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

Description: Algebraic Number Theory for Beginners by John Stillwell This book, meant for undergraduate mathematics students and teachers, introduces algebraic number theory through problems from ordinary number theory that can be solved with the help of algebraic numbers, using a suitable generalization of unique prime factorization. The material is motivated by weaving historical information throughout. FORMAT Hardcover LANGUAGE English CONDITION Brand New Publisher Description 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 Dedekinds 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. Author Biography John Stillwell is the author of many books on mathematics; among the best known are Mathematics and its History, Naive Lie Theory, and Elements of Mathematics. He is a member of the inaugural class of Fellows of the American Mathematical Society and winner of the Chauvenet Prize for mathematical exposition. Table of Contents 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. Review 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 VirginiaStillwell, 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. Stillwells 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 CruzThis 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, ChoiceIt 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 Promotional A concise and well-motivated introduction to algebraic number theory, following the evolution of unique prime factorization through history. Review Quote 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. Stillwells 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 Promotional "Headline" A concise and well-motivated introduction to algebraic number theory, following the evolution of unique prime factorization through history. Details ISBN1316518957 Author John Stillwell Short Title Algebraic Number Theory for Beginners Publisher Cambridge University Press Language English Year 2022 ISBN-10 1316518957 ISBN-13 9781316518953 Format Hardcover Pages 250 Imprint Cambridge University Press Place of Publication Cambridge Country of Publication United Kingdom Publication Date 2022-08-11 AU Release Date 2022-08-11 NZ Release Date 2022-08-11 UK Release Date 2022-08-11 Subtitle Following a Path From Euclid to Noether Alternative 9781009001922 DEWEY 512.74 Illustrations Worked examples or Exercises Audience Professional & Vocational We've got this At The Nile, if you're looking for it, we've got it. 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