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Algebraic Number Theory
J. W. S. Cassels and A. Fröhlich, editors
Publisher: London Mathematical Society (2010)
Details: 366 pages, Paperback
Topics: Number Theory, Algebraic Number Theory
MAA Review[Reviewed by Allen Stenger, on 06/29/2010]
This is a reprint of a much-admired 1967 text, printed with 8 pages of new errata. The errata were compiled by Kevin Buzzard with the assistance of an army of checkers and are available for download from his web page.
The book is the proceedings of a conference held at Brighton, UK in 1965, but it is much more useful than the typical such collection. This is partly because the conference was organized specifically for teaching, partly because the papers were written by some of the biggest names in the field, and partly because the papers were reworked very carefully by the authors before being published here. The editors included an unusual disclaimer: “neither the lecturers nor the note-takers have any responsibility for any inaccuracies which may remain: they are an act of God.” And indeed, despite the length of the errata, as Buzzard says “almost all of them are utterly trivial.”
The book is still remarkably up-to-date. It covers nearly all areas of the subject, although its approach is slanted somewhat toward class field theory. Some more recent texts with a similar approach and coverage include Lang’s Algebraic Number Theory and Weil’s misnamed Basic Number Theory.
The book’s approach is very abstract and there is very little here on the classic problems that have driven the development of the theory. For example, Fermat’s last theorem, higher-order reciprocity laws, and rational points on elliptic curves are not mentioned except indirectly through some discussion of cyclotomic fields and of complex multiplication. Some more recent texts that give good coverage of these topics include van der Poorten’s Notes on Fermat’s Last Theorem, Ireland & Rosen’s A Classical Introduction to Modern Number Theory, and Silverman & Tate’s Rational Points on Elliptic Curves.
Despite the top-notch contributors and the historical significance of the present book, it is not a good book for beginners in algebraic number theory, primarily because of the lack of applications. It would be quite possible to study the present book and learn a lot from it without learning that the subject had anything to do with numbers. Good books for beginners include Marcus’s Number Fields and Pollard & Diamond’s Theory of Algebraic Numbers, in addition to the application-oriented books listed above. However, these books do not deal with the more advanced topics such as cohomology and class field theory that dominate the present work. This seems like a lot of books to recommend for one subject, but as Lang writes in the Foreword to his book, after recommending a long list of books, “It seems that over the years, everything that has been done has proved useful theoretically or as examples, for the further development of the theory. Old, and seemingly isolated special cases have continuously acquired renewed significance, often after half a century or more.”
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 MathNerds.org, a math help site that fosters inquiry learning.