problems.
I will only read as many problems as are required; if you submit more,
I will start from the front of your solution set, read the appropriate number
of problems, then stop.

The only exam will be a take-home final exam. The exam will be distributed
Monday, December 3, and will be due in my department mailbox at noon on
Monday, December 8. (The last class is on December 7, but no material after
December 3 will be on the exam.)
The exam will be open notes and open text,
but no use of references other than the text, my typed notes and
your lecture notes, and no conferring
with anyone other than me.

## Course grading

Course grades will be based on the following formula: 75% homework,
25% final exam. Late homeworks will be accepted for full credit at my
discretion, otherwise for half credit for one week, then not at all.
("At my discretion" means you see me *in advance* with an explanation
of the need for more time.)
## Lecture notes

I plan to distribute lecture
notes in class and on the course web site,
though these may lag slightly behind the lectures.

## Topics

### Number fields

The Gaussian integers and applications. Algebraic numbers, algebraic
integers and their basic properties. Some examples of number fields.
Ideals. Examples of and counterexamples to unique factorization.
### Lattices and applications

Minkowski's theorem. Interlude on algorithms (LLL method) and applications
outside number theory. Applications to algebraic numbers: discriminant
bound, class number bound, Dirichlet's units theorem.
### Multiplicative structure of ideals

Unique factorization of ideals. Ideal class group, class numbers.
Applications (e.g., cases of Fermat's last theorem).
### Relative properties of number fields

Prime decomposition,
ramification, discriminant, different, decomposition and inertia
groups, residue fields.
### Examples: cyclotomic and quadratic fields

Calculation of invariants, applications in classical number theory.
### Localization and completion

*p*-adic numbers, local fields. Hensel's lemma. Extensions and
ramification of local fields.
### Analytic methods (if time permits)

Zeta functions of number fields. Dirichlet's theorem on primes in
arithmetic progressions.

Kiran S. Kedlaya (`kedlaya(at)math.berkeley.edu`)