Algebra I: Basic Notions of Algebra

260 Pages·1990·14.9 MB·English

by  A. I. Kostrikin

AL Kostrikin IR. Shafarevich (Eds.)

Algebra I

Basic Notions of Algebra

With 45 Figures

Springer-Verlag
Borlin Ucidelberg New York
London Paris Tokyo
Hong Kong
Encyclopaedia of
Mathematical Sciences

Volume 11

Editor-in-Chief: R.V. Gamkrelidze
Basic Notions of Algebra

LR. Shalarevich

‘Translated ftom the Russian
by M. Reid

Contents

Preface

SL

53

What is Algebra? rere
The ides ef coownatiwicn’ Fess dinars ol quuateor chemical
scordinatca.oa Ete malls ockenoeeaioms ame palm

Fields

eid axio'e, Boosey, Feld of rational funciona indepen vate
Fnetion fi ef-s plane orb cure, Bild of Linon aves ad font Lautert
Conumutarive Rings see
mg: wos zero very and iecgral dep. Fe uations Polynomial
Ring ef putyucaalfuriens on olive sha eveve Riug of pow: serie and
Feens: pone serits,Hevieca ngs Dine! sums ol roge. Ring uf artis ations
autorun; nn lac ocnion demain, xetnesf UPD
Horomerphisms and Ideals

ivnenaoephins ea, quotient rings he hove mains theorem” Ths esc
‘in enone in Tey of fetes, Principal al drs elaine wh
UD. Prov oF cole Charaerst ofa Geld. tenon -n which sven pak
nomi bus 8 soot. Alpe-aicly deed Leste. Finite els, Represeroy elements
a yrnra tng ae feeticit an smash and pre idle. Tages af fhstions
Uiteaprdacs ge renstamce aaulyi, Caaatig erent peat
Mudules vecestee

Dirce sums ant “er mates Testor prods. Tous sytin tic ard ster ve powers
Sfemodulethe dual zodule Fysivalen! elem bemomrbism of oodile, Medes
of ioral ors am eet delds, ahe of esti sae al ees,
Algebraic Aspects of Dimension :
Rasikofa module Mecuies utile te: Meise of frie 2ype 6. a pipe
loon. Noctlasim ecules ngs Nostheian rig aa sr gel ons TAS
uot of graded sage. Transcendence caper of eet Pte eer ins

u

8

4
p

§8

945

The Algebraic View of Infinitesimull Nations.

Fubsooe module second ender wnlesimals uo the (ageat space of a muni
Singur poins, Vector fee amd list ardor aercnla) operators Higher lor
nif, Jes and ier: opecatare Complecas oF tags, aks ena,

[Nunn ele Yl uans the Goku ratonal aubers atl rational feces
The adic mento ld in rumber theory

‘Noneommutative Rings .
Dues daGritios, 4 yehris er ange ing wPeudmonphisns of ¢ module Gzoup

slyphen Qusternions rat dive er alybram "ester vat. Endemoepiems of
‘idhensonn aston spas over levi alga, Tensor alsa ane th: Goo-
“Someta ssn ng, Eatlie agsbra sapsrlge ras: CLiford be, Siar
Frm gas all a igh hah the wtdsiephie ng of a wetor space
tora darn agen

Modules over Noncennmutative Rings

IModet and eseeneataions, Reposesatons of elgwbnrs Ie ax form. Simple
‘ules cor-i ete the Jan Feler nearer, Lzugthe«cag2> cH
Endomarplists ola audute Sears nt

Somisimpls Modules ral Rings
Demin ity. A ycoup algebra sersiaple Modules overs sept ig, Ser
Simple ings fine cut: Woddorhora yshoorea, Spl rings of ie length art
the lara hsosen of pisces geomet. Favor wed cont san geomet
eisai ages of ie rank over am aiebrasely closed eld Aopen ms
"eprasntations ft groupe.

Division Algebras of Finite Rank
Diino algsbree of Fits 13k over & 02 ove cts feds Isens theorem aac
‘us-lgebralcally closed ld, Ceuta vsiou lgsbras Frye aver te pare
sl rational Belle

Tho Notion of u Group

Freasformaisa groups symmetry aulo-nompheons, Semmetsoe of dys amc ye
tanned coercion as. Syomneties of Physica ws. Soups, Le =rgulaeaeion,
Sungroups, normal sucarwaps qucten:areups Orde ol aa elemont The eal cas
sroup. Gruap ofextensams ofe edie. Braver group. Die! prof uel of to soups
Examples of Groups: Finite Groups

Symomeri ard ukermating groups Synrty op legals ar nepal
bho Seey grap fae. strep ce Pr ee

Examples of Groups: Ifivite Discrete Groups

Diserove team gros Crlallqunpic proupe: Duce gious of metion
ef the Lahnenerty pline ‘he seular group. Free gecups. Spey geoup by
[eraratore vod eelatone Loyal problems elonearental rn, Ceres Ka
Braid eoup

Taumples of Groups: Lie Groups and Algebraic Groups.

Lis group. Tories, Theis oe Lie’ there

‘A. Compact Lie Groups

‘Tho susial compas granza art seca o he Yello ben he

B, Complex Analytic Lie Groups

“Those! corgi Lio groups Soma thor Lie groupe. Ihe Lovet? erp

©. Algebnic Groups

Alosbrite props thee grees Fa gi worl

2”

a

”

96

los

Le

rr
Ms
ut
180
98,

19.

Consats

Goneral Results of Group Theory
ines. proluets. The Weddertun-Reaake-Simid theorem, Gmpsitinn sti ths
lordan-Heldarshepram. Slope grou, sohable groupe Simple imps! Le s70¥pS
Sinplecamnples Tie wre Sunple ie groups, elasifcalinn

Group Representations

A. Representations of Finite Groups

Repuesentions. Orhaganality relatinrs,

RL Representations of Compact Lis Grup

Represntatons af compre! grips Integevag over a 2roup, Helio her:
Characters comps: Abe tan yougs and | ence eis: Wey We ven
<uzansional Riemantiaa goounely. Reprevelalins of SUIZiand SOM Faeroe eee
©. Representations of the Clussicul Complex Lie Groups
Represeutarions of cucu Tie genpps Cuz ieeduetiity of ipa ons
oF ieuliseniogalcassal complex Lie RFD

Some Applications of Grays

A. Galois Theory

‘Galois hooey Soleing equativrs iy vada

B. Tho Galois ‘Theory of Linear Dicerential Equations (Picard
Vessiot Theory}

C. Classification of Unramified Covers

Chasieaven uf uscumiod evar and thelundarcen wp

D. invariant Theory.

"he fs, unluscete: theater: of insariae sory

F. Group Representations sind the Clussificulion of Blementary
Particles

Lie Algebras and Nenassoeiative Algebra

A. Lie Algebras

enon breckets as an eeanple of Ties
B, Lie Theory

i alae ua Li gr,

©. Applications of Lie Aleebras
Lic grees al iit bal ena,

1. Other Nonassociative Algebras
The Ciyley mumbets, Awol comple s:rusose Gn Felson ylonsmlah ot
‘apuce. Nomociatve sl ison starts

Categories

ingens nl eitegarics Universal maping sake actor
tepolog: oep spaces, skepensiva, Group a
Homoiogical Algebra i.
A, Topotogical Oxigins of the Notions of Llomological Algebra
Cariploeee age their Neate. Hemluas and eukomologs of polysodeons, basa
pout ths sm, Dilfental farms £38 do Ram codorcotogy; de Rham's there?
Teme anual echamnlogy sieve,

B. Cohomology of Modules and Groups

Cobomologr af medals. Geup oohomvngy Lopologial messing of tbe sake
mology # diac: prone

C. Sheol Cohornotogy

Shewves; beat cobomologe. Failums Hhecreme, Rita Rock thearem,

ra. Lie nays and Li lgehs.

“ater ecg ia
In eategcriss. Norevlony grap

14
4 Preface

§22. K-theory 230

A. Topological K-theory BO
leaned Pena 4. Perasealy adhere one A RGU nl

2o inikedancasiogal te group. The sym im elie serra

Thoiadertscrers

B, Alucbraie K-Iheory 2M

Ihe gran ees of clave module. Ky, Ky ac Kang. Ko fl am

lstlations wb Ge Braver poun, Kothew's sad ashton

Comments on the Literature . 239
References 2d
Index of Names 29
Sunyeet index. : : 351

Preface

This hook aims lo present a acneral survey of algebra, ofits basic notions and
hes, Now what language should we choose for this! la rapky to the
question ‘What docs mathomalies study”, 1 is hardly acceptable to answer
Sinus! or sels with specified rekaaone’ for among the myriad conceivable
surucures or sets with specified relations, only a very sul discrete subset is of
seal interest (a mathernaticiqns. and the whole poi uf the question ix ta
ndersland the special value of this infinicesimal fraction dotted among the
amorphous masses. In the same way. [he meaning af « mathematical motion is
hry no means confined ca its formal definition; in Lact, iC may he rather better
expressed by a (generally fairly sonal) sample of the basie examples, which serve
the mathematician as the motivation and the substuolive daGrition, and al the
ste fir us the rent weaning of the wotion,

Perhaps the same kind of dificulty arises if we attempt fo characterise in tert
‘of peneral properties any phenomenon which has any degree of individuality
For example, it doesn't make sense ta give a definition of Lhe Germans or the
Frente; one ean only describe theie history or their way of life, Ln the same
it's not possible to give » definilion afin individ! human beings one can oly
either sive his‘passport data or attempt to describe his appearance and charac
ter, and relate a mmaber of typical events from his biography. This is the pan
We ulLempl lo Fallow in this book, applicd to algebra, Thus the bok secom-
modates the axiomatic and logical development of the subjeet together with more
dscriptive materia: « careful treuument of the key exumples and of points of
onliel hetvioen algebra and alher branches of mathematics and the natural
seianecs. The choice of material here is of course strongly influenced by the
anther’s personal opinions unt sles
Prsfioe s

As readers, I hae in mind students of mathemati in the first yeurt of an
underyrachate course, or theoretical physicists oy muthematicians from outside
algebra wanting %¢ yet an impression of the spirit of algebra amd its place in
pathermalies, Those purls of the book devoted to Ihe syslemutic treatment af
notions and results of algebra auke very limited demands on The reader: se
presuppose only that the roder knows calculus, anafytic geometry and linear
algebra in the form taught in many high schgels and colleges, The extent of te
prerequisites required in our treatment of examples is harder co state; an a
‘quaintance with projective space, wopological spaces, differentiable and complex.
analytic manifolds and the basic theory of functions of a complex variable is
desirable, hut the reader should hear in mind that dilfeultes arising in the
treatment of some specilic example are likely to he puroly Jocal in nature, aud
snot (o alloc the understanding of the rest of the book.

This book makes no pretence to teach algebra: itis merely an attempt to talk
‘about it. Thave attempted 10 compensale uf least te sume extent for this by giving
1 detailed bibliography: iu the comments preceding this, the reader can find
references lo hooks from which he ean study Ihe questions rvised in this book,
‘and aise some other areas of algcbra which lack of space has not allowed us to
eat

'S preliminary ve:sion of this boul has been read by FA, Bogomtoloy, RY.
Gumkaelide, SP. Déushkin, AT. Kosirikin, YuT, Manin, VY, Nikulin, AN,
Parshit, ME. Polysanov, V.L. Popov, AB. Roiter and A.N. Tyurin: Tam
‘grateful to them for their comments und supgestigns which, have heen incor
porated in the book

Tam extremely grateful to %T Shafsrevich for hier enormous help with the
manuscript and for many valuable comments

Moscow, 1984 LR, Sha revich

have taken the opportunity in the English translation co cotroet a oumber
of errors und inaccuracies which remained undeweled in the original; Lam very
arataiul to E.8, Vinberg, A.M. Valkhonskil and R. Zagie for poiriting these oul
Tam especially gruleful 1 the (eanshitor M, Reid for innumerable improvements
of the teat

Moscow, 1987 LR. Shubrevieh
* 51, Wha gtr

§1. What is Algebra?

‘What is algebra? Ista branch of mutbcmaties. u method ora frame of mind?
Such qvestions do not of course ac ether short or unambiguous answer,
‘One can altompl u description ofthe plaos accupice! by algehra in mathematics
by drumixg attention lo the process for which Hermann Weel eained the um
pronoutceable word ‘coordinatiation” es [HL Weyl 109 (1939) Chop. 1.24).
‘An indivicual might fad his way about the orld relying exctusisey on his sense
‘organs, sight, feeling. ou his experience of manipulaang obyeds in the werkt
outside aad on the intoiion resulting from Ihis, However, these Is another
Pools apptuach: by means of meusiremens, subjective impressions cim he
Irusvfnined into ohjctiva marks inte numbers. which arc then capuble of ing,
preserved indctniely, of being communicated to otherindividuals who huve not
‘experienced the sume impressions, and mos! importactly, which can be operated
fon te provile new information concerning the chjects ofthe measurement

“The uldest example isthe iden of courting (coordinatisation) and colctdauon
{operation}, which allow us tn draw cnofusions. the eumber oLobjects without
handling them ali omoe. Atempts to “measure or (‘express as a number a
saety of objoos guve rise to fractions and negative murbrs in adition the
whole numbers. The attomat co express tne diggemal of a square of side 1 4s a
Dumber led (0 @ Funous etisis of the mathematics of carly anciqoity and to the
construction of ierafionsl numbers

Measurement devermines the pointe ofa five by eal numbers, andl mieh more
widely. expreseee many physical quabtities ax avmbers, To Gallleo is ue the
ros! extreme statement im hiv sime of the ik. of vourlinatisatin: “Measure
everything thac is measuruble, und make meusurable everything satis not ycl
40, The sucess of his ie, slarting from the tive of Galileo, was brian. The
cxcation of aualslic yooraety allowed us to seprescat pein The plane hy patrs
of numbers, and poinis of spaee fy triples, and by means of operations with
numbers. led lo the ssuovery ol ever new geurnctie Facts, However, the sucerss
‘of analytic geometry is mainly based on the fet that i reduces toy ques not
‘only pons, bn alse teres, suofases and on, For example, a curvein the plane
isyiven by an equation Pts, y}= 0: in dhe ease of line, Fis lineae polynomial,
‘uid determined by is Seaelficients: the cocfficiens of x und vand the constut
term, In tha case ofa conic section sc have a Curve of degree 2, determined by
its 6 omlfceurs Eis n polynomial of degree then it seaey case thac it has
Jn-+ Di 2) cwolfcienlsy the sorresponding corve is determincd by these
<ocliions in the same way that a point is given by its coordinates

To over to expres as uuunbees the roots of sn eqitation, the complex numbers
sere futroduosd, and this fakes a step inte a completely new branch of mathe-
nates, whieh includes epic fonctions and Riemann succes

Foor a long time it might have seemed that the path indicated by Galileo
consisted af measuring “everything” an tcems of a known and undisputed vollec-
GL Wha: Alger? +

lion of rumbers, und that the problem consists just of crcating more and more
stuhele methods of measurements, such as Cartesian coordinates or new physical
instruments. Admitiedly, from time to time the numbers considered as known
for simpiy called numbers) turned out to be inadequate: this led to a‘sisis whieh
had (6 be resolved by extending the notion of numb. ureating a new form of
numbers, which theraselves soon came to he eonsidaved as the unique possibility.
Inanycasc, as a rule, at any given orent (he notion of number was considered
to bo completely clear, and the development moved only in the direction of
extenling it:

"1, 2, many” = niclural mambers-= integers
= rilionils + coals complex numbers

Rut matrixes, for example, form a completely independent world of ‘number-
like objects. which cannot he included in this chain, Simultuncously with them,
‘qualernions were discovered, and then other hypercomplex systems’ (now ealled
algebras). Infinitesimal transformations led te differential operators, for which
the nalural operation curns eut te be something completely new, the Poisson
bracket. Pinite fields lured up in algebra, and p-adic numbers in number theory:
Gradually, il bovame clear thac cle atternpl Ws lind a unified oll-embrucing
concept of number is absolutely hopeless. In this situation the principle declared
by Galleo could be accused of intolerance; for the requirement to ‘mike mei
surable crersthing which is not yet so* clutly discriminates against anything
which stubborly refuses to he measurable, excluding it from the sphere of
interest of science, and possibly even of rouson Gand thus becomes a secondary
quality or secwndie ewuse i the lerminotogy of Galilco). Hven if, more modestly.
the polemic term ‘everything’ is restricted to objects of physics and mathematics,
more and more of these tumned up hich could not be ‘measured’ in terms of
‘ordinary numbers,

The principle of cvmdinutisation can nevertheless he preserves, provide ye
admit that the set of “ouraber-like objects’ by means af which coordinatisation
is achiovad can be just as diverse as the World of physical and mathematical
objects they coordinative. The objects which scree as ‘onordinates should satisfy
onily certain enrditions ofa very yonsral character,

‘They musi be individually distinguishable. For example, schereas ull points of
fing have identical properties (the line is homogeneous}, and a point can only
be fixed hy putting a finger on it, numbers are al) individuat: 3, 7/3, 72, 2 and 80
on, (The same principle is applied when nescbom puppiss, indzstingurshable Lo
the owner, bave different coloured ribbons tied round their necks (o distinguish
them}

They should be sulficrently abstract to reflect properties common 10 a wide
circle of phenomenons,

Contain fundamental aspects of the simations under study should be reflected
iu operations thal can he carried out on the abjvets being coordinatised: addition,
multiplication, comparison of magnitudes, differentiation, forming Poisson
brackets and so on,
8 92, What is algshaa?

We can now founulate the point we are making in more deta, as follows:

‘Thesis, Amsthing which is she ubjoct af marhemutical study (eaves und surfaces,
maps, symmcerics, crystals, quantum mechamical queatities imu so an) can be
“eordinatised’ or ‘measured’. Towerer, for such a suordinatisatian the ‘ordinary
numbers ore by no means adeutte

Conversely, when we meee a ne type of abject, we are forced 10 construct (or
a discocer) new éupes of ‘quamities’ ta coordinarise them. The construction and
the study of the quantities arising in this way is what characterises the plsce of
algebra in worhematies fof course. cory approximately)

From this point of view, the development afyny ranch af algebra consists of
‘wo slages. ‘The first of these isthe birth of che new type of algebraic abjects oul
of some problem of eootdinatisation. The second is their subsequent career, that
is, the systematic development af the theory of this class of objects, this is
sometimes eloely reluted, and sometimes ilmost completely unrelated lo thc
area in coonection with which the objects arose. In what Follows ee yall try Hot
to lose sight of these ewe stages. But since algebra courses are often exclusively
conoemed with the sovord stage, we will maintain the balance by paying a litte
‘mote attention to the frst,

We conclude thig ution with (wo exumples of courdinutisution whieh are
somewhat lose standard thas those considered up to naw.

Example 2. The Dictionary of Quantuen Mechanies. In quantum mechanics,
the basic physical rotions are soordinatised’ by mathematical ebjects, as follows,

a

ula phase sytem

iene space

om

Seba ph Saljint operstor

Smrstuncousy measurable

Commuting epeators

saushies
Quancistakings posse | Opsraue: Faring wae pense

tae ia ata WE eevee

Sec alas of amis

obtainable by msatuzement | SPA ef an wpeator

easily of amaton

Irom stats Hoste y Mega

sly ain

Example 2, Finite Models for Systems of Incidence and Parallelism Axioms,
We start with a stall digression. Th the axiomatic construction af goometry. we
‘often consirer not the whole set of axioms, bul just some purt of them: &@ he