Table Of ContentS M M
pringer onographs in athematics
Springer-Verlag Berlin Heidelberg GmbH
Georges Gras
Class Field Theory
From Theory to Practice
123
Georges Gras
University of Franche-Comté
Faculty of Sciences
Laboratory of Mathematics and CNRS
16, route de Gray
25030 Besançon Cedex, France
e-mail: [email protected] or [email protected]
Translator of the original French manuscript
Henri Cohen
University of Bordeaux I
Mathematics and Computer Sciences
351, Cours de la Libération
33405 Talence Cedex, France
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A` Marie-Nicole
A` mes parents
Preface
This book is intended for students, for researchers, and for all those who
are familiar with classical algebraic number theory. Its aim is to help in the
practical use and understanding of the principles of global class field theory
for number fields, without any attempt to give proofs of the foundations
or their chronological appearance. It does give some historical landmarks,
however, for a theory which began at the beginning of the twentieth century
and involved many first-rank mathematicians.
More precisely, we will assume the existence and functorial properties
of local and global reciprocity maps, as well as the existence of the Galois
correspondence of class field theory (between class groups of K and abelian
extensions of K, called class fields). These results are essentially given in
Theorems II.1.4 and II.3.3, as well as Theorems II.1.5 and II.3.5 of Chapter
II, and their proofs can easily be found in the literature.
Even though the proofs of the four basic results are omitted, we give
detailedjustificationsfortheconsequencesthatwededucefromtheseresults.
Thisismoreeffectivesincetheproofsofthefoundationsrelyonalogicalchain
of reasoning which is different from that of their use. The space that we gain
in proceeding in this manner is used to give examples, remarks, to insist on
certain technical aspects, and to give practical applications, and finally to
give some new or little-known results (such as the precise description of the
Galois group of the abelian closure of a number field).
This is done at an elementary level, but with a sufficient degree of gen-
erality. For example, for a fixed number field K, the generalized ideal class
groupsC(cid:2)S (indexedbyanintegralidealm(cid:2)=0ofsupportT correspondingto
m
ramification,andasetofplacesS correspondingtodecomposition(i.e.,split-
ting)), which will play a central role, include both the ordinary class group
C(cid:2)ord andtherestricted(ornarrow)classgroupC(cid:2)res whichdonotcomefrom
twodifferenttheories,butasingleonethankstoasuitablechoiceofmandS.
The presence of the decomposition parameter S is also essential because of
the symmetry theorems between ramification and decomposition occurring
in the theory.
In addition, so as not to create a methodological bias, we will often com-
pare the two classical formalisms: on the one hand that of id`ele class groups,
andontheotherhandthatofgeneralizedidealclassgroupswhichareGalois
viii Preface
groups of ray class fields. Concerning this, we note that a number of exposi-
tions of class field theory begin with the formalism of congruence groups in-
troducedbyWeber,whichisessentiallythatofgeneralizedidealclassgroups
as above after lifting to the group of fractional ideals of K. However, once
we know the existence of ray class fields containing a given abelian number
field (assumption which we can make from the beginning, contrary to most
textbooks), we obtain a much nicer and more efficient Galois-theoretic de-
scriptionoftheclassificationofabelianextensions:thisisthegeneralizationof
theKronecker–Webertheorem,whererayclassfields(overQ)arecyclotomic
fields, and introduces the conductor. At this finite level, the idelic viewpoint
differs little from the generalized ideal class viewpoint. We will, however, use
both techniques since some theoretical computations are much simpler in
idelicterms,andsomearithmeticalinterpretationsaremorenaturalinterms
of ideal classes. The situation is different for limiting processes.
We assume known the classical theory of algebraic number fields (finite
extensionsofQ),andthatoflocalfieldsobtainedascompletionsoftheabove,
thus giving finite extensions of the fields Q and R. We also assume known
(cid:2)
the elementary properties of profinite groups. Since we will use idelic and
profinite objects, we will naturally have to use topological arguments. We
will also use elementary results of the representation theory of finite groups.
We give a number of exercises, together with their solutions. They are
usually results which are used elsewhere. Thus, we will consider them as
supplementaryresultswithproofs.Sometimesseveralsolutionsareproposed,
as well as a number of comments.
Inadditiontogivingthemainreferencesconcerningthetopicswewillbe
treating,wewillalmostsystematicallyquotethebookofKoch[e,Ko3]which
for each subject gives a “main reference”. This will be more useful for the
reader who does not intend to read a large number of papers. In addition,
Koch’s book contains material going beyond class field theory; this is also
the case for the book of Manin–Panchishkin [e, MP], which deals with more
geometric aspects of number theory.
For the foundations of class field theory we have not tried to cite all the
originalpapers,whichareverynumerousandoftennoteasilyaccessible.This
isperhapsunfair,andcannotdojusticetotheremarkableworkofthepioneers
and their descendants, which has led to the modern multifaceted approach
to the theory, as well as its extensions (local class field theory in arbitrary
dimensions, the Langlands programme for nonabelian class field theory, and
higherclassfieldtheoryinthecaseofarithmeticgeometry).Amongthemain
pioneers,fromtheveryendofthenineteenthcenturytothefirstthirdofthe
twentieth, we must at least mention D. Hilbert, L. Kronecker, H. Weber,
P. Furtwa¨ngler, T. Takagi, E. Artin. They have been followed by H. Hasse,
F.K. Schmidt, C. Chevalley, J. Herbrand, A. Scholz, O. Taussky, S. Iyanaga,
whose contributions precede the major references used today (Artin–Tate
[d, AT], Cassels–Fr¨ohlich [d, CF], Iyanaga [d, Iy1], Lang [d, Lang1], Serre
Preface ix
[d, Se2], Weil [d, We1]), all published in the sixties, which contain (except
Lang’s book, more in accordance with our point of view) the cohomologi-
cal setting of class field theory developed by Hochschild–Nakayama, Tate,
Sˇafareviˇc,Weil,Serre,...Thereadermayreconstructthehistoryofclassfield
theory by using the references of Section (i) of the bibliography, Chapter XI
of [d, CF], Appendix 2 of [d, Iy1]; in [h, Iy2], in addition to interesting bio-
graphicalinformation(inparticularonTakagi),onecanfindamorepersonal
and detailed description of the scientific and human relationships between
several mathematicians, around class field theory, in France, Germany, and
Japan,duringalargepartofthetwentiethcentury.Atthe1998international
conference held in Tokyo, many talks where devoted to the history of class
field theory (see in [i, Miy0] the articles by Iyanaga, Frei, Koch, Roquette).
More recent books extend beyond the subject of class field theory alone,
which is then considered as a basic tool, and deal with Galois cohomology
(Section (g) of the bibliography) or higher K-theory. They are at a much
higher technical level than that of the present book.
Finally, we would like to stress that we have used notations which seem
tobeasmuchaspossibleconsideredthemostcommon.Theyaretakenfrom
several “cultures”, and are sufficiently precise so as to be understandable
without reference to their definition, using some natural general principles.
However, we have dropped a number of bad notations which have unfortu-
×
natelybecomeclassical,suchasK andJ fortheS-units(globalandidelic
S S
respectively), since for S =∅ they do not represent K× and J, but the unit
groups (global and idelic respectively) E and U!
I would like to express my warmest thanks to all the people who have
looked at the manuscript of the book, at different stages of writing, and/or
who have helped in its publication; in particular, in a rough chronological
order, to Paul Rey for his pertinent remarks, for his persistence in finding
incorrect statements, Michel Waldschmidt for his editorial action and his
bibliographic comments, E´va Bayer-Fluckiger for her friendly help, Thong
Nguyen Quang Do for fruitful technical exchanges, Jean Cougnard, Chazad
Movahhedi, Franz Lemmermeyer for a number of simplifications and im-
provements,ChristianMaireformanyinterestingdiscussionsonpro-p-groups
andrestrictedramification,Jean-Franc¸oisJaulentaboutthelogarithmicclass
group, also to Henri Lombardi, Detlev Hoffmann for valuable help, anony-
mous colleagues for their suggestions, then especially to Henri Cohen for
the work of translating the first draft of the book into English and for his
influence through his books on computational number theory.
La B´erarde, August 2002 Georges Gras
Table of Contents
Preface ................................................... vii
Introduction to Global Class Field Theory................ 1
I. Basic Tools and Notations ................................ 7
§1 Places of K............................................ 9
§2 Embeddings of a Number Field in its Completions.......... 12
§3 Number and Ideal Groups ............................... 21
a) The Local Case: The Group K× ................... 21
v
b) The Global Case: Numbers, Ideals, and Units ........ 23
§4 Id`ele Groups — Generalized Class Groups ................. 27
a) Id`ele Groups — Topology ......................... 28
b) Generalized Class Groups — Rank Formulas......... 37
§5 Reduced Id`eles — Topological Aspects .................... 45
a) The Fundamental Exact Sequence .................. 45
b) Topological Lemmas .............................. 49
c) Characters of Profinite Groups..................... 53
§6 Kummer Extensions .................................... 54
a) Algebraic Kummer Theory ........................ 54
b) Arithmetic Aspects of Kummer Theory ............. 59
II. Reciprocity Maps — Existence Theorems................. 65
§1 The Local Reciprocity Map — Local Class Field Theory..... 65
a) Decomposition of Places: Local and Global Cases..... 66
b) Local Class Field Theory Correspondence ........... 74
c) Local Conductors and Norm Groups ................ 80
d) Infinite Local Class Field Theory ................... 86
§2 Id`ele Groups in an Extension L/K........................ 91
a) Canonical Injection of C in C ................... 91
K L
b) Relations Between Local and Global Norms.......... 92
c) Galois Structure of J : Semi-Local Theory .......... 94
L
d) Local Norm Groups — The Non-Galois Case......... 98
§3 Global Class Field Theory: Idelic Version.................. 104
a) Global Reciprocity Map — The Product Formula —
Global Class Field Theory Correspondence .......... 104
