Algebraic Number Theory for Beginners
Following a Path From Euclid to Noether

A concise and well-motivated introduction to algebraic number theory, following the evolution of unique prime factorization through history.

John Stillwell (Author)

9781316518953, Cambridge University Press

Hardback, published 11 August 2022

242 pages
23.5 x 15.7 x 2 cm, 0.49 kg

'It goes without saying that the exposition is as clear as possible. There are a few exercises in each section, historical comments, a decent bibliography and an index.' Franz Lemmermeyer, zbMATH Open

This book introduces algebraic number theory through the problem of generalizing 'unique prime factorization' from ordinary integers to more general domains. Solving polynomial equations in integers leads naturally to these domains, but unique prime factorization may be lost in the process. To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces, and modules. It was left to Emmy Noether to encapsulate the properties of rings that make unique prime factorization possible, in what we now call Dedekind rings. The book develops the theory of these concepts, following their history, motivating each conceptual step by pointing to its origins, and focusing on the goal of unique prime factorization with a minimum of distraction or prerequisites. This makes a self-contained easy-to-read book, short enough for a one-semester course.

Preface
1. Euclidean arithmetic
2. Diophantine arithmetic
3. Quadratic forms
4. Rings and fields
5. Ideals
6. Vector spaces
7. Determinant theory
8. Modules
9. Ideals and prime factorization
References
Index.

Subject Areas: Number theory [PBH], Algebra [PBF]