A Field Guide to Algebra

199 Pages·2007·1.623 MB·English

by  Antoine Chambert-Loir

Undergraduate Texts in Mathematics
Editors
S. Axler
F.W. Gehring
K.A. Ribet
Antoine Chambert-Loir
A Field Guide
to Algebra
With 12 Illustrations
AntoineChambert-Loir
Universite´ deRennes1
IRMAR,CampusdeBeaulieu
35042RennesCedex
France
EditorialBoard
S.Axler F.W.Gehring K.A.Ribet
MathematicsDepartment MathematicsDepartment Departmentof
SanFranciscoState EastHall Mathematics
University UniversityofMichigan UniversityofCalifornia
SanFrancisco,CA94132 AnnArbor,MI48109 atBerkeley
USA USA Berkeley,CA94720-3840
USA
Pictured on the cover: Constructing the square root of a real number (Pythagoras’s
theorem).Seepage2fordiscussion.
MathematicsSubjectClassification(2000):12-01,12Fxx,11Rxx,13Bxx
LibraryofCongressCataloging-in-PublicationData
Chambert-Loir,Antoine.
Afieldguidetoalgebra/AntoineChambert-Loir.
p.cm.—(Undergraduatetextsinmathematics,ISSN0172-6056)
Includesbibliographicalreferencesandindex.
ISBN0-387-21428-3(hardback:alk.paper)
1. Algebraicfields. I. Title. II. Series.
QA247.C48 2004 512′.3—dc22 2004048103
ISBN0-387-21428-3 Printedonacid-freepaper.
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Then this is a kind of knowledge which legislation may fitly
prescribe;andwemustendeavourtopersuadethosewhoareto
be the principal men of our State to go and learn arithmetic,
not as amateurs, but they must carry on the study until they
see the nature of numbers with the mind only; nor again, like
merchants or retail-traders, with a view to buying or selling,
but for the sake of their military use, and of the soul herself;
and because this will be the easiest way for her to pass from
becoming to truth and being.
Plato, Republic, Book VII. Engl. transl. by B. Jowett
J’ai fait en analyse plusieurs choses nouvelles.
E´variste Galois, letter to A. Chevalier (29 mai 1832)
Contents
1 Field extensions ........................................... 1
1.1 Constructions with ruler and compass...................... 1
1.2 Fields.................................................. 3
1.3 Field extensions ......................................... 9
1.4 Some classical impossibilities.............................. 15
1.5 Symmetric functions ..................................... 19
1.6 Appendix: Transcendence of e and π ....................... 22
Exercises ................................................... 26
2 Roots...................................................... 31
2.1 Ring of remainders ...................................... 31
2.2 Splitting extensions...................................... 34
2.3 Algebraically closed fields; algebraic closure................. 35
2.4 Appendix: Structure of polynomial rings.................... 40
2.5 Appendix: Quotient rings................................. 43
2.6 Appendix: Puiseux’s theorem ............................. 45
Exercises ................................................... 50
3 Galois theory .............................................. 55
3.1 Homomorphisms of an extension in an algebraic closure ...... 55
3.2 Automorphism group of an extension ...................... 59
3.3 The Galois group as a permutation group................... 64
3.4 Discriminant; resolvent polynomials........................ 68
3.5 Finite fields............................................. 73
Exercises ................................................... 75
4 A bit of group theory...................................... 83
4.1 Groups (quick review of basic definitions)................... 83
VIII Contents
4.2 Subgroups.............................................. 84
4.3 Group actions........................................... 86
4.4 Normal subgroups; quotient groups ........................ 87
4.5 Solvable groups; nilpotent groups.......................... 90
4.6 Symmetric and alternating groups ......................... 92
4.7 Matrix groups........................................... 96
Exercises ...................................................100
5 Applications ...............................................107
5.1 Constructibility with ruler and compass ....................107
5.2 Cyclotomy..............................................108
5.3 Composite extensions ....................................113
5.4 Cyclic extensions ........................................116
5.5 Equations with degrees up to 4............................118
5.6 Solving equations by radicals..............................125
5.7 How (not) to compute Galois groups.......................129
5.8 Specializing Galois groups ................................132
5.9 Hilbert’s irreducibility theorem............................139
Exercises ...................................................145
6 Algebraic theory of differential equations ..................151
6.1 Differential fields ........................................151
6.2 Differential extensions; construction of derivations ...........154
6.3 Differential equations ....................................158
6.4 Picard-Vessiot extensions.................................160
6.5 The differential Galois group; examples.....................164
6.6 The differential Galois correspondence .....................169
6.7 Integration in finite terms, elementary extensions ............170
6.8 Appendix: Hilbert’s Nullstellensatz ........................177
Exercises ...................................................179
Examination problems.........................................181
References.....................................................189
Index..........................................................191
Preface
This is a small book on algebra where the stress is laid on the structure of
fields, hence its title.
Youwillhearaboutequations,bothpolynomialanddifferential,andabout
the algebraic structure of their solutions. For example, it has been known for
centuries how to explicitely solve polynomial equations of degree 2 (Babylo-
nians, many centuries ago), 3 (Scipione del Ferro, Tartaglia, Cardan, around
1500 a.d.), and even 4 (Cardan, Ferrari, xvith century), using only algebraic
operations and radicals (nth roots). However, the case of degree 5 remained
unsolveduntilAbelshowedin1826thatageneralequationofdegree5cannot
be solved that way.
Soon after that, Galois defined the group of a polynomial equation as
the group of permutations of its roots (say, complex roots) that preserve all
algebraicidentitieswithrationalcoefficientssatisfiedbytheseroots.Examples
ofsuchidentitiesaregivenbytheelementarysymmetricpolynomials,foritis
well known that the coefficients of a polynomial are (up to sign) elementary
symmetric polynomials in the roots. In general, all relations are obtained by
combining these, but sometimes there are new ones and the group of the
equation is smaller than the whole permutation group.
Galois understood how this symmetry group can be used to characterize
the solvability of the equation. He defined the notion of solvable group and
showed that if the group of the equation is solvable, then one can express its
roots with radicals, and conversely.
Tellingthisstorywillleadusalonginterestingpaths.Youwill,forexample,
learn why certain problems of construction by ruler and compass which were
posed by the ancient Greeks and remained unsolved for centuries have no
solution. On the other hand, you will know why (and maybe discover how)
one can construct certain regular polygons.
X Preface
There is an analogous theory for linear differential equations, and we will
introduce a similar group of matrices. You will also learn why the explicit
(cid:1)
computation of certain indefinite integrals, such as exp(x2), is hopeless.
On the menu are also some theorems from analysis: the transcendance
of the number π, the fact that the complex numbers form an algebraically
closed field, and also Puiseux’s theorem that shows how one can parametrize
the roots of polynomial equations, the coefficients of which are allowed to
vary.
Therearesomeexercicesattheendofeachchapter.Pleasetakesometime
to look at them. There is no better way to feel at ease with the topics in this
book. Don’t worry, some of them are even easy!
I downloaded the portraits of mathematicians from theMacTutor History
of Mathematics site, http://www-groups.dcs.st-andrews.ac.uk/˜history/. I
encouragethoseofyouwhoareinterestedinHistoryofMathematicstobrowse
this archive. Reading the books in the bibliography, like the small [4], is also
highly recommended. I found the the scans of mathematical stamps at the
address http://jeff560.tripod.com/ — those interested in that subject will be
delighted to browse the book [13].
I taught most of this book at E´cole polytechnique (Palaiseau, France).
I would like to take the opportunity here to acknowledge all the advice and
comments I received from my colleagues, namely, Jean-Michel Bony, Jean
Lannes, David Renard and Claude Viterbo. I would also like to thank Sarah
Carr for her help in polishing the English translation.
1
Field extensions
We begin with the geometric problem of constructions with ruler and compass.
We then introduce the notions of fields, of field extensions, and of algebraic
extensions. This will quickly give us the key to the impossibility of some clas-
sical problems. In Chapter 5 we will be able to see how Galois theory gives a
definitive criterion allowing us to decide if a geometric construction is, or is
not, feasible with ruler and compass.
1.1 Constructions with ruler and compass
For the Ancient Greeks, the concepts of numbers and of lengths were inti-
mately linked. The problem of geometric constructions of remarkable num-
bers was then naturally posed. Generally, they were allowed to use only ruler
andcompass,buttheysometimesdevisedingeniousmechanicaltoolstodraw
more general curves than lines and circles (cf. [4] and the notes of [9]).
Let us give this problem a formal mathematical definition.
Definition 1.1.1. Let Σ be a set of points in the plane R2. One says that
a point P is constructible with ruler and compass from Σ if there is an
integer n and a sequence of points (P ,...,P ) with P = P and such that
1 n n
for any i∈{1,...,n}, denoting Σi =Σ∪{P1,...,Pi−1}, one of the following
holds:
– there are four points A, B, A(cid:1) and B(cid:1) ∈Σ such that P is the intersec-
i i
tion point of the two nonparallel lines (AB) and (A(cid:1)B(cid:1));
– there are four points A, B, C, and D ∈ Σ such that P is one of the
i i
(at most) two intersection points of the line (AB) and the circle with center
C and radius CD;