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Number Fields

Daniel A. Marcus
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This edition of the book has been superseded by a newer edition.


This is an interesting and easy-to-read introduction to algebraic number theory, focusing on the properties of ideals and of ideal classes in subfields of the complex numbers, and using Galois theory as its primary tool. The book assumes no prior knowledge of number theory, and assumes a modest background in linear algebra and abstract algebra (the needed Galois theory is summarized in an appendix). It includes some applications to classical number theory, such as solvability of particular diophantine equations, and proofs of quadratic reciprocity and of Dirichlet’s theorem on primes in arithmetic progressions, but these are sidelines and the book’s focus is on the algebraic structures rather than on applications to the rational integers. It does include Dirichlet’s classical formula for the number of ideal classes, and explores several alternative forms of it. It doesn't go into class field theory, except for a hint of what it deals with.

The book is well-stocked with exercises. Many of these are very concrete and deal with extracting the properties of particular number fields, usually either real quadratic fields or cyclotomic fields. It also includes many results that did not fit in the main narrative.

This 1977 book is still up-to-date, but is an “antique” in one sense: It was printed from camera-ready typewritten copy and doesn’t have the nice appearance that 30 years of TeX have accustomed us to. The typescript is extremely clean and is readable, though small, and there are a few very tiny special characters that you have to squint at.

Two somewhat similar books are Pollard & Diamond’s Carus Monograph The Theory of Algebraic Numbers and Murty & Esmonde’s Problems in Algebraic Number Theory. Pollard & Diamond is a brief introduction to the subject, and as the title indicates it concentrates on algebraic numbers. It has a modest amount on ideals and no coverage of the higher structures such as ideal classes or the unit group. It makes little use of abstract algebra, and requires little previous knowledge of number theory. Its last chapter is a fairly detailed application of these ideas to Fermat’s Last Theorem. Murty & Esmonde cover generally the same topics as the present book, but in less depth. They make no use of Galois theory, partly through using ad hoc approaches and partly through not working in as much generality as the present book.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.


 A special case of Fermat's conjecture.- Number fields and number rings.- Prime decomposition in number rings.- Galois theory applied to prime decomposition.- The ideal class group and the unit group.- The distribution of ideals in a number ring.- The Dedekind zeta function and the class number formula.- The distribution of primes and an introduction to class field theory.