Table Of ContentSpecial Notation
Set Theory and Number Theory
(cid:0) naturalnumbers........................................1
(cid:1) integers................................................1
n binomialcoefficient....................................18
r
x greatestintegerinx....................................25
b(cid:0) c(cid:1)
8 .x/ dthcyclotomicpolynomial.............................29
d
(cid:30).n/ Euler(cid:30)-function ...................................... 30
(cid:2) rationalnumbers...................................... 34
(cid:3) realnumbers..........................................34
(cid:4) complexnumbers......................................35
a b a isadivisorofb......................................37
j
.a;b/ gcdofa andb.........................................37
a;b lcmofa andb ........................................ 55
T U
a b modm a congruenttobmodm................................57
(cid:17)
X Y X issubsetofY .......................................81
(cid:18)
X Y X ispropersubsetofY.................................81
(cid:5)
(cid:6) emptyset.............................................81
X Y cartesianproduct......................................84
(cid:2)
1 identityfunctiononset X ..............................84
X
X numberofelementsinfiniteset X.......................84
j j
im f imageoffunction f ................................... 85
f a b f.a/ b.............................................86
V 7! D
a b a isequivalenttob .................................... 96
(cid:17)
a equivalenceclassofa .................................97
T U
a congruenceclassofa.................................168
T U
(cid:7) m integersmodulom....................................168
i
x ;::;x ;::;x x ;:::;x withx deleted.............................330
1 i n 1 n i
(cid:14) Kroneckerdelta......................................370
ij
b
Group Theory
S symmetricgrouponset X.............................103
X
S symmetricgrouponn letters.......................... 103
n
sgn.(cid:11)/ signumofpermutation(cid:11)..............................117
GL.n;k/ generallineargroup..................................128
Isom.(cid:3) 2/ groupofisometriesoftheplane........................136
O2.(cid:3) / orthogonalgroupoftheplane..........................136
D dihedralgroupoforder2n.............................141
2n
6.2;R/ stochasticgroup......................................144
V four-group...........................................145
H G H issubgroupofG...................................145
(cid:20)
H < G H ispropersubgroupofG............................145
A alternatinggrouponn letters..........................147
n
aH coset................................................151
G H indexof H inG......................................153
T V U
SL.n;k/ speciallineargroup...................................154
G H isomorphic.......................................... 156
(cid:24)
D
ker f kernelof f ..........................................160
H G H isnormalsubgroupofG............................161
(cid:0)
Z.G/ centerofgroupG ....................................163
Q quaterniongroupoforder8 ........................... 164
G=H quotientgroup.......................................176
H K directproduct........................................183
(cid:2)
G stabilizerofx ........................................194
x
(cid:1) .x/ orbitofx ............................................194
C .a/ centralizerofa G ..................................195
G
2
GL.V/ allautomorphismsofvectorspaceV ...................381
H K directsum...........................................473
n(cid:8)S sumofsubgroups .................................... 477
i 1 i
nD S directsum...........................................477
i 1 i
PND.H/ normalizerof H G.................................489
G
L (cid:20)
UT.n;k/ unitriangulargroup...................................493
Commutative Rings and Linear Algebra
I or I identitymatrix.......................................128
n
(cid:1) i Gaussianintegers.....................................217
T U
(cid:2) .(cid:3) / ringoffunctionson(cid:3) ................................ 222
(cid:0) .X/ Booleanring.........................................226
U.R/ groupofunitsinring R...............................226
(cid:2) .R/ ringoffunctionsonring R............................227
(cid:1) p; (cid:1) q finitefieldhaving p,orq,elements .................... 228
Frac.R/ fractionfieldofdomain R.............................231
R nonzeroelementsinring R............................232
(cid:2)
deg.f/ degreeofpolynomial f.x/............................233
k x polynomialringoverk................................237
T U
k.x/ fieldofrationalfunctionsoverk .......................238
k x powerseriesringoverk...............................240
TT UU
R S isomorphic.......................................... 241
(cid:24)
D
.a ;:::;a / idealgeneratedbya ;:::;a ..........................246
1 n 1 n
.a/ principalideal........................................246
R S directproduct........................................249
(cid:2)
a I coset................................................290
C
R=I quotientring.........................................291
k.z/ adjoiningz tofieldk..................................297
A B Hadamardproduct....................................306
(cid:14)
A B Kroneckerproduct....................................309
(cid:10)
Mat .k/ alln n matricesoverk..............................323
n
AT transp(cid:2)ose............................................325
Row.A/ rowspaceofmatrix A ................................329
Col.A/ columnspaceofmatrix A.............................329
dim.V/ dimensionofvectorspace.............................336
E=k fieldextension.......................................341
E k degreeoffieldextension E=k..........................341
T V U
Hom .V;W/ alllineartransformationsV W ..................... 367
k
!
T matrixoftransformationT relativetobases X,Y ........370
Y X
T U
det.A/ determinant..........................................385
tr.A/ trace................................................392
Supp.w/ supportofw kn ....................................408
2
Gal.E=k/ Galoisgroupof E=k..................................452
Var.I/ algebraicsetofideal I ................................540
Id.V/ idealofalgebraicsetV ............................... 544
pI radicalofideal I .....................................545
DEG.f/ multidegreeofpolynomial f.x1;:::;xn/...............559
A FIRST COURSE
IN ABSTRACT ALGEBRA
Third Edition
JOSEPH J. ROTMAN
University of Illinois
at Urbana-Champaign
PRENTICEHALL,UpperSaddleRiver,NewJersey07458
To mytwo wonderfulkids,
DannyandElla,
whomIloveverymuch
Contents
SpecialNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Preface tothe ThirdEdition . . . . . . . . . . . . . . . . . . . . . viii
Chapter1 NumberTheory . . . . . . . . . . . . . . . . . . . . . 1
Section1.1 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Section1.2 BinomialCoefficients . . . . . . . . . . . . . . . . . . . 16
Section1.3 GreatestCommonDivisors . . . . . . . . . . . . . . . . 34
Section1.4 TheFundamentalTheoremofArithmetic . . . . . . . . . 53
Section1.5 Congruences . . . . . . . . . . . . . . . . . . . . . . . . 57
Section1.6 DatesandDays . . . . . . . . . . . . . . . . . . . . . . 72
Chapter2 GroupsI . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Section2.1 SomeSetTheory . . . . . . . . . . . . . . . . . . . . . . 80
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
EquivalenceRelations . . . . . . . . . . . . . . . . . . . . . . . . 96
Section2.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . 103
Section2.3 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Section2.4 SubgroupsandLagrange’sTheorem . . . . . . . . . . . 144
Section2.5 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . 155
Section2.6 QuotientGroups . . . . . . . . . . . . . . . . . . . . . . 168
Section2.7 GroupActions . . . . . . . . . . . . . . . . . . . . . . . 189
Section2.8 CountingwithGroups . . . . . . . . . . . . . . . . . . . 205
Chapter3 CommutativeRings I . . . . . . . . . . . . . . . . . 214
v
vi CONTENTS
Section3.1 FirstProperties . . . . . . . . . . . . . . . . . . . . . . . 214
Section3.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Section3.3 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 232
Section3.4 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . 240
Section3.5 GreatestCommonDivisors . . . . . . . . . . . . . . . . 250
EuclideanRings . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Section3.6 UniqueFactorization. . . . . . . . . . . . . . . . . . . . 272
Section3.7 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . 278
Section3.8 QuotientRingsandFiniteFields . . . . . . . . . . . . . 288
Section3.9 Officers,Magic,Fertilizer,andHorizons . . . . . . . . . 305
Officers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Magic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Fertilizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Chapter 4 LinearAlgebra . . . . . . . . . . . . . . . . . . . . . 321
Section4.1 VectorSpaces . . . . . . . . . . . . . . . . . . . . . . . 321
GaussianElimination . . . . . . . . . . . . . . . . . . . . . . . . 345
Section4.2 EuclideanConstructions . . . . . . . . . . . . . . . . . . 354
Section4.3 LinearTransformations . . . . . . . . . . . . . . . . . . 367
Section4.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . 385
Section4.5 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
BlockCodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
LinearCodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
Chapter 5 Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Section5.1 ClassicalFormulas . . . . . . . . . . . . . . . . . . . . . 430
Vie`te’sCubicFormula . . . . . . . . . . . . . . . . . . . . . . . 443
Section5.2 InsolvabilityoftheGeneralQuintic . . . . . . . . . . . . 447
FormulasandSolvabilitybyRadicals . . . . . . . . . . . . . . . 458
TranslationintoGroupTheory . . . . . . . . . . . . . . . . . . . 460
Section5.3 Epilog . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
Chapter 6 GroupsII . . . . . . . . . . . . . . . . . . . . . . . . . 473
Section6.1 FiniteAbelianGroups . . . . . . . . . . . . . . . . . . . 473
Section6.2 TheSylowTheorems . . . . . . . . . . . . . . . . . . . 487
Section6.3 OrnamentalSymmetry . . . . . . . . . . . . . . . . . . . 498
Chapter 7 CommutativeRingsII . . . . . . . . . . . . . . . . . 516
Section7.1 PrimeIdealsandMaximalIdeals . . . . . . . . . . . . . 516
Section7.2 UniqueFactorization. . . . . . . . . . . . . . . . . . . . 522
Section7.3 NoetherianRings . . . . . . . . . . . . . . . . . . . . . 532
CONTENTS vii
Section7.4 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 538
Section7.5 Gro¨bnerBases . . . . . . . . . . . . . . . . . . . . . . . 556
MonomialOrders . . . . . . . . . . . . . . . . . . . . . . . . . . 557
GeneralizedDivisionAlgorithm . . . . . . . . . . . . . . . . . . 564
Gro¨bnerBases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
AppendixA Inequalities . . . . . . . . . . . . . . . . . . . . . . . 581
AppendixB Pseudocodes . . . . . . . . . . . . . . . . . . . . . . 583
HintsforSelected Exercises . . . . . . . . . . . . . . . . . . . . . 587
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
Preface to the Third Edition
A First Course in Abstract Algebra introduces groups and commutative rings.
GrouptheorywasinventedbyE.Galoisintheearly1800s,whenheusedgroups
tocompletelydeterminewhentherootsofpolynomialscanbefoundbyformulas
generalizingthequadraticformula. Nowadays,groupsaretheprecisewaytodis-
cussvarioustypesofsymmetry,bothingeometryandelsewhere. Besidesintro-
ducingGalois’ideas,wealsoapplygroupstosomeintricatecountingproblems
aswellastotheclassificationoffriezesintheplane. Commutativeringsprovide
the proper contextin which to study number theory as well as manyaspects of
thetheoryofpolynomials. Forexample,generalizationsofideassuchasgreatest
commondivisorandmodulararithmeticextendeffortlesslytopolynomialrings
overfields. Applicationsincludepublicaccesscodes,finitefields,magicsquares,
Latinsquares,andcalendars. Wethenconsidervectorspaceswithscalarsinar-
bitrary fields (not just the reals), and this study allows us to solve the classical
Greek problems concerning angle trisection, doubling the cube, squaring the
circle, and construction of regular n-gons. Linear algebra over finite fields is
appliedtocodes,showinghowonecanaccuratelydecodemessagessentovera
noisy channel (for example, photographs sent to Earth from Mars or from Sat-
urn). Here, one sees finite fields being used in an essential way. In Chapter 5,
we give the classical formulas for the roots of cubic and quartic polynomials,
after which both groups and commutative rings together are used to prove Ga-
lois’ theorem (polynomials whose roots are obtainable by such formulas have
solvableGaloisgroups)andAbel’s theorem(thereis nogeneralizationof these
formulastopolynomialsofhigherdegree).ThisisonlyanintroductiontoGalois
theory; readers wishing to learn more of this beautiful subject will have to see
a more advancedtext. For those readers whose appetiteshavebeen whettedby
theseresults,thelasttwochaptersinvestigategroupsandringsfurther: weprove
the basis theorem for finite abeliangroups and the Sylowtheorems, and we in-
troduce the study of polynomials in several variables: varieties; Hilbert’s basis
viii
PREFACETOTHETHIRDEDITION ix
theorem, the Nullstellensatz, and algorithmic methods associated with Gro¨bner
bases.
Letmementionsomenewfeaturesofthisedition. Ihaverewrittenthetext,
addingmoreexercises,andtryingtomaketheexpositionmoresmooth. Thefol-
lowingchangesinformatshouldmakethebookmoreconvenienttouse. Every
exerciseexplicitlycitedelsewhereinthetextismarkedbyanasterisk;moreover,
everycitationgivesthepagenumberonwhichthecitedexerciseappears. Hints
for certain exercisesare in a section at the endof the book so that readers may
consider problems on their own before reading hints. One numbering system
enumerates all lemmas, theorems, propositions, corollaries, and examples, so
thatfindingbackreferencesiseasy. ThereareseveralpagesofSpecialNotation,
givingpagenumberswherenotationisintroduced.
Today, abstractalgebra is viewedas a challengingcourse; manybright stu-
dents seem to have inordinate difficulty learning it. Certainly, they must learn
to think in a new way. Axiomatic reasoning may be new to some; others may
bemorevisuallyoriented. Somestudentshaveneverwrittenproofs;othersmay
haveoncedoneso,buttheirskillshaveatrophiedfromlackofuse. Butnoneof
theseobstaclesadequatelyexplainstheobserveddifficulties. Afterall,thesame
obstacles exist in beginning real analysis courses, but most students in these
coursesdolearnthematerial,perhapsaftersomeearlystruggling. However,the
difficulty of standard algebra courses persists, whether groups are taught first,
whetherringsaretaughtfirst,orwhethertextsarechanged. Ibelievethatama-
jor contributing factor to the difficulty in learning abstract algebra is that both
groupsandringsareintroducedinthefirstcourse;assoonasastudentbeginsto
be comfortable withone topic, it is droppedto study the other. Furthermore, if
one leaves group theory or commutativering theory before significant applica-
tionscanbegiven,thenstudentsareleftwiththefalseimpressionthatthetheory
iseitherofnorealvalueor,morelikely,thatitcannotbeappreciateduntilsome
future indefinite time. (Imagine a beginning analysis course in which both real
and complex analysis are introduced in one semester.) If algebra is taught as
a one-year (two-semester) course, there is no longer any reason to crowd both
topics into the first course, and a truer, more attractive, picture of algebra is
presented. Thisoptionismorepracticaltodaythaninthepast,forthemanyap-
plicationsofabstractalgebrahaveincreasedthenumbersofinterestedstudents,
manyofwhomareworkinginotherdisciplines.
Ihaverewrittenthistextfortwoaudiences. Thisneweditioncanserveasa
textfor thosewhowishto continueteachingthecurrentlypopulararrangement
ofintroducingbothgroupsandringsin thefirst semester. As usual,onebegins
bycoveringmostofChapter1,afterwhichonechoosesselectedpartsofChap-
ters2and3,dependingonwhethergroupsorcommutativeringsaretaughtfirst.
Chapters2and3havebeenrewritten,andtheyarenowessentiallyindependent
