Math 586: Introduction to Algebraic Number Theory.

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Catalog Information

Title

Introduction to Algebraic Number Theory.

Credit Hours

3

Prerequisite

Math 372 or equivalent; instructor's consent.

Description

Algebraic integers; different and discriminant; decomposition of primes; class group; Dirichlet unit theorem; Dedekind zeta function; cyclotomic fields; valuations; completions.

Desired Learning Outcomes

Prerequisites

Math 372 is a prerequisite for this course. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical career, students should be skilled at proving theorems by themselves.

Minimal learning outcomes

Students should achieve an advanced mastery of the topics listed below. This means that they should know all relevant definitions, correct statements and proofs of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving difficult problems related to these concepts, and by proving theorems about the below concepts, even if the theorems go beyond the material in the text.

  1. Number Fields
    • Algebraic Numbers
    • Algebraic Integers
    • Cyclotomic Fields
    • Trace and Norm
    • Discriminants
    • Integral Bases
    • Computing Integral Bases
  2. Prime decomposition in rings of integers
    • Ideal theory of Dedekind domains
    • Splitting, ramification, inertia of primes
    • Computing prime decompositions
    • Decomposition and inertia groups
    • Frobenius maps
    • Functorial properties of the Frobenius
  3. Ideal Class Group
    • Finiteness of the class group
    • Minkowski bounds
    • Distribution of ideals in ideal classes
    • Class group computations in quadratic fields
  4. Dirichlet's unit theorem
    • Computation of fundamental units in quadratic fields
  5. Cebotarev Density Theorem (Statement)
  6. Dedekind zeta function
    • Class number formula

Textbooks

Possible textbooks for this course include (but are not limited to):

  • Daniel Marcus, Number Fields
  • Serge Lang, Algebraic Number Fields
  • Pierre Samuel, Algebraic Theory of Numbers
  • Gerald Janusz, Algebraic Number Fields

Additional topics

As time permits, additional topics that might be considered include Galois representations, class field theory, module theory over Dedekind domains, etc.

Courses for which this course is prerequisite

This course is not a prerequisite for any other courses.