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Local Fields

Jean-Pierre Serre
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 67
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Michael Berg
, on
Corps Locaux, or, as the title above has it, Local Fields, is a masterpiece.   It was my good fortune, as an advanced undergraduate and, later, PhD student at UCLA and UCSD, to be shepherded by wise professors in the direction of class field theory.  I did my doctoral work on modular forms, but it was the late V. S. Varadarajan’s seminar on class field theory that actually started me in the direction of algebraic number theory.  I had already done a reading course with him, using the exquisite book, Elliptic Functions According to Eisenstein and Kronecker, by the redoubtable André Weil, then hot off the presses, and I was hooked.  Varadarajan was doing a seminar on class field theory soon after this, and I jumped aboard.  And so it was that I was introduced to what might perhaps be called the Bourbaki tradition in algebraic number theory.  A friend and fellow traveler was a big fan of Pierre Samuel’s Algebraic Theory of Numbers, and I soon became a fan, too. Thus, when I was starting out as a serious mathematics student, now well over thirty years ago, I was in happy contact with fabulous French sources in the field.
And things got better and better --- after Samuel (whose book I owned in bootleg (Xeroxed) form: I have it still), I got hold of one of the greatest books I have ever read: Jean-Pierre Serre’s Cours d’Arithmétique, i.e. A Course of Arithmetic.   I realized even at that tender age that this was mathematical writing at its very best.  I stand by this opinion, which is of course universally agreed on: Serre is …, well, it’s Serre --- it cannot be imagined that anyone should match him for elegance and clarity in exposition, modulo a serious commitment on the reader’s part to work hard when reading Serre’s prose.  It is very beautiful, and once understood, possessed of a quality of, as it were, inevitability.  Serre says it all so well, it cannot really be said better.  To wit, as recently as a year or so back I was called to teach our senior seminar and I chose to do modular forms.  I picked for my text an acknowledged classic explicitly on this subject, but as the course unfolded it became clear to me that I needed something else, even if it were done much more compactly and tersely: to A Course of Arithmetic I went.  I was right to do so.
Local Fields fits in this line with a vengeance. In my opinion, it is the best source on this material, at the very heart of algebraic number theory (being one of the two lungs of class field theory: the other is, naturally, global class field theory), and is non-negotiable as a text for any and all serious students and practitioners of this craft.  Other excellent sources exist, of course, e.g. Cassels-Fröhlich’s compact first chapter, or Iwasawa’s Local Class Field Theory.  I’d suggest that a fledgling algebraic number theorist acquire all of these books, and others besides, but Serre needs to be given a certain priority in this list.  He is unquestionably superb as an expositor, indeed unsurpassed, as well as being a fabulous mathematician, of course.  And once you have acquired the taste for his writing, go at his Faisceaux Algébriques Cohérents: everybody needs to know what a sheaf is, after all.  In other words, it’s like with P. G. Wodehouse or Agatha Christie: just get the books and read them.  You will never be disappointed.


Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

 Basic results on discrete valuation rings, Dedekind domains, and completions.-
Ramification theory: discriminant, different, ramification subgroups, Hasse-Arf theorem, Artin representations.-
Group cohomology, with emphasis on arithmetical applications: theorems of Tate and Nakayama, Galois cohomology, class formations.-
Local class field theory, presented from the cohomological point of view. The main result is the determination of the topological Galois group of the maximal abelian extension.