Galois Theory of P-Extensions

by Koch, Helmut

• ISBN13: 9783540436294
• ISBN10: 3540436294
• Format: Hardcover
• Copyright: 2002-11-01
• Publisher: Springer Nature

Summary

First published in German in 1970 and translated into Russian in 1973, this classic now becomes available in English. After introducing the theory of pro-p groups and their cohomology, it discusses presentations of the Galois groups G S of maximal p-extensions of number fields that are unramified outside a given set S of primes. It computes generators and relations as well as the cohomological dimension of some G S, and gives applications to infinite class field towers.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. In a "Postscript" Helmut Koch and Franz Lemmermeyer give a survey on the development of the field in the last 30 years. Also, a list of additional, recent references has been included.

Table of Contents

Introduction	1	(2)
Profinite Groups	3	(14)
Projective Limits of Groups and Rings	3	(4)
Profinite Groups	7	(2)
Subgroups and Quotient Groups	9	(1)
Abelian Profinite Groups, Pontryagin's Duality Theory	10	(1)
Discrete Modules	10	(2)
The Category C	12	(1)
Inductive Limits in C	13	(4)
Galois Theory of Infinite Algebraic Extensions	17	(4)
The Galois Group of Infinite Extensions	17	(2)
The Main Theorem of Galois Theory	19	(2)
Cohomology of Profinite Groups	21	(20)
Definition of Cohology Groups	21	(3)
Group Extensions	24	(1)
Dimension Shifting	24	(4)
Shapiro's Lemma	28	(1)
Restriction and Corestriction	28	(1)
The Transfer	29	(1)
Inflation and Transgression	30	(5)
Inductive Limits of Cohomology Groups	35	(2)
Cup Products	37	(4)
Free pro-p Groups	41	(8)
Construction of Free pro-p Groups	41	(1)
The Magnus Group Algebra	42	(1)
Abelian por-p Groups	43	(1)
First Characterization of Free pro-p Groups	44	(3)
Second Characterization of Free pro-p Groups	47	(2)
Cohomological Dimension	49	(4)
Definition of Cohomological Dimension	49	(1)
Euler-Poincare Characteristic	50	(3)
Presentation of pro-p Groups	53	(6)
The Generator Rank	53	(1)
Relation Systems	54	(5)
Group Algebras of pro-p Groups	59	(18)
Complete Group Algebras	59	(1)
Discrete and Compact G-Modules	60	(1)
Pro-p Groups of Dimension ≤ 2	61	(3)
Filtrations	64	(2)
Computing with Commutators and Powers	66	(2)
Group Rings of Free pro-p Groups	68	(1)
The Theorem of Golod-Shafarevich	69	(3)
Relation Structure and Cup Product	72	(5)
Results from Algebraic Number Theory	77	(16)
Algebraic Number Theory for Infinite Extensions	77	(1)
Normal Extensions	77	(2)
The Frobenius Automorphism	79	(1)
Local and Global Fields	79	(2)
The Multiplicative Group of a Local Field	81	(1)
Finite Class Field Theory	82	(2)
Infinite Class Field Theory	84	(1)
The Principal Ideal Theorem	85	(2)
Cohomology of the Formation Module	87	(2)
Cohomology of the Multiplicative Group	89	(1)
Norm Residue Symbol	90	(3)
The Maximal p-Extension	93	(6)
Fields of Characteristic p	93	(1)
Fields Containing the p-th Roots of Unity	94	(2)
Fields not Containing the p-th Roots of Unity	96	(3)
Local Fields of Finite Type	99	(12)
The Case X(p) ≠ p	99	(2)
The Case X(p) = p, δ(k) = 0	101	(2)
The Case X(p) = p, δ(k) = 1	103	(8)
Global Fields of Finite Type	111	(22)
The Maximal p-Extension	111	(3)
The Maximal p-Extension with Restricted Ramification	114	(5)
Generator Rank	119	(3)
Explicit Computation of Generators and Relations	122	(7)
The Structure of Gs in Special Cases	129	(4)
On p-Class Groups and p-Class Field Towers	133	(16)
A Criterion for Class Number Prime to p	133	(3)
The p-Class Field Tower of Cyclic Extensions of Degree p	136	(8)
A Criterion for Infinite Class Field Towers	144	(5)
The Cohomological Dimension of Gs	149	(14)
Cohomology of S-Unit Groups	149	(5)
The Case δ(k) = 1	154	(3)
The Case δ(k) =0	157	(6)
References	163	(4)
Biliography	167	(6)
Notation	173	(2)
Postsctipt	175	(6)
Additional References	181	(6)
Author Index	187	(2)
Subject Index	189