Table Of ContentOtto Endler
Valuation Theory
Springer-Verlag
Berlin Heidelberg New York 1972
Otto Endler
Mathematisches Institut der UniversiUit 80nn
and Instituto de Matematica P. e A, Rio de Janeiro
AMS Subject Classifications (1970)
Primary 12J20, 12J10, 10M10, 13A15
Secondary 13815,12810,13820,10840, 13F05, 13F15, 13J15, 14A05,
13J05, 13J10, 14H05
ISBN-13: 978-3-540-06070-3 e-ISBN-13: 978-3-642-65505-0
001: 10.1007/978-3-642-65505-0
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction
by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copy
right Law where copies are made for other than private use, a fee is payable to the publisher, the amount
of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1972.
Library of Congress Catalog Card Number 72-92285.
To the Memory of
WOLFGANG KRULL
26 August 1899 - 12 April 1971
Preface
These are the revised notes of a course for graduate
students and some seminar talks which I gave at the University of
Rochester during Fall Term 1969/70. They would not have been
written without the encouragement and the aid which I received,
during all stages of the work, by friends from Rochester, Rio de
Janeiro, and Bonn. I wish to thank all of them: Barbara Grabkowicz
encouraged me to write these notes in English and read carefully
parts of a preliminary manuscript, as did Gervasio G. Bastos, Yves
A. E. Lequain, Walter Strubel, and Antonio J. Engler. Many valuable
suggestions were given me by Yves A. E. Lequain, and several
improvements of theorems and proofs are due to him. I am particularly
grateful to Linda C. Hill for her criticism in reading the last
version and for improving and smoothing many of my formulations.
Last but not least I thank Wilson Goes for the excellent typing.
Most of this book was elaborated when I stayed in Rio de
Janeiro as a Visiting Professor at IMPA (Institute for Pure and
Applied Mathematics) and as a Pesquisador-Conferencista of CNPq
(National Research Council). Thanks are also due to these institu
tions.
Rio de Janeiro, September 1972
Otto Endler
Contents
v
Preface
Symbols Used in Text VIII
Introduction XI
Chapter I Valuations
§ 1 Valuations • 1
§ 2 Completions and extensions of valuations 8
§ 3 Non-archimedean valuations 18
§ 4 Discrete exponential valuations 25
§ 5 Complete discretely valued fields 30
Chapter II Valuation Rings
§ 6 Valuation rings 40
§ 7 Krull valuations 45
§ 8 Places , 52
§ 9 The extension theorem 59
§ 10 Integrally closed domains 66
§ 11 Prtlfer rings, Approximation theorems 72
§ 12 Krull rings and Dedekind rings 82
Chapter III Extep_sion of Valuation Rings
§ 13 The case of an algebraic field extension 94
§ 14 The case of a normal field extension , 104
§ 15 Decomposition group and decomposition field 109
§ 16 Henselian valuation rings 113
§ 17 Extension of valuation rings and henselization 123
§ 18 The equality r ei,fi = n 136
§ 19 Inertia group and inertia field 144
§ 20 Ramification group and ramification field 154
§ 21 Higher ramification groups 173
§ 22 Unramified and tamely ramified extensions 178
-VII-
Chapter IV Fields with Prescribed Valuations
§ 23 Introduction and notation • • 185
§ 24 Topological preliminaries • 191
§ 25 Solvable (~l, ••• ,~k)-prescriptions 197
§ 26 Henselian and antihenselian valuations 200
§ 27 Prescription of value groups and residue fields 206
§ 28 The case o£ infinite field extensions •••• 215
Exercises 222
Bibliography 238
Index 240
Symbols Used in Text
=:> implication
~ if and only if
8 -+ T map from the set 8 into the set T
x ..... Y map assigning y to x
18 identical map of the set 8
1 identical imbedding of 8 into T, where 8 is a
8,T
subset of T
1oL18 restriction of the map ~ to the subset 8
V diagonal map
~ empty set
!;;; inclusion
c proper inclusion
E is an element of
8\T difference set (set of all x E 8 such that x ~ T)
81 x ••• x 8 n product of the sets 81, ••• ,8n
lXE I 81 product of the family of sets (8t ) t E I
II number of elements (cardinality)
IN set of all non-negative rational integers
!L, ring of all rational integers
III field of all rational numbers
IR field of all real numbers
C field of all complex numbers
IF prime field of characteristic p (~O)
P
IAI lp field of all p-adic numbers
IR set of all non-negative real numbers
+
IR* multiplicative group of all positive real numbers
+
K = K U {'" } projective field obtained from the field K
ac(K) algebraic closure of the field K
sc(K) separable closure of the field K
K maximal separable subextension of a given algebraic
sep
field extension LIK
K. maximal purely inseparable subextension of a given
·:1.ns
normal field extension NIK
[L:K] separability degree of the algebraic extension LIK
sep
-IX-
[L:K] . inseparability degree of the algebraic extension LIK
lnsep
Char K characteristic of the field K
Mon{LIK) set of all K-monomorphisms from L into a given
algebraically closed field 0
set of all classes of K-conjugate K-monomorphisms
from L into 0
Aut{NIK) group of all K-automorphisms of N (Galois group
of NIK)
k{H) fixed field corresponding to the subgroup H of
Aut{NIK)
g{L) subgroup of Aut{NIK) corresponding to the sub
extension L of NIK
(G:H) group index of H in G
Hom{A,B) group of homomorphisms from A into B (Where A
and B are abelian groups)
p-character group of X {where X is an abelian
torsion group and p Char 0
integral closure of R in S
absolute value of ~
absolute value of R
absolute value of ~
p-adic valuation of ~
v p-adic exponential valuation of ~
P
Tep topology defined by the valuation ep
(K ,ep ) valued field (Where ep is a valuation of K)
(K,v) valued field (Where v is an exponential valuation
of K)
(K,A) valued field (Where A is a valuation ring of K )
(K ,cP ) completion of (K ,ep)
(K,~) completion of (K,v)
(K,A) completion of (K,A)
~ (L;A) set of all valuation rings of L which lie over
the valuation ring A
ramification index of B over K
residue degree of B over K
-x-
GZ decomposition group
KZ decomposition field
GT inertia group
KT inertia field
GV ramification group
KV ramification field
deg F degree of the polynomial F
p minimal polynomial of x over K
x\K
5. Kronecker symbol
~,j
Introduction
Giving a course or writing a book on Valuation Theory, one
is faced with the problem whether more emphasis should be given to
usual valuations (by some authors called "absolute values"), which
include archimedean valuations and in particular the usual absolute
value, or to Krull valuations, which correspond to valuation rings
of arbitrary rank and generalize non-archimedean valuations only.
If one is interested only in applications to Algebraic Number Theory
it suffices to consider usual valuations and one may even restrict
oneself to archimedean and discrete non-archimedean valuations,
since these are the only ones which occur in algebraic number fields
(i.e., finite extensions of ~). On the other hand, arbitrary Krull
valuations, or rather the corresponding valuation rings, play an
important role in Commutative Algebra and its application to
Algebraic Geometry, as well as in the theory of Diophantine equations.
In this book, we try to meet both demands. The standard
material on usual (archimedean and non-archimedean) valuations is
given rather briefly in Chapter I; more can be found in any book on
Algebraic Number Theory using valuation-theoretical methods (e .g.
Weiss [35J). The basic theory of valuation rings, Krull valuations,
and places is given in Chapter II, which includes also some results
on Prfifer rings, Krull rings, and Dedekind rings. Chapter III is
concerned with extensions of valuation rings; it contains Infinite
Ramification Theory and emphasizes the importance of henselizations.
Some more advanced topics of Valuation Theory, such as the theory
of maximal valued fields and Ribenboim's generalization of the
approximation theorem, are not presented in this book (cf Ribenboim
[30] ) . In Chapter IV we consider again usual valuations. Krull's
