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AdvancedModern AlgebraThird Edition, Part 2
Joseph J. Rotman
GRADUATE STUDIESIN MATHEMATICS 180
American Mathematical Society
Advanced Modern AlgebraThird Edition, Part 2
Joseph J. Rotman
American Mathematical SocietyProvidence, Rhode Island
GRADUATE STUDIES IN MATHEMATICS 180
10.1090/gsm/180
EDITORIAL COMMITTEE
Dan AbramovichDaniel S. Freed (Chair)
Gigliola StaffilaniJeff A. Viaclovsky
The 2002 edition of this book was previously published by Pearson Education, Inc.
2010 Mathematics Subject Classification. Primary 12-01, 13-01, 14-01, 15-01, 16-01,18-01, 20-01.
For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-180
Library of Congress Cataloging-in-Publication Data
Rotman, Joseph J., 1934–Advanced modern algebra / Joseph J. Rotman. – Third edition.
volumes cm. – (Graduate studies in mathematics ; volume 165)Includes bibliographical references and index.ISBN 978-1-4704-1554-9 (alk. paper : pt. 1)ISBN 978-1-4704-2311-7 (alk. paper : pt. 2)1. Algebra. I. Title.
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Contents
Foreword vii
Preface to Third Edition: Part 2 ix
Chapter C-1. More Groups 1
C-1.1. Group Actions 1Graphs 16Counting 20
C-1.2. Sylow Theorems 24
C-1.3. Solvable and Nilpotent Groups 33Solvable Groups 34Nilpotent Groups 43
C-1.4. Projective Unimodular Groups 50General Linear Group GL(n, k) 50Simplicity of PSL(2, q) 52Simplicity of PSL(n, q) 58
C-1.5. More Group Actions 66Projective Geometry 67Multiple Transitivity 74PSL Redux 77
C-1.6. Free Groups and Presentations 81Existence and Uniqueness of Free Groups 82Presentations 92
C-1.7. Nielsen–Schreier Theorem 97
C-1.8. The Baer–Levi Proof 102The Categories Simp and Simp∗ 102Fundamental Group 104Covering Complexes 110Co-Galois Theory 115
iii
iv Contents
C-1.9. Free Products and the Kurosh Theorem 118
C-1.10. Epilog 124
Chapter C-2. Representation Theory 127
C-2.1. Artinian and Noetherian 127
C-2.2. Jacobson Radical 130
C-2.3. Group Actions on Modules 135
C-2.4. Semisimple Rings 137
C-2.5. Wedderburn–Artin Theorems 146
C-2.6. Introduction to Lie Algebras 161
C-2.7. Characters 168
C-2.8. Class Functions 176
C-2.9. Character Tables and Orthogonality Relations 180
C-2.10. Induced Characters 186
C-2.11. Algebraic Integers Interlude 193
C-2.12. Theorems of Burnside and of Frobenius 200
C-2.13. Division Algebras 208
Chapter C-3. Homology 223
C-3.1. Introduction 223
C-3.2. Semidirect Products 226
C-3.3. General Extensions and Cohomology 236
C-3.4. Complexes 255
C-3.5. Homology Functors 262
C-3.6. Derived Functors 271
C-3.7. Right Derived Functors 285
C-3.8. Ext and Tor 292
C-3.9. Cohomology of Groups 309
C-3.10. Crossed Products 326
C-3.11. Introduction to Spectral Sequences 333
Chapter C-4. More Categories 339
C-4.1. Additive Categories 339
C-4.2. Abelian Categories 344
C-4.3. g-Sheaves 359
C-4.4. Sheaves 368
C-4.5. Sheaf Cohomology 378
C-4.6. Module Categories 384
C-4.7. Adjoint Functor Theorem for Modules 392
Contents v
C-4.8. Algebraic K-Theory 403The Functor K0 404The Functor G0 408
Chapter C-5. Commutative Rings III 419
C-5.1. Local and Global 419Subgroups of Q 419
C-5.2. Localization 427
C-5.3. Dedekind Rings 445Integrality 446Algebraic Integers 455Characterizations of Dedekind Rings 467Finitely Generated Modules over Dedekind Rings 477
C-5.4. Homological Dimensions 486
C-5.5. Hilbert’s Theorem on Syzygies 496
C-5.6. Commutative Noetherian Rings 502
C-5.7. Regular Local Rings 510
Bibliography 527
Index 537
Foreword
My friend and UIUC mathematics department colleague Joe Rotman was com-pletely dedicated to his series of books on algebra. He was correcting his draft ofthis revision of Advanced Modern Algebra during what sadly turned out to be hisfinal hospital visit. At that time, Joe and his family asked me to do what I couldto see this close-to-finished version to publication.
Two more friends and colleague of Joe’s, Jerry Janusz and Paul Weichsel, joinedthe project. Jerry did a meticulous line-by-line reading of the manuscript, and allthree of us answered questions posed by the AMS editorial staff, based on ArleneO’Sean’s very careful reading of the manuscript.
It is clear that this book would have been even richer if Joe had been able tocontinue to work on it. For example, he planned a chapter on algebraic geome-try. We include the first paragraph of that chapter, an example of Joe’s distinctlypersonal writing style, as a small memorial to what might have been.
Mathematical folklore is the “standard” mathematics “everyone”knows. For example, all mathematics graduate students today arefamiliar with elementary set theory. But folklore changes withtime; elementary set theory was not part of nineteenth-centuryfolklore. When we write a proof, we tacitly use folklore, usuallynot mentioning it explicitly. That folklore depends on the calendarmust be one of the major factors complicating the history of math-ematics. We can find primary sources and read, say, publicationsof Ruffini at the beginning of the 1800s, but we can’t really followhis proofs unless we are familiar with his contemporary folklore.
I want to express my thanks to Sergei Gelfand and Arlene O’Sean of the AMSand to Jerry Janusz and Paul Weichsel of UIUC for all their help. Our overrid-ing and mutual goal has been to produce a book which embodies Joe Rotman’sintentions.
Bruce Reznick
May 31, 2017
vii
Preface to Third Edition:Part 2
The second half of this third edition of Advanced Modern Algebra has Part 1 as pre-requisite. This is not to say that everything there must be completely mastered, butthe reader should be familiar with what is there and should not be uncomfortableupon seeing the words category, functor, module, or Zorn.
The format of Part 2 is standard, but there are interactions between the dif-ferent chapters. For example, group extensions and factor sets are discussed in thechapter on groups as well as in the chapter on homology. I am reminded of myexperience as an aspiring graduate student. In order to qualify for an advanceddegree, we were required to take a battery of written exams, one in each of algebra,analysis, geometry, and topology. At the time, I felt that each exam was limitedto its own area, but as I was wrestling with an algebra problem, the only way Icould see to solve it was to use a compactness argument. I was uncomfortable:compactness arguments belong in the topology exam, not in the algebra exam! Ofcourse, I was naive. The boundaries of areas dividing mathematics are artificial;they really don’t describe what is being studied but how it is being studied. Itis a question of attitude and emphasis. Doesn’t every area of mathematics studypolynomials? But algebraists and analysts view them from different perspectives.After all, mathematics really is one vast subject, and all its parts and emphases arerelated.
A word about references in the text. If I mention Theorem C-1.2 or ExerciseC-1.27 on page 19, then these are names in Part 2 of the third edition. Referencesto names in Part 1 will have the prefix A- or B- and will say, for example, TheoremA-1.2 in Part 1 or Exercise B-1.27 on page 288 in Part 1. In an exercise set, anasterisk before an exercise, say, *C-1.26, means that this exercise is mentionedelsewhere in the text, usually in a proof.
Thanks goes to Ilya Kapovich, Victoria Corkery, Vincenzo Acciaro, and StephenUllom.
ix
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Index
Page numbers in italic refer to Part 1.
Abel, N. H., 7, 219
abelian category, 346
abelian group, 128
divisible, 496
free, 328
ordered, 444
primary, 362
rank 1, 420
reduced, 502
torsion, 380
torsion-free, 380
totally ordered, 444
abelian Lie algebra, 163
absolute Galois group, 480
ACC, 128, 282, 300
Accessory Irrationalities, 199
action of group, 5, 152
r-transitive, 76
transitive, 6, 187
acyclic, 262
sheaf, 381
additive category, 340
additive functor, 340, 465
additive notation, 130
Adelard of Bath, 4
Adian, S. I., 125
adjacency, 127
adjacency matrix, 17
adjacent (graph), 16
adjoining a unit, 39
adjoining to field, 79
adjoint
functors, 666
linear transformation, 431, 437
matrix, 584
Adjoint Functor Theorem, 401
adjoint functors, 392
counit, 402
left, right, 393
unit, 393
Adjoint Isomorphism, 526, 527
Ado, I. D., 163
affine group, 139
affine variety, 594
afforded by, 173
Albert, A. A., 219, 327
Alberti, L. B., 68
algebra, 284, 543
central simple, 209
crossed product, 328
cyclic, 328
division, 209, 331
enveloping, 548
finitely generated, 604
generated by n elements, 558
graded, 550
not-necessarily-associative, 161
algebra map, 543
algebraic
closure, 341
element, 79
extension, 79
integer, 193, 449, 455
conjugate, 196
number field, 449
numbers, 340
algebraically
closed, 341
dependent, 345
algorithm
Buchberger, 646
Euclidean, 17
537
538 Index
into disjoint cycles, 118almost all, 319
almost split, 302
alternatingbilinear form, 418
group, 141
multilinear, 563
space, 418alternating sum, 26
Amitsur, S. A., 209, 328, 560
annihilator, 133, 508element, 379
module, 381
antanairesis, 16anti-isomorphism, 293
Apollonius, 4
Archimedes, 4Arf invariant, 429
Arf, C., 429
Artin, E., 58, 149
Artin, M., 429artinian ring, 128, 286
ascending central series, 44
ascending chain condition, 282associated polynomial function, 593
associated prime ideal, 620
associates, 52associativity, 29, 128
generalized, 131, 553
Atiyah, M., 415atlas, 525
augmentation, 145, 338
augmentation ideal, 145, 338Auslander, M., 160, 223, 302, 490, 496, 518,
522
Auslander–Buchsbaum Theorem, 520, 522
automorphism
field, 180group, 155, 228
inner, 8
outer, 9Axiom of Choice, 313
(B,N) pair, 81
b-adic digits, 23
Baer sum, 244, 302Baer Theorem, 537
Baer, R., 97, 235, 425, 494
bar resolution, 316normalized, 318
Barr, M., 87, 401
Barratt, M. G., 271Barratt–Whitehead Theorem, 271
base b, 23
base of topology, 675
basepoint, 104, 463basic subgroup, 521
basisdependence, 349free abelian group, 328free algebra, 556free group, 82free module, 329, 481free monoid, 92ideal, 283standard, 253vector space
finite-dimensional, 252infinite-dimensional, 319
Basis Theoremfinite abelian groups, 367, 499Hilbert, 286
Bass, H., 300, 415, 498Bautista, R., 160Beltrami, E., 594
Besche, H. U., 4biadditive, 509bidegree, 334Bifet, E., 225bifunctor, 521bijection, 241bilinear form, 417
alternating, 418nondegenerate, 420skew, 418symmetric, 417
negative definite, 426positive definite, 426
bilinear function, 417, 509bimodule, 470binary operation, 29Binomial Theorem
commutative ring, 32exterior algebra, 569
birational map, 627Bkouche, R., 488block in G-set, 77Block, R. E., 166
Boole, G., 129Boolean group, 129Boolean ring, 33, 41Boone, W. W., 125boundaries, 262bouquet, 110bracelet, 24bracket, 162Brauer group, 218
relative, 220Brauer, R., 159, 190, 216, 219Brauer–Thrall conjectures, 159Bruck, R. H., 72Bruck–Ryser Theorem, 72Brunelleschi, F., 68Buchberger’s algorithm, 646
Index 539
Buchberger’s Theorem, 643Buchberger, B., 629, 640
Buchsbaum, D. A., 223, 349, 518, 522
Burnside Basis Theorem, 48Burnside ring, 199
Burnside’s Lemma, 20, 183
Burnside’s problem, 125Burnside’s Theorem, 168, 202
Burnside, W., 20, 125
C∞-function, 35
cancellation lawdomain, 34
group, 130
Cardano, G., 5Carmichael, R., 35
Carnap, R., 461
Cartan, E., 66, 161, 166Cartan, H., 380, 475, 538
Cartan–Eilenberg Theorem, 538
cartesian product, 235castle problem, 8
Casus Irreducibilis, 189
categorical statement, 355category, 443
G-category, 408
abelian, 346additive, 340
cocomplete, 385
cogenerator, 398
composition, 443enough injectives, 378
enough projectives, 378
exact, 349generator, 385
morphism, 443
noetherian, 391objects, 443
opposite, 465
pre-additive, 446small, 525
virtually small, 405
Cauchy sequence, 654Cauchy’s Theorem, 10
Cauchy, A.-L., 7
Cayley graph, 18Cayley Theorem, 2
Cayley, A., 2, 3, 140
Cayley–Hamilton Theorem, 392Cech, E., 380
center
group, 155Lie algebra, 168
matrix ring, 268, 281
ring, 277
centerless, 155central extension, 314
universal, 314central simple algebra, 209centralizer
group element, 8subgroup, 8subset of algebra, 212
chain, 314chain map, 259
over f , 275change of rings, 475character, 173, 203
afforded by, 173degree, 173generalized, 179induced, 187irreducible, 173kernel, 184linear, 173
regular, 176restriction, 191table, 180trivial, 176
character group, 532characteristic of field, 60characteristic polynomial, 390characteristic subgroup, 32chessboard, 23Chevalley, C., 65, 161, 332Ch’in Chiu-shao, 8Chinese Remainder Theorem
Z, 25k[x], 89
circle operation, 280circle group, 129circuit, 17, 107Claborn, L., 471class
function, 176group, 471number, 471sums, 155
class equation, 11class group, 540class of nilpotent group, 44Classification Theorem of Finite Simple
Groups, 66, 176, 227Clifford algebra, 572Clifford, W. K., 572coboundary, 241cocomplete, 385cocycle identity, 238codiagonal, 302, 341coefficients, 41cofactor, 584cofinal subset, 318cofinite, 41, 596cogenerator, 398
540 Index
Cohen, I. S., 317, 450coherent sheaf, 525Cohn, P. M., 474cohomological dimension, 323cohomology
sheaf, 379cohomology group, 241cohomology groups of G, 309coinduced module, 326cokernel, 297
additive category, 343Cole, F., 55colimit (see direct limit), 658colon ideal, 603coloring, 21Columbus, 4column space of matrix, 270commensurable, 13
common divisor, 10in Z, 10several polynomials, 103two polynomials, 66
commutative, 128commutative diagram, 305commutative ring, 32
Dedekind, 467domain, 34DVR, 421euclidean ring, 98factorial, 104field, 37integers in number field, 449Jacobson, 610local
regular, 514PID, 101polynomial ring, 42
several variables, 45reduced, 598UFD, 105valuation ring, 444
commutator, 35subgroup, 35
compact, 674companion matrix, 385Comparison Theorem, 273complement, 40, 325
of subgroup, 231complete factorization, 120complete graph, 17completely decomposable, 425completely reducible, 170completion, 655complex, 257
acyclic, 262component, 109connected, 105
covering, 111de Rham, 574differentiations, 257dimension, 103direct sum, 354modulus, 129pointed, 104quotient, 104, 260simplicial, 103simply connected, 108subcomplex, 260zero, 257
component, 109composite integer, 11composite of functions, 239composition factors, 195composition series, 195, 302
length, 195
composition, category, 443compositum, 209congruence mod I, 55congruence class, 244congruent mod m, 19congruent matrices, 419conjugacy class, 8, 157conjugate
algebraic integers, 196elements in field extension, 459group elements, 154intermediate fields, 207
conjugate subgroups, 8conjugation
Grassmann algebra, 567groups, 154quaternions, 276
connected, 105connecting homomorphism, 265constant
g-sheaf, 360presheaf, 369sheaf, 370
constant function, 236constant functor, 462constant polynomial, 44constant term, 44content, 109continuous, 675contracting homotopy, 265contraction of ideal, 449contragredient, 198contravariant functor, 464convolution, 274, 282coordinate field, 625coordinate list, 253coordinate ring, 597Copernicus, 4coproduct
Index 541
family of objects, 452two objects, 447
corestriction, 320
Corner, A. L. S., 425Correspondence Theorem
groups, 165
modules, 298rings, 279
coset
ideal, 55
subgroup, 144cosyzygy, 489
covariant functor, 461
covering complex, 111fiber, 111
intermediate, 116
lifting, 111projection, 111
regular, 117
sheets, 111universal, 116
covering space, 359
Cramer’s Rule, 586crossed homomorphism, 248
crossed product algebra, 328
Cubic Formula, 5cubic polynomial, 44, 188
Culler, M., 126
cyclehomology, 262
permutation, 117
cycle structure, 120cyclic
group, 141
module, 296
cyclic algebra, 328cyclotomic field, 461
cyclotomic polynomial, 93
DCC, 128, 286, 301De Morgan laws, 41
De Morgan, A., 41
de Rham complex, 168, 574de Rham, G., 168, 574
Dean, R. A., 39
decomposable, 425Dedekind ring, 467, 535
Dedekind Theorem, 204
Dedekind, R., 174, 204, 446degree
G-set, 5
character, 173euclidean ring, 98
extension field, 78
graded map, 550
homogeneous element, 550polynomial, 42
several variables, 631representation, 169several variables, 630
degree-lexicographic order, 634deleted resolution, 273derivation, 587
group, 248Lie algebra, 162principal, 248ring, 161
derivative, 46derived series
groups, 37Lie algebra, 165
Derry, D., 425Desargues, G., 68Descartes, R., 3, 7descending central series, 43
Lie algebra, 165determinant, 576diagonal, 341diagonal map, 302diagonalizable, 394, 401diagram, 305
commutative, 305diagram chasing, 308Dickson, L. E., 58, 63, 122, 222, 327dicyclic group, 96Dieudonne, J., 558differential form, 574differentiations, 257dihedral group, 136
infinite, 34, 123dilatation, 59dimension, 255, 322dimension shifting, 273Diophantus, 4, 445direct image, 374direct limit, 658direct product
commutative rings, 54
groups, 167modules, 323, 451rings, 275
direct sumadditive category, 341complexes, 354matrices, 384modules, 323, 324, 451
external, 324, 326internal, 326
vector spaces, 259, 268direct summand, 325direct system, 657
transformation, 662directed graph, 18
connected, 18
542 Index
directed set, 659Dirichlet, J. P. G. L., 368, 446, 466, 471
discrete, 678
discrete valuation, 332, 444discrete valuation ring, 421
discriminant, 223
bilinear form, 420
of OE , 467of cubic, 224
of quartic, 230
disjoint permutations, 117disjoint union, 452
distributivity, 29
dividescommutative ring, 36
in Z, 9divisible module, 496division algebra, 209
Division Algorithm
k[x], 62
k[x1, . . . , xn], 637in Z, 10
division ring, 275
characteristic p, 331divisor
in Z, 9Dlab, V., 160domain
commutative ring, 34
DVR, 421morphism, 443
of function, 236
PID, 101regular local ring, 516
UFD, 105
Double Centralizer Theorem, 212
double coset, 81double cover, 370
dual basis, 269
dual space, 260, 269duals in category, 450
DVR, 421
Dye, R. L., 429Dynkin diagrams, 167
Dynkin, E., 167
Eckmann, B., 310, 314
edge, 16trivial, 17
Eick, B., 4
eigenvalue, 388eigenvector, 388
Eilenberg, S., 310, 358, 417, 441, 475, 491,538
Eisenstein Criterion, 95
Eisenstein integers, 32Eisenstein, G., 95
elementary cancellation, 84elementary divisors
finite abelian group, 373matrix, 397
elementary matrix, 410elimination ideal, 648empty word, 83endomorphism
abelian group, 274module, 294ring, 274
Engel’s Theorem, 165Engel, F., 165enlargement of coset, 62, 165, 298enough injectives, 378enough projectives, 378enveloping algebra, 548epimorphism (epic), 345
equal subsets, 236equality of functions, 118equalizer
condition, 369equivalence
inverse, 356equivalence class, 244equivalence relation, 243equivalent
categories, 356extensions, 241, 295filtration, 302height sequences, 423matrices, 406normal series, 197representations, 171series, groups, 197
Eratosthenes, 4etymology
K-theory, 415abelian, 219abelian category, 346acyclic complex, 262
adjoint functors, 392, 666affine, 594affine space, 627alternating group, 141artinian, 149automorphism, 155canonical form, 386coherent ring, 525commutative diagram, 305cubic, 44cycle, 117dihedral group, 136domain, 34exact sequence, 575Ext, 295exterior algebra, 562
Index 543
factor set, 237field, 37flat, 529free group, 92functor, 461homology, 225homomorphism, 47ideal, 446isomorphism, 47kernel, 50Latin square, 71left exact, 469nilpotent, 165polyhedron, 136power, 130profinite, 477pure subgroup, 364quadratic, 44
quasicyclic, 503quaternions, 276quotient group, 162radical, 598rational canonical form, 386regular representation, 203ring, 29symplectic, 424syzygy, 486, 492Tor, 307torsion subgroup, 359variety, 594vector, 248
Euclid, 4Euclid’s Lemma, 69, 98, 101
integers, 12Euclidean Algorithm I
integers, 17Euclidean Algorithm II
integers, 18Euclidean Algorithm, k[x], 70euclidean ring, 98Eudoxus, 4
Euler φ-function, 142Euler Theorem, 148Euler, L., 19, 446Euler–Poincare characteristic, 110, 262evaluation homomorphism, 49even permutation, 124exact
abelian category, 347category, 349functor, 355, 469
left, 467right, 517
hexagon, 325sequence, 305
almost split, 302complexes, 260
factored, 310short, 306
splice, 310
triangle, 268Exchange Lemma, 256
exponent
group, 376module, 381
extension
central, 314
universal, 314groups, 227
modules, 295, 306
of ideal, 449extension field, 78
algebraic, 79
degree, 78finite, 78
Galois, 207, 475
inseparable, 182normal, 190
pure, 187
purely transcendental, 345radical, 187
separable, 182
simple, 214extension sheaf, 375
exterior algebra, 562
exterior derivative, 574exterior power, 562
FAC, 384, 525
factor groups, 192
factor modules, 302factor set, 237
factorial ring (see UFD), 104
faithful G-set, 6faithful module, 292
family of supports, 380
Fano plane, 69Fano, G., 69
Feit, W., 34, 219
Feit–Thompson Theorem, 219Fermat Little Theorem, 22
Fermat prime, 96
Fermat’s Theorem, 148Fermat, P., 445
Ferrari, Lodovici, 5
FFR, 500fiber, 111
Fibonacci, 4, 590
field, 37algebraic closure, 341
algebraically closed, 341
finite, 186
fraction, 38Galois, 88
544 Index
perfect, 401prime, 59rational functions, 44
15-puzzle, 124, 126filtration, 302, 333
length, 302refinement, 302
filtrationsequivalent, 302
finiteextension, 78order (module), 379topology, 479, 679
finite index topology, 675finite-dimensional, 251finitely generated
algebra, 604ideal, 283
module, 296finitely generated group, 93finitely presented group, 93finitely presented module, 488Finney, Jr., R. L., 488First Isomorphism Theorem
abelian category, 358commutative rings, 58complexes, 260groups, 163modules, 297vector spaces, 269
Fitting subgroup, 49Fitting, H., 49Five Lemma, 309fixed field, 202fixed points, 253fixed-point-free, 203fixes, 117, 180flabby sheaf, 381flat dimension, 491flat module, 529flat resolution, 491
forgetful functor, 462formal power series
one variable, 41several variables, 515
Formanek, E., 560four-group, 137fraction field, 38fractional ideal, 469, 539Fraenkel, A. A. H., 442Frattini Argument, 38Frattini subgroup, 47Frattini Theorem, 47free
abelian group, 328algebra, 556commutative algebra, 558, 671
group, 82module, 329, 481monoid, 92resolution, 255
free product, 118freeness property, 330Freudenthal, H., 310Freyd, P., 355Frobenius
automorphism, 186complement, 204group, 204kernel, 205Reciprocity, 191Theorem
Frobenius kernels, 206real division algebras, 215
Frobenius group, 46
Frobenius, F. G., 20, 24, 174, 187, 198, 203,215, 327, 374
full functor, 355full subcategory, 349fully invariant, 32function, 236
bijection, 241constant, 236identity, 236inclusion, 237injective, 238polynomial, 44rational, 45restriction, 239surjective, 238
functoradditive, 340, 465constant, 462contravariant, 464contravariant Hom, 464covariant, 461covariant Hom, 461exact, 355, 469
forgetful, 462full, 355identity, 461left exact, 467, 468representable, 351, 528right exact, 517two variables, 521
fundamental group, 87, 463Fundamental Theorem
Arithmetic, 198finite abelian groups
elementary divisors, 374invariant factors, 376
finitely generated abelian groupselementary divisors, 374invariant factors, 377
Index 545
Galois Theory, 211, 479modules
elementary divisors, 382
invariant factors, 382symmetric functions, 208
symmetric polynomials, 208, 639
G-category, 408
G-domain, 606G-ideal, 608
g-map, 365
G-set, 5block, 77
degree, 5
faithful, 6primitive, 77
g-sheaf, 360
constant, 360stalk, 360
zero, 361
Gabriel, P., 160, 166, 386Galligo, A., 487
Galois extension, 207, 475
Galois field, 88Galois group, 181, 475
absolute, 480
Galois Theorem, 86Galois, E., 8, 146
Gaschutz, W., 254
Gauss Theorem
R[x] UFD, 110cyclotomic polynomial, 96
Gauss’s Lemma, 111
Gauss, C. F., 215Gaussian elimination, 409
Gaussian equivalent, 410
Gaussian integers, 32gcd, 10
Gelfond, A., 347
Gelfond-Schneider Theorem, 347general linear group, 50, 128
general polynomial, 84
generalized associativity, 131, 553generalized character, 179
generalized matrix, 142
generalized quaternions, 82, 254generate
dependence, 349
generatorcategory, 403
cyclic group, 141
of ModR, 385generators and relations, 92, 403
algebra, 556
Gerard of Cremona, 4
germ, 364germs, 371
global dimension, left, 490global section, 362gluing, 369Godement resolution, 382Godement, R., 382Going Down Theorem, 453Going Up Theorem, 453Goldman, O., 604Goodwillie, T. G., 590Gordan, P., 285graded algebra, 550graded map, 550graph, 16
adjacency matrix, 17adjacent, 16automorphism, 16Cayley, 18complete, 17
connected, 16directed, 18edge, 16isomorphism, 16labeled, 18vertex, 16
Grassmann algebra, 566Grassmann, H. G., 68, 566greatest common divisor
domain, 97in Z, 10several polynomials, 103two polynomials, 66
Green, J. A., 314Griess, R., 168Griffith, P. A., 518Grobner, W., 640Grobner basis, 640Gromov, M., 126Grothendieck group, 404, 406, 408
Jordan–Holder, 412reduced, 407
Grothendieck, A., 336, 402, 412, 415, 441,
592group
p-group, 11abelian, 128additive notation, 130affine, 139algebra, 274alternating, 141axioms, 128, 138Boolean, 129circle group, 129conjugacy class, 157cyclic, 141dicyclic, 96dihedral, 136finitely generated, 93
546 Index
finitely presented, 93four-group, 137
free, 82
free abelian, 328Frobenius, 46, 204
fundamental, 87
Galois, 181general linear, 128
generalized quaternions, 82
hamiltonian, 156
Heisenberg, 46infinite dihedral, 34
Mathieu, 75
maximal condition, 35minimal generating set, 48
modular, 123, 173
nilpotent, 44normalizer condition, 46
perfect, 36, 78
polycyclic, 34Prufer, 503
projective unimodular, 51
quasicyclic, 503quaternions, 156
quotient, 162
simple, 173solvable, 34, 192
special linear, 50, 140
special unitary, 437special unitary group, 235
stochastic, 139
symmetric, 117, 128topological, 461, 678
torsion, 359
torsion-free, 359
unitary, 437unitriangular, 30
group algebra, 274
group object, 460group of units, 37
Gruenberg, K. A., 481
Grushko Theorem, 120Grushko, I. A., 120
Gutenberg, 4
Hall subgroup, 39Hall Theorem, 38, 40
Hall, P., 38, 245
Hamel basis, 321Hamel, G. K. W., 321
Hamilton, W. R., 156, 276, 327, 392
hamiltonian, 156Haron, A. E. P., 355
Hasse, H., 219, 429
Hasse–Minkowski Theorem, 429
Hausdorff, F., 676height
abelian group, 422
prime ideal, 502
height (rational function), 353
height sequence, 422
Heisenberg group, 46
Heller, A., 415
Herbrand quotient, 325
Herbrand, J., 325
hereditary ring, 474
Hermite, C., 122
hermitian, 437
Herstein, I. N., 208
higher center, 44
Higman, D. G., 159
Higman, G., 125, 215
Hilbert, D., 29, 209, 232, 285
Basis Theorem, 286
Nullstellensatz, 600, 612
Theorem 90, 217, 327
Theorem on Syzygies, 500
Hipparchus, 4
Hirsch length = Hirsch number, 41
Hirsch, K. A., 41
Hirzebruch, F. E. P., 415
Hochschild, G. P., 336
Holder, O., 198
Hom functor
contravariant, 464
covariant, 461
homogeneous element, 550
homogeneous ideal, 550
homology, 262
homology groups of G, 309
homomorphism
R-homomorphism, 291
algebra, 543
commutative ring, 47
graded algebra, 550
group, 150
conjugation, 154
natural map, 162
Lie algebra, 164
ring, 279
homotopic, 265
homotopic paths, 105
homotopy
contracting, 265
Hopf’s formula, 314
Hopf, H., 310, 314
Hopkins, C., 139
Hopkins–Levitzki Theorem, 139
Houston, E., 218
Hume, J., 3
Hurewicz, W., 305, 310
hyperbolic plane, 424
hypersurface, 596
Index 547
IBN, 483ideal, 50, 278
augmentation, 145, 338basis of, 283colon, 603commutative ring, 50elimination, 648finitely generated, 283fractional, 539generated by subset, 53homogeneous, 550invertible, 469, 539left, 278Lie algebra, 164maximal, 74minimal, 129minimal left, 287monomial, 645
nilpotent, 614order, 379primary, 617prime, 75principal, 51proper, 50radical, 598right, 278two-sided, 278
ideal generated by X, 280idempotent, 132, 177identity
function, 236functor, 461group element, 128morphism, 443
imageabelian category, 346function, 236linear transformation, 260module homomorphism, 296
inclusion, 237increasing p ≤ n list, 565
indecomposable, 333, 425Independence of Characters, 203independent list, 252
maximal, 257indeterminate, 43index of subgroup, 147induced
character, 187class function, 189module, 187representation, 187
induced map, 461, 464homology, 264
induced module, 326induced topology, 676induction (transfinite), 345
infinite order, 133, 379infinite-dimensional, 251inflation, 321initial object, 459injections
coproduct, 447, 452direct sum of modules, 327
injective, 238dimension, 488limit (see direct limit), 658module, 492object in category, 348
injective resolution, 256injectively equivalent, 489inner automorphism, 8, 155inner product, 417
matrix, 419space, 417
inseparableextension, 182polynomial, 182
integers, 9integers mod m, 31integers, algebraic number field, 449integral
basis, 461closure, 448element, 446extension, 446
integral closure, 604integral domain (see domain), 34integrally closed, 448intermediate covering complex, 116intermediate field, 207Invariance of Dimension, 255, 256invariant (of group), 152invariant basis number, 483invariant factors
finite abelian group, 376matrix, 386
invariant subspace, 295
inversecommutative ring, 36function, 241Galois problem, 232group element, 128image, 61limit, 653right, 282system, 651
inverse image, 375inverse image (simplicial map), 111invertible ideal, 469, 539invertible matrix, 585invertible sheaf, 526irreducible
character, 173
548 Index
element, 67
module (see simple module), 299
representation, 156, 170
variety, 614
irredundant, 620
union, 616
Isaacs, I. M., 343
isometry, 135, 429
isomorphic
commutative rings, 47
groups, 150
modules, 291
stably, 411
isomorphism
R-isomorphism, 291
category, 445
complexes, 259
groups, 150
modules, 291
rings, 47
vector spaces, 259
Ivanov, S. V., 125
Iwasawa Theorem, 79
Iwasawa, K., 79
Jacobi identity, 49
Lie algebra, 163
Jacobi, C. G. J., 49
Jacobson radical, 130
Jacobson ring, 610
Jacobson semisimple, 130
Jacobson, N., 164, 610
Janusz, G. J., 150, 222
Jategaonkar, A. V., 490
Jonnson, B., 425
Jordan canonical form, 397
Jordan, C., 24, 55, 63, 198
Jordan, P., 166
Jordan–Dickson Theorem, 63
Jordan–Holder category, 412
Jordan–Holder Theorem
Grothendieck group, 412
groups, 198
modules, 303
Jordan–Moore Theorem, 55
juxtaposition, 83
k-algebra, 543
k-linear combination, 250
k-map, 343
Kaplansky Theorem, 535
Kaplansky, I., 52, 223, 282, 411, 434, 501,522, 560
Kepler, J., 68
kernel
additive category, 343
character, 184
group homomorphism, 153Lie homomorphism, 164
linear transformation, 260
module homomorphism, 296ring homomorphism, 50, 279
Killing, W. K. J., 66, 161, 166
Klein, F., 68Kronecker delta, 30
Kronecker product, 520
Kronecker Theorem, 83
Kronecker, L., 24, 374Krull dimension, 502
Krull Theorem, 609
Krull, W., 159, 318, 479, 504Krull–Schmidt Theorem, 159
Kulikov, L. Yu., 521
Kummer, E., 446Kurosh, A. G., 425, 448
labeled graph, 18
Lady, E. L., 427Lagrange Theorem, 146
Lagrange, J.-L., 7, 146
Lam, C., 72Lame, G., 446
Lambek, J., 533
Landau, E., 14, 139Laplace expansion, 583
Laplace, P.-S., 583
Lasker, E., 620
Latin square, 71, 157, 466orthogonal, 71
lattice, 210
Laurent polynomials, 281, 443Laurent, P. A., 281
law of inertia, 427
Law of Substitution, 128, 237laws of exponents, 132
Lazard, M., 666
leading coefficient, 42least common multiple
commutative ring, 72
in Z, 14least criminal, 40
Least Integer Axiom, 9
left adjoint, 393left derived functors, 275
left exact functor, 467
left hereditary ring, 535left noetherian ring, 284
left quasi-regular, 131
lengthcomposition series, 195
cycle, 117
filtration, 302
module, 303normal series, 192
Index 549
word, 83Leonardo da Pisa (Fibonacci), 4Leray spectral sequence, 380Leray Theorem, 381Leray, J., 380, 381Levi, F., 97Levitzki, J., 139, 560lexicographic order, 631Lichtenbaum, S., 416Lie algebra, 162Lie’s Theorem, 165Lie, M. S., 161lifting, 228, 483limit (see inverse limit), 653Lindemann, F., 347linear
fractional transformation, 353functional, 473
polynomial, 44representation, 170transformation, 259
nonsingular, 259linear character, 173linear combination
in Z, 10module, 296vector space, 250
linearly dependent list, 252linearly independent infinite set, 319linearly independent list, 252list, 250
coordinate, 253increasing p ≤ n, 565linearly dependent, 252linearly independent, 252
local homeomorphism, 359local ring
regular, 514localization
map, 428, 435module, 435
ring, 428locally closed, 375locally connected, 379locally constant, 370locally free sheaf, 525locally isomorphic, 422Lodovici Ferrari, 7long exact sequence, 267�Los, J., 454lower central series, 43lowest terms
in Q, 12in k[x], 69
Lubkin, S., 355Luroth, J., 355Luroth’s Theorem, 355
Luther, M., 4Lying Over, 450
Lyndon, R. C., 336
Lyndon–Hochschild–Serre, 336
m-adic topology, 676Mac Lane, S., 310, 441, 461, 553
Mann, A., 12, 97
mapping problem, universal, 449Maschke’s Theorem, 140, 337
Maschke, H., 140, 337
Mathieu, E., 75Matlis, E., 491
matrix
elementary, 410generalized, 142
linear transformation, 263
nilpotent, 401nonsingular, 128
permutation, 170
scalar, 158, 268strictly triangular, 269
unitriangular, 30
maximalnormal subgroup, 14
tree, 108
maximal conditiongroups, 35
maximal element
poset, 314
maximal ideal, 74maximal independent list, 257
maximum condition, 128, 283
Mayer, W., 271Mayer–Vietoris Theorem, 271
McKay, J. H., 12
McLain, D. H., 12Merkurjev, A. S., 416
metric space, 673
Milnor, J. W., 125, 416minimal
left ideal, 129, 287
polynomialmatrix, 393
prime ideal, 318
prime over ideal, 504minimal generating set, 48
minimal normal subgroup, 37
minimal polynomialalgebraic element, 80
minimum condition, 128, 287
Minkowski, H., 429, 471minor, 581
Mitchell, B., 355, 386
Mobius, A. F., 68, 86
modular group, 123, 173modular law, 300
550 Index
module, 288bimodule, 470
cogenerator, 398
cyclic, 296divisible, 496
faithful, 292
finitely generated, 296finitely presented, 488
flat, 529
free, 329, 481
generator, 385injective, 492
left, 288
primary, 381projective, 484
quotient, 297
right, 289simple, 299
small, 385
torsion, 380torsion-free, 359, 380
trivial, 136
modulus, 129Molien, T., 154, 338
monic polynomial, 42
several variables, 631monkey, 27
monoid, 133
W+(Ω), 632free, 92
monomial ideal, 645
monomial order, 630degree-lexicographic order, 634
lexicographic order, 631
monomorphism (monic), 344
Monster, 168Moore Theorem, 88
Moore, E. H., 55, 88
Moore, J., 358, 491Morita equivalent, 388
Morita, K., 388
morphism, 443epic, 345
identity, 443
monic, 344Motzkin, T. S., 101
moves, 117
multilinear function, 552alternating, 563
multiplication by r, 291
multiplication table, 150multiplicative subset, 428
multiplicity, 72
Munshi, R., 613
Nagata, M., 223Nakayama’s Lemma, 131
Nakayama, T., 131natural
isomorphism, 523transformation, 523
natural map, 57groups, 162modules, 297rings, 279vector spaces, 269
natural numbers, 9, 141Navarro, G., 369N/C Lemma, 9Neumann, B. H., 215Neumann, H., 215Niccolo Fontana (Tartaglia), 4Nielsen, J., 97nilpotent
element, 598
ideal, 132Lie algebra, 165matrix, 401
nilpotent group, 44class, 44
nilpotent ideal, 614nilradical, 608Nobeling, G., 537Noether, E., 163, 215, 219, 284, 620noetherian, 128, 284, 301
category, 391nondegenerate, 420
quadratic form, 429nonderogatory, 394nongenerator, 47nonsingular
linear transformation, 259matrix, 128
nontrivial subgroup, 139norm, 216, 456
algebraic integer, 196euclidean ring, 98
normal
basis, 463extension, 190series, 192
factor groups, 192length, 192refinement, 197
subgroup, 153generated by X, 158
Normal Basis Theorem, 466normal series
refinement, 34normal subgroup
maximal, 14minimal, 37
normalized bar resolution, 318normalizer, 8
Index 551
normalizer condition, 46
not-necessarily-associative algebra, 161
Novikov, P. S., 124, 125
nullhomotopic, 264
Nullstellensatz, 600, 612
weak, 599, 612
number field
algebraic, 449
cyclotomic, 463
quadratic, 455
O’Brien, E. A., 4
objects of category, 443
obstruction, 292
odd permutation, 124, 126
Ol’shanskii, A. Yu., 508
Ol′shanskii, A. Yu., 125
one-to-one
(injective function), 238
one-to-one correspondence
(bijection), 241
onto function
(surjective function), 238
Open(X), 353
opposite category, 465
opposite ring, 292
orbit, 6
orbit space, 6
order
group, 135
group element, 133
power series, 46
order ideal, 300, 379
order-reversing, 210
ordered abelian group, 444
totally ordered, 444
ordered pair, 235
orthogonal
basis, 425
complement, 421
direct sum, 424
group, 431
matrix, 158
orthogonal Latin squares, 71
orthogonality relations, 182
orthonormal basis, 425
outer automorphism, 9, 155
φ-function, 142
p′-complement, 39
p-adic topology, 675
p-adic integers, 655
p-adic numbers, 655
p-complement, 39
p-group, 11, 31
p-primary, 481
p-primary abelian group, 362
(p)-primary module, 381pairwise disjoint, 245Papp, Z., 498Pappus, 4parallelogram law, 248parity, 19, 124Parker, E. T., 72partially ordered set, 209
chain, 314directed set, 659discrete, 652well-ordered, 316
partition, 55, 245partition of n, 377Pascal, B., 68path
circuit, 17, 107reduced, 17, 107
path class, 106Peirce decomposition, 134Peirce, C. S., 134perfect field, 56, 401perfect group, 36, 78periodic cohomology, 315permutation, 116
adjacency, 127complete factorization, 120cycle, 117disjoint, 117even, 124odd, 124, 126parity, 124signum, 125transposition, 117
permutation matrix, 16, 170PI-algebra, 560PID, 101Pigeonhole Principle, 261Plucker, J., 68Poincare, H., 42, 150, 224, 225pointed complex, 104
pointed spaces, 463pointwise operations, 35Polya, G., 23polycyclic group, 34polynomial, 42
n variables, 45commuting variables, 559cyclotomic, 93function, 593general, 84irreducible, 67monic, 42noncommuting variables, 556reduced, 224separable, 182skew, 275
552 Index
zero, 42polynomial function, 44, 593polynomial identity, 560Poncelet, J. V., 68Pontrjagin duality, 501Pontrjagin, L. S., 333poset, 209, 314positive definite, 426positive word, 91power series, 41, 515powers, 130pre-additive category, 446Premet, A., 166presentation, 92preserves
finite direct sums, 342preserves multiplications, 276presheaf, 671
constant, 369direct image, 374inverse image, 375map, 353of sections, 363
primary component, 362, 381primary decomposition, 362
commutative rings, 481, 620irredundant, 620
primary ideal, 617belongs to prime ideal, 618
Prime Avoidance, 509prime element, 105prime factorization
in Z, 11polynomial, 72
prime field, 59prime ideal, 75
associated, 620belongs to primary ideal, 618minimal, 318minimal over ideal, 504
primitive
element, 66theorem, 214
polynomial, 108associated, 109
ring, 158root of unity, 92
primitive G-set, 77primitive element, 85principal
kG-module, 136character (see trivial character), 176ideal, 51ideal domain, 101
principal derivation, 248Principal Ideal Theorem, 504product
categoricalfamily of objects, 452
two objects, 450
directgroups, 167
modules, 323, 451
rings, 275product topology, 678
profinite completion, 656
profinite group, 680
projection (covering complex), 111projections
direct sum of modules, 327
product, 450, 452projective
dimension, 486
general linear group, 73hyperplane, 69
limit (see inverse limit), 653
line, 69module, 484
object in category, 348
plane, 166space, 69
subspace, 69
unimodular group, 51projective resolution, 255
projective space, 68, 69
projective unimodular group, 402projectively equivalent, 487
projectivity, 72
properclass, 442
divisor, 106
ideal, 50
subgroup, 139submodule, 295
subring, 32
subset, 237subspace, 249
Prufer, H., 365, 503
Prufer group, 503Prufer topology, 676
pullback, 455
pureextension, 187
subgroup, 364
submodule, 370purely inseparable, 164
purely transcendental, 345
pushout, 456Pythagorean triple, 15, 623
primitive, 15
Pythagorus, 4
Qin Jiushao, 8quadratic field, 455
Index 553
quadratic form, 428equivalence, 429
nondegenerate, 429
quadratic polynomial, 44Quartic Formula, 7
quartic polynomial, 44, 189
resolvent cubic, 7quasi-isomorphic, 426
quasicyclic group, 503
quasiordered set, 445
quaternions, 156division ring, 276
generalized, 82, 254
Quillen, D., 349, 416, 487quintic polynomial, 44
quotient
(Division Algorithm)k[x], 63
(Division Algorithm) in Z, 10complex, 104, 260group, 162
Lie algebra, 164
module, 297space, 258
quotient ring, 57, 278
r-chart, 525r-cycle, 117
R-homomorphism, 291
R-isomorphism, 291
R-linear combination, 296R-map, 291
R-module, 288
R-sequence, 507r-transitive, 75
sharply r-transitive, 76
Rabinowitz trick, 600radical extension, 187
radical ideal, 598
Rado, R., 369rank, 419
free abelian group, 329
free module, 482linear transformation, 269
matrix, 270
rank (free group), 89rational canonical form, 386
rational curve, 625
rational functions, 44rational map, 626
Razmyslov, Yu. P., 560
real projective space, 68realizes the operators, 232
Recorde, R., 3
reduced
abelian group, 502basis, 648
commutative ring, 598mod {g1, . . . , gm}, 636polynomial, 224
reduced path, 17, 107reduced word, 84reduction, 84, 636Ree, R., 66Rees, D., 498, 504refinement, 34, 197, 302reflexive relation, 243regular character, 176regular covering complex, 117regular G-set, 203regular local ring, 514regular map, 626regular on module, 498regular representation, 169Reisz Representation Theorem, 422
Reisz, M., 422Reiten, I., 160, 302relation, 243relative Brauer group, 220relatively prime
k[x], 69in Z, 12integers, 12UFD, 107
remainder, 10k[x], 63k[x1, . . . , xn], 637mod G, 637
repeated roots, 74representable functor, 351, 528representation
character, 173completely reducible, 170group, 135irreducible, 156, 170linear, 170regular, 169
representation of ring, 292
representation on cosets, 2representative of coset, 144residually finite, 90residue field, 511resolution
bar, 316deleted, 273flat, 491free, 255injective, 256projective, 255
resolvent cubic, 7, 229restriction, 239
cohomology, 320representation, 191
restriction sheaf, 375
554 Index
resultant, 225retract, 325
retraction, 102, 325
right R-module, 289right adjoint, 393
right derived functors, 285, 288
right exact functor, 518ring, 29, 273
artinian, 128, 286
Boolean, 33, 41
commutative, 32Dedekind, 535
division ring, 275
quaternions, 276endomorphism ring, 274
group algebra, 274
hereditary, 474Jacobson, 610
left hereditary, 535
left noetherian, 128, 284opposite, 292
polynomial, 42
self-injective, 499semisimple, 150, 335
simple, 144
skew polynomial, 42unique factorization domain, 522, 541
von Neumann regular, 493
zero, 31ring extension, 446
Ringel, C., 160
Roiter, A. V., 160root
multiplicity, 72
polynomial, 64
root of unity, 92, 129primitive, 92
Rosset, S., 42, 560
Rotman, J. J., 427, 488roulette wheel, 23
Ruffini, P., 7
Russell paradox, 442Russell, B. A. W., 442
Ryser, H. J., 72
S-polynomial, 641Salmeron, L., 160
Sarges, H., 286
Sasiada, E., 454saturated, 444
scalar
matrix, 158, 268multiplication, 247
module, 288
transformation, 268
scalar-closed, 508Schanuel’s Lemma, 489
dual, 500Schanuel, S., 223, 351, 415Schering, E., 24, 374Schmidt, O. Yu., 159Schneider, T., 347Schottenfels Theorem, 64Schottenfels, I. M., 64, 402Schreier Refinement Theorem, 34
groups, 197modules, 302
Schreier transversal, 98Schreier, O., 97Schur’s Lemma, 146, 200Schur, I., 245Scipio del Ferro, 4Second Isomorphism Theorem
groups, 164modules, 297
secondary matrices, 417section, 362
global, 362zero, 362
Seidenberg, A., 450self-adjoint, 436self-injective, 499self-normalizing, 27semidirect product, 230semigroup, 133semisimple
Jacobson, 130ring, 150
semisimple module, 334semisimple ring, 335separable
element, 182extension, 182polynomial, 182
seriescomposition, 302factor modules, 302
Serre, J.-P., 97, 223, 336, 384, 441, 487,
525, 592Serre–Auslander–Buchsbaum Theorem, 518sesquilinear, 436set, 442sgn, 125Shafarevich, I., 232Shapiro’s Lemma, 324Shapiro, A., 324sharply r-transitive, 76sheaf
abelian groups, 370acyclic, 381coherent, 525cohomology, 379constant, 370direct image, 374
Index 555
double cover, 370extension, 375extension by zero, 375flabby, 381germs, 371inverse image, 375locally free, 525map, 371restriction, 375sheet, 359skyscraper, 370space, 360structure, 361
sheafification, 372g-sheaf, 365presheaf, 366
sheet, 359sheets, 111
Shelah, S, 309short exact sequence, 306
almost split, 302split, 307
shuffle, 571signature, 427signum, 125similar matrices, 154, 267Simmons, G. J., 86simple
extension, 214group, 173Lie algebra, 164module, 299, 334ring, 144transcendental extension, 353
simple components, 149simplex, 103
dimension, 103simplicial map, 104
inverse image, 111simply connected, 108Singer, R., 95
single-valued, 237skeletal subcategory, 384skeleton, 103skew field, 275skew polynomial ring, 42skew polynomials, 275Skolem, T., 215Skolem–Noether Theorem, 215skyscraper sheaf, 370slender, 454small category, 525small class (= set), 442small module, 385Small, L., 288, 332, 474, 535smallest
element in partially ordered set, 316
subspace, 250Smith normal form, 411Smith, H. J. S., 411solution
linear system, 249universal mapping problem, 449
solution space, 144, 249solvable
by radicals, 188group, 192Lie algebra, 165
solvable group, 34spans, 250
infinite-dimensional space, 319Spec(R)
topological space, 615special linear group, 50, 140special unitary group, 235, 437
Specker, E., 537spectral sequence, 334splice, 310split extension
groups, 230modules, 295
split short exact sequence, 307splits
polynomial, 72, 84splitting field
central simple algebra, 211polynomial, 84
squarefree integer, 15stabilizer, 6stabilizes an extension, 246stably isomorphic, 411stalk, 671
g-sheaf, 360Stallings, J. R., 120, 324standard basis, 253standard polynomial, 560Stasheff, J., 553Steinberg, R., 66
Steinitz Theorem, 214, 484Steinitz, E., 214Stevin, S., 3Stickelberger, L., 24, 374Strade, H., 166string, 373strongly indecomposable, 426structure constants, 328structure sheaf, 361, 525subalgebra, Lie, 163subbase of topology, 675subcategory, 349, 446
full, 349skeletal, 384
subcomplex, 260inverse image, 111
556 Index
subfield, 38generated by X, 59prime field, 59
subgroup, 139basic, 521center, 155centralizer of element, 8centralizer of subgroup, 8characteristic, 32commutator, 35conjugate, 8cyclic, 141Fitting, 49Frattini, 47fully invariant, 32generated by X, 143Hall, 39, 245index, 147
nontrivial, 139normal, 153
generated by X, 158normalizer, 8proper, 139pure, 364self-normalizing, 27subnormal, 46, 192Sylow, 25torsion, 359
submatrix, 581submodule, 295
cyclic, 296generated by X, 296proper, 295torsion, 379
subnormal subgroup, 46, 192subpresheaf, 371subquotient, 333subring, 32, 277subring generated by X, 280subsheaf, 371subspace, 249
invariant, 295proper, 249smallest, 250spanned by X, 250
subword, 83superalgebra, 572support, 323surjective, 238Suslin, A. A., 416, 487Suzuki, M., 66Swan, R. G., 325, 411Sylow subgroup, 25Sylow Theorem, 26, 27Sylow, L., 24Sylvester, J. J., 426symmetric
algebra, 559bilinear form, 417
function, 208
group, 117space, 417
symmetric difference, 33, 129
symmetric functionselementary, 84, 180
symmetric group, 128, 242
symmetric relation, 243
symmetry, 135symplectic
basis, 424
group, 431syzygy, 486
tangent half-angle formula, 624
target, 236, 443, 463Tarry, G., 72
Tarski monsters, 508
Tarski, A., 508Tartaglia, 4
tensor algebra, 556
tensor product, 510sheaves, 525
terminal object, 459
Thales of Miletus, 4Theatetus, 4
Third Isomorphism Theorem
groups, 165
modules, 298Thompson, J. G., 34, 46, 203, 207, 219
Thrall, R. M., 159
Three Subgroups Lemma, 50T.I. set, 208
Tietze, H. F. F., 107
top element, 670topological group, 461, 678
topological space
metric space, 673topology, 675
p-adic, 675
base, 675compact, 674
discrete, 678
finite index, 675generated by S, 675Hausdorff, 676
induced, 676product, 678
Prufer, 676
subbase, 675torsion
group, 359
module, 380
subgroup, 359submodule, 379
Index 557
torsion-free, 359, 380totally ordered abelian group, 444
trace, 172, 222, 456
trace form, 456Trace Theorem, 222
transcendence basis, 349
transcendence degree, 351transcendental element, 79
transcendental extension, 353
transfer, 314
transfinite induction, 345transformation of direct system, 662
transgression, 321
transition functionslocally free sheaf, 526
transition matrix, 264
transitiver-transitive, 75
group action, 6
sharply r-transitive, 76transitive relation, 243
transpose, 248
transposition, 117transvection
2× 2 matrix, 53
n× n matrix, 53linear transformation, 59
transversal, 97
Schreier, 98tree, 17, 107
maximal, 108
triangulated space, 224trivial character, 176
trivial edge, 17
trivial module, 136
twin primes, 16type
abelian group, 423
type T(X | R), 93type (pure extension field), 187
UFD, 105
Ulm, H., 372unimodular matrix, 50
unique factorization domain, 105
unique factorization, k[x], 71unit, 36
noncommutative ring, 133
unit (adjoint functors), 393unitary
group, 437
matrix, 437transformation, 437
unitriangular, 30
universal
central extension, 314Coefficients Theorem, 307
universal covering complex, 116
universal mapping problem, 449
solution, 449
upper bound, 210, 314
upper central series, 44
valuation ring, 444
discrete, 421
van der Waerden trick, 87
van der Waerden, B. L., 215
van Kampen, E. R., 109
Vandermonde matrix, 589
Vandermonde, A.-T., 589
variety, 594
affine, 594
irreducible, 614
vector bundle, 361
vector space, 247
vertices, 103
Viete, F., 3, 6
Vietoris, L., 271
virtually small, 405
Vogtmann, K., 126
von Dyck, W., 82
von Neumann regular, 493
von Neumann, J., 493
von Staudt, K. G. C., 68
Watts, C. E., 396–398, 663
weak dimension, 492, 493
Wedderburn Theorem
finite division rings, 146, 215
Wedderburn, J. M., 146, 327
Wedderburn–Artin Theorem, 149, 154
wedge of p factors, 562
Weierstrass, K., 347
weight, 630
well-defined, 237
well-ordered, 316
Weyl, H., 166
Whitehead’s problem, 309
Whitehead, J. H. C., 271
Widman, J., 3
Wielandt, H., 27
Wiles, A. J., 441, 445, 593
Williams, K. S., 102
Wilson’s Theorem, 149
Wilson, J., 149
Wilson, R. L., 166
Witt, E., 146
Wolf, J. A., 125
word
empty, 83
length, 83
positive, 91
reduced, 84
word on X, 83
558 Index
yoke, 492Yoneda Lemma, 350Yoneda, N., 291, 302, 350, 528
Zaks, A., 278Zariski
closure, 602topology
on kn, 596on Spec(R), 615
Zariski topology, 524Zariski, O., 524, 596Zassenhaus Lemma, 195
modules, 302
Zassenhaus, H., 195, 245Zermelo, E. E. F., 442zero complex, 257zero divisor, 34zero g-sheaf, 361zero object, 459zero of polynomial, 593zero polynomial, 42zero ring, 31zero section, 362zero-divisor, 288
on module, 498ZFC, 442Zorn’s Lemma, 314Zorn, M., 314
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