Cyclic Galois Extensions of Commutative Rings
The structure theory of abelian extensions of commutative rings is a subjectwhere commutative algebra and algebraic number theory overlap. This exposition is aimed at readers with some background in either of these two fields. Emphasis is given to the not
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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Cornelius Greither
Cyclic Galois Extensions of Commutative Rings
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Cornelius Greither Mathematisches Institut der Universitat Munchen Theresienstr. 39 W-SOOO Miinchen 2, Germany
Mathematics Subject Classification (1991): llRlS, llR23, llR33, lIS15, 13B05, 13B15,14E20
ISBN 3-540-56350-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56350-4 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author 46/3140-543210 Printed on acid-free paper
CONTENTS
Introductlon
vii
Chapter 0: Galolll theory of commutatlYe ring.
§1 §2 §3 §4 §5 §6 §7 §8
Definitions and basic properties The main theorem of Galois theory Functoriality and the Harrison product Ramification Kummer theory and Artin-Schreier theory Normal bases and Galois module structure Galois descent Zp-extensions
1
6 8 17 19
25 28
30
Chapter I: Cyclotomic deaoent
§1 §2 §3
Cyclotomic extensions Descent of normal bases Cyclotomic descent: the main theorems
32
38 45
Chapter D: COI'UtrIctlon IIDd "HJlbert'. Theorem 90"
§1 §2 §3 §4
Corestriction Lemmas on group cohomology "Hilbert 90": the kernel and image of the corestriction Lifting theorems
55 60 62 64
Chapter 01: Calculations with UD1tI
§1 §2 §3
Results on twisted Galois modules Finite fields and t,-adic fields Number fields
67 70
73
Chapter IV: Cyclic p-exteD8ions IIDd Zp-extensions of number flelda §1 §2 §3 §4 §5 §6
Cp» -extensions and ramification Z p -extensions The asymptotic order of P(R,Cp"l Calculation of q K: examples Torsion points on abelian varieties with complex multiplication Further results: a short survey
77 79 83 88 91 95
vi
ChIIpter V: Geometric theory: Cyclic extenaloDII of fln1tely generated fJelda
§1 §2 §3
Geometric prerequisites l.p -extensions of absolutely finitely generated fields A finiteness result
CIuIpter VI: Cyclic GslalB theory without the cond1t1on. "p-1
§1
§2 §3 §4 §5 §6
Witt rings and Artin-Schreier theory for rings of characteristic p Patching results Kummer theory without the condition "p -1 E R" The main result and Artin-Hasse exponentials Proofs and examples Application: Generic Galois extensions
E
97 101 106
R" 109
113 116
120 126 135
References
140
Index
14
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