Algebraic Number Theory for Beginners

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.

Provides a short, self-contained, and readable introduction to the field for beginners, reviewing even basic linear algebra from the viewpoint of number theory
Integrates historical information into the mathematical development, conveying to students where concepts come from and dispelling any mystery around mathematical terms
Includes approximately 300 timely and interesting exercises, testing students' understanding as new concepts occur, but leading to new results
Prerequisites are only a familiarity with the concept of matrices, as well as proofs and abstraction
An ideal main text for a course in algebraic number theory, or as supplementary material for a course in abstract algebra or number theory

Reviews & endorsements

'In Algebraic Number Theory for Beginners, John Stillwell once again displays his remarkable talent for using the history of mathematics to motivate and explore even the most abstract mathematical concepts at an accessible, undergraduate level. This book is another gem of the genre Stillwell has done so much to enhance.' Karen Hunger Parshall, University of Virginia

'Stillwell, more than any author I know, helps us understand mathematics from its roots. In this book, he leads us into algebraic number theory along a historical route from concrete to abstract. In doing so, Stillwell makes a strong pedagogical case for flipping a typical algebraic number theory course — that students will understand number theory better if questions about numbers come before and throughout the abstract theory of rings and ideals. The treatments of mathematics and its history are crystal clear and meticulous. Stillwell's text is particularly well-suited for an advanced undergraduate or early graduate-level course in number theory. Experts also will find this text to be an incredible resource for its historical approach and well-motivated exercises. Stillwell has written another gem, this time for readers interested in number theory, abstract algebra, and their intertwined history.' Martin Weissman, University of California, Santa Cruz

'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

'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

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