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)

9781009001922, Cambridge University Press

Paperback / softback, published 11 August 2022

250 pages
22.8 x 15.1 x 1.3 cm, 0.36 kg

'This book is sure to be welcomed by advanced students and their instructors … A helpful index and an extensive list of references conclude the text … Highly recommended.' J. Johnson, Choice

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]