This is an introduction to the subject, covering both global and local fields. The prerequisite is a standard graduate course in algebra.
This is a text for a basic course in algebraic number theory, written in accordance with the following objectives.
1. Provide reasonable coverage for a one-semester course.
2. Assume as prerequisite a standard graduate course in algebra, but cover integral extensions and localization before beginning algebraic number theory. For general algebraic background, see my online text “Abstract Algebra: The Basic Graduate Year”, which can be downloaded from my web site www.math.uiuc.edu/∼ r-ash/ The abstract algebra material is referred to in this text as TBGY.
3. Cover the general theory of factorization of ideals in Dedekind domains, as well as the number field case.
4. Do some detailed calculations illustrating the use of Kummer’s theorem on lifting of prime ideals in extension fields.
5. Give enough details so that the reader can navigate through the intricate proofs of the Dirichlet unit theorem and the Minkowski bounds on element and ideal norms.
6. Cover the factorization of prime ideals in Galois extensions.
7. Cover local as well as global
Table of Contents
- Norms, Traces and Discriminants
- Dedekind Domains
- Factoring of Prime Ideals in Extensions
- The Ideal Class Group
- The Dirichlet Unit Theorem
- Cyclotomic Extensions
- Factoring of Prime Ideals in Galois Extensions
- Local Fields
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