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

Algebraic number theory is introduced through the problem of generalizing unique prime factorization from ordinary integers to more general domains. Dedekind's concept of ideals is needed to restore unique prime factorization, and Emmy Noether's theory of Dedekind rings encapsulates the properties of rings that make it possible. The book develops the theory of these concepts, following their history and motivating each conceptual step by pointing to their origins, with a focus on unique prime factorization.

Format: Hardback
Length: 250 pages
Publication date: 11 August 2022
Publisher: Cambridge University Press

This comprehensive book delves into the realm of algebraic number theory, offering a comprehensive exploration of the problem of generalizing unique prime factorization from ordinary integers to more diverse domains. While solving polynomial equations in integers naturally leads to these domains, the process can sometimes result in the loss of unique prime factorization. To reclaim this essential feature, Dedekind's concept of ideals becomes indispensable. However, it is crucial to have a solid understanding of algebraic number fields, algebraic integers, rings, vector spaces, and modules as supporting foundations. It was left to the brilliant Emmy Noether to encapsulate the properties of rings that enable unique prime factorization, which is now recognized as Dedekind rings. Throughout the book, the theory of these concepts is developed, tracing their historical origins, providing motivating insights, and maintaining a clear focus on the ultimate goal of achieving unique prime factorization with minimal distractions or prerequisites. This approach ensures that the book remains self-contained, making it accessible and easy to read, while also being suitable for a one-semester course. By delving into the intricacies of algebraic number theory, this book offers a valuable resource for students, scholars, and enthusiasts alike, providing a deeper understanding of the mathematical foundations underlying this fascinating field.

Weight: 494g
ISBN-13: 9781316518953