Number Fields

1. Front Matter

   Pages i-xviii

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2. A Special Case of Fermat’s Conjecture

   • Daniel A. Marcus

   Pages 1-8

3. Number Fields and Number Rings

   • Daniel A. Marcus

   Pages 9-38

4. Prime Decomposition in Number Rings

   • Daniel A. Marcus

   Pages 39-67

5. Galois Theory Applied to Prime Decomposition

   • Daniel A. Marcus

   Pages 69-90

6. The Ideal Class Group and the Unit Group

   • Daniel A. Marcus

   Pages 91-110

7. The Distribution of Ideals in a Number Ring

   • Daniel A. Marcus

   Pages 111-127

8. The Dedekind Zeta Function and the Class Number Formula

   • Daniel A. Marcus

   Pages 129-158

9. The Distribution of Primes and an Introduction to Class Field Theory

   • Daniel A. Marcus

   Pages 159-177

10. Back Matter

    Pages 179-203

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Requiring no more than a basic knowledge of abstract algebra, this textbook presents the basics of algebraic number theory in a straightforward, "down-to-earth" manner. It thus avoids local methods, for example, and presents proofs in a way that highlights key arguments. There are several hundred exercises, providing a wealth of both computational and theoretical practice, as well as appendices summarizing the necessary background in algebra.

Now in a newly typeset edition including a foreword by Barry Mazur, this highly regarded textbook will continue to provide lecturers and their students with an invaluable resource and a compelling gateway to a beautiful subject.

 

From the reviews:

“A thoroughly delightful introduction to algebraic number theory” – Ezra Brown in the Mathematical Reviews

“An excellent basis for an introductory graduate course in algebraic number theory” – Harold Edwards in the Bulletin of the American Mathematical Society

• number fields
• number rings
• prime decomposition in number rings
• Galois theory applied to prime decomposition
• ideal class group
• unit group
• distribution of ideals
• Dedekind zeta function and the class number formula
• distribution of primes
• class field theory
• MSC (2010): 12-01, 11Rxx, 11Txx

“This volume has stood the test of time. It is both demanding of and rewarding for anyone willing to work through it.” (C. Baxa, Monatshefte für Mathematik, Vol. 201 (2), 2023)

“It is well structured and gives the reader lots of motivation to learn more about the subject. It is one of the rare books which can help students to learn new stuff by themselves by solving the numerous exercises which cover very deep and important results … . The prerequisites for the reader are kept to a minimum making this book accessible to students at a much earlier stage than usual textbooks on algebraic number theory.”

“A book unabashedly devoted to number fields is a fabulous idea. … it goes without saying that the exercises in the book — and there are many — are of great importance and the reader should certainly do a lot of them; they are very good and add to the fabulous experience of learning this material. … it’s a wonderful book.” (Michael Berg, MAA Reviews, October 22, 2018)

• Columbus, USA

  Daniel A. Marcus

Daniel A. Marcus received his PhD from Harvard University in 1972. He was a J. Willard Gibbs Instructor at Yale University from 1972 to 1974 and Professor of Mathematics at California State Polytechnic University, Pomona, from 1979 to 2004. He published research papers in the areas of graph theory, number theory and combinatorics. The present book grew out of a lecture course given by the author at Yale University.

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