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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