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Cohomology of Number Fields

J. Neukirch, A. Schmidt, and K. Wingberg
Publication Date: 
Number of Pages: 
Grundlehren der mathematischen Wissenschaften 323
[Reviewed by
Fernando Q. Gouvêa
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One of the fascinating things about number theory is the vast array of mathematical tools that can be (and have been) used to study it. Since the mid-twentieth century, one of these tools has been cohomology. Almost as soon as the cohomology of groups was invented in the 1930s, number theorists realized that when the group is a Galois group, the cohomology carries arithmetic information. Soon Galois cohomology was playing a central role in algebraic number theory. I tried to give a sketch of this historical process in my review of the first edition of this book (Bulletin of the AMS, 39, 101–107). The publication of a second edition gives me a chance to once again emphasize what an important book it is.

Neukirch, Schmidt, and Wingberg focus on Galois cohomology, rather than dealing with its more difficult generalizations (étale cohomology, for example). They also stick to the more classical Galois modules: typically, submodules of number fields. Given these restrictions, however, they do a very thorough job. In particular, they provide complete proofs of several important theorems whose original proofs "were spread over many original articles, some of which contained serious mistakes, and some even remained unpublished." Just this would make the book a necessary part of the number theorist's library. That it's also well written, clear, and systematic is a very welcome bonus.

A new edition always allows for corrections, but in this case there has also been significant expansion, as a comparison of the tables of contents shows immediately. The preface to the second edition gives a summary:

In the algebraic part you will find new sections on filtered cochain complexes, on the degeneration of spectral sequences and on Tate cohomology of profinite groups. Amongst other topics, the arithmetic part contains a new section on duality theorems for unramified and tamely ramified extensions, a careful analysis of 2-extensions of real number fields and a complete proof of Neukirch's theorem on solvable Galois groups with given local conditions.

There are many goodies here, from a proof of Poitou-Tate duality to a complete account of Shafarevich's theorem on solvable groups as Galois groups. There is a chapter on Iwasawa Theory of number fields that can serve as a good introduction to that circle of ideas. The final chapter deals with Grothendieck's "Anabelian Geometry" and the conjectures that surround it.

Cohomology of Number Fields is not easy reading, but it is an indispensable book for anyone working in number theory. I'm not sure how well it would serve as a text for novices — unless they are very unusual novices — but that is not really the intended audience for a book in the Grundlehren series. Neukirch, Schmidt, and Wingberg have, in fact, produced something close to the archetypal book in that series: authoritative, complete, careful, and sure to be a reliable reference for many years.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.

I Algebraic Theory: Cohomology of Profinite Groups.- Some Homological Algebra.- Duality Properties of Profinite Groups.- Free Products of Profinite Groups.- Iwasawa Modules
II Arithmetic Theory: Galois Cohomology.- Cohomology of Local Fields.- Cohomology of Global Fields.- The Absolute Galois Group of a Global Field.- Restricted Ramification.- Iwasawa Theory of Number Fields; Anabelian Geometry.- Literature.- Index.