Galois Theory of p-Extensions / Edition 1

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Overview

Helmut Koch's classic is now available in English. Competently translated by Franz Lemmermeyer, it introduces the theory of pro-p groups and their cohomology. The book contains a postscript on the recent development of the field written by H. Koch and F. Lemmermeyer, along with many additional recent references.

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Editorial Reviews

From the Publisher
From the reviews:

Most algebraic number theorists have had occasion to refer to Helmut Koch's Galloissche Theorie der p-Erweiterungen, first published some thirty years ago and still an essential reference on the subject of p-extensions (that is, Galois extensions whose Galois groups are p-groups) of algebraic number fields. But readers like myself have found ourselves hobbled by (a) the difficulty of finding a copy and (b) the inadequacy of our German reading skills.

No more. Thanks to Franz Lemmermeyer, here is Koch's Galois Theory of p-Extensions in English, supplemented by a "Postscript" that summarizes in a few pages what has happened since 1969. This new edition will keep Koch's work available for many years yet, and I'm sure many readers will be grateful that Springer has sprung for it.

"This book … has been acclaimed as an excellent complement to J. P. Serre’s ‘Cohomologie galoisienne’. … The monograph is devoted to the Galois theory of p-extensions on the basis of cohomological techniques. … this classic can be recommended to any reader interested in this field." (G. Kowol, Monatshefte für Mathematik, Vol. 141 (2), 2004)

"This is an English translation (an excellent job by F. Lemmermeyer) of the German edition … . Koch’s clear and precise exposition may … serve as an excellent guide to any newcomer. … The goal is to compute the generators and relations of Gs, its cohomological dimension and to apply these results to the construction of infinite class field towers." (B. Kunyavskii, Zentralblatt MATH, Vol. 1023, 2003)

"First published in German in 1970 and translated into Russian in 1973, this classic now becomes available in English. … The book demonstrates that the cohomology of groups is very useful for studying Galois theory of number fields; at the same time, it offers a down to earth introduction to the cohomological method." (L’ Enseignement Mathematique, Vol. 48 (3-4), 2002)

From the Publisher

From the reviews:

Most algebraic number theorists have had occasion to refer to Helmut Koch's Galloissche Theorie der p-Erweiterungen, first published some thirty years ago and still an essential reference on the subject of p-extensions (that is, Galois extensions whose Galois groups are p-groups) of algebraic number fields. But readers like myself have found ourselves hobbled by (a) the difficulty of finding a copy and (b) the inadequacy of our German reading skills.

No more. Thanks to Franz Lemmermeyer, here is Koch's Galois Theory of p-Extensions in English, supplemented by a "Postscript" that summarizes in a few pages what has happened since 1969. This new edition will keep Koch's work available for many years yet, and I'm sure many readers will be grateful that Springer has sprung for it.

"This book … has been acclaimed as an excellent complement to J. P. Serre’s ‘Cohomologie galoisienne’. … The monograph is devoted to the Galois theory of p-extensions on the basis of cohomological techniques. … this classic can be recommended to any reader interested in this field." (G. Kowol, Monatshefte für Mathematik, Vol. 141 (2), 2004)

"This is an English translation (an excellent job by F. Lemmermeyer) of the German edition … . Koch’s clear and precise exposition may … serve as an excellent guide to any newcomer. … The goal is to compute the generators and relations of Gs, its cohomological dimension and to apply these results to the construction of infinite class field towers." (B. Kunyavskii, Zentralblatt MATH, Vol. 1023, 2003)

"First published in German in 1970 and translated into Russian in 1973, this classic now becomes available in English. … The book demonstrates that the cohomology of groups is very useful for studying Galois theory of number fields; at the same time, it offers a down to earth introduction to the cohomological method." (L’ Enseignement Mathematique, Vol. 48 (3-4), 2002)

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Product Details

  • ISBN-13: 9783642078170
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 12/3/2010
  • Series: Springer Monographs in Mathematics Series
  • Edition description: Softcover reprint of hardcover 1st ed. 2002
  • Edition number: 1
  • Pages: 191
  • Product dimensions: 0.44 (w) x 9.21 (h) x 6.14 (d)

Table of Contents

1. Profinite Groups.- 2. Galois Theory of Infinite Algebraic Extensions.- 3. Cohomology of Profinite Groups.- 4. Free pro-p Groups.- 5. Cohomological Dimension.- 6. Presentation of pro-p Groups.- 7. Group Algebras of pro-p Groups.- 8. Results from Algebraic Number Theory.- 9. The Maximal p-Extension.- 10. Local Fields of Finite Type.- 11. Global Fields of Finite Type.- 12. On p-Class Groups and p-Class Field Towers.- 13. The Cohomological Dimension of GS.- References.- Notation.- Postscript.- Additional References.- Author Index.

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