Advanced Modern Algebra Third Edition, Part 2

GRADUATE STUDIES IN MATHEMATICS 180

Joseph J. Rotman

American Mathematical Society 10.1090/gsm/180

GRADUATE STUDIES IN MATHEMATICS 180

Advanced Modern Algebra Third Edition, Part 2

Joseph J. Rotman

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staﬃlani Jeﬀ 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, visit www.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. QA154.3.R68 2015 512–dc23 2015019659

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Foreword vii

Preface to Third Edition: Part 2 ix

Chapter C-1. More Groups 1 C-1.1. Group Actions 1 Graphs 16 Counting 20 C-1.2. Sylow Theorems 24 C-1.3. Solvable and Nilpotent Groups 33 Solvable Groups 34 Nilpotent Groups 43 C-1.4. Projective Unimodular Groups 50 General Linear Group GL(n, k)50 Simplicity of PSL(2,q)52 Simplicity of PSL(n, q)58 C-1.5. More Group Actions 66 Projective Geometry 67 Multiple Transitivity 74 PSL Redux 77 C-1.6. Free Groups and Presentations 81 Existence and Uniqueness of Free Groups 82 Presentations 92 C-1.7. Nielsen–Schreier Theorem 97 C-1.8. The Baer–Levi Proof 102 The Categories Simp and Simp∗ 102 Fundamental Group 104 Covering Complexes 110 Co-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 403 The Functor K0 404 The Functor G0 408 Chapter C-5. Commutative Rings III 419 C-5.1. Local and Global 419 Subgroups of Q 419 C-5.2. Localization 427 C-5.3. Dedekind Rings 445 Integrality 446 Algebraic Integers 455 Characterizations of Dedekind Rings 467 Finitely 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 completely dedicated to his series of books on algebra. He was correcting his draft of this revision of Advanced Modern Algebra during what sadly turned out to be his final hospital visit. At that time, Joe and his family asked me to do what I could to see this close-to-finished version to publication. Two more friends and colleague of Joe’s, Jerry Janusz and Paul Weichsel, joined the project. Jerry did a meticulous line-by-line reading of the manuscript, and all three of us answered questions posed by the AMS editorial staff, based on Arlene O’Sean’s very careful reading of the manuscript. It is clear that this book would have been even richer if Joe had been able to continue to work on it. For example, he planned a chapter on algebraic geometry. We include the first paragraph of that chapter, an example of Joe’s distinctly personal 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 are familiar with elementary set theory. But folklore changes with time; elementary set theory was not part of nineteenth-century folklore. When we write a proof, we tacitly use folklore, usually not mentioning it explicitly. That folklore depends on the calendar must be one of the major factors complicating the history of mathematics. We can find primary sources and read, say, publications of Ruffini at the beginning of the 1800s, but we can’t really follow his proofs unless we are familiar with his contemporary folklore. I want to express my thanks to Sergei Gelfand and Arlene O’Sean of the AMS and 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’s intentions. Bruce Reznick May 31, 2017

vii

Preface to Third Edition: Part 2

The second half of this third edition of Advanced Modern Algebra hasPart1aspre- requisite. This is not to say that everything there must be completely mastered, but the reader should be familiar with what is there and should not be uncomfortable upon seeing the words category, functor, module,orZorn. The format of Part 2 is standard, but there are interactions between the different chapters. For example, group extensions and factor sets are discussed in the chapter on groups as well as in the chapter on homology. I am reminded of my experience as an aspiring graduate student. In order to qualify for an advanced degree, 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 limited to its own area, but as I was wrestling with an algebra problem, the only way I could 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! Of course, 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. It is a question of attitude and emphasis. Doesn’t every area of mathematics study polynomials? But algebraists and analysts view them from different perspectives. After all, mathematics really is one vast subject, and all its parts and emphases are related. A word about references in the text. If I mention Theorem C-1.2 or Exercise C-1.27 on page 19, then these are names in Part 2 of the third edition. References to names in Part 1 will have the prefix A- or B- and will say, for example, Theorem A-1.2 in Part 1 or Exercise B-1.27 on page 288 in Part 1. In an exercise set, an asterisk before an exercise, say, *C-1.26, means that this exercise is mentioned elsewhere in the text, usually in a proof. Thanks goes to Ilya Kapovich, Victoria Corkery, Vincenzo Acciaro, and Stephen Ullom.

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Index

Page numbers in italic refer to Part 1.

Abel,N.H.,7, 219 adjoint functors, 392 abelian category, 346 counit, 402 abelian group, 128 left, right, 393 divisible, 496 unit, 393 free, 328 Adjoint Isomorphism, 526, 527 ordered, 444 Ado, I. D., 163 primary, 362 affine group, 139 rank 1, 420 affine variety, 594 reduced, 502 afforded by, 173 torsion, 380 Albert, A. A., 219, 327 torsion-free, 380 Alberti, L. B., 68 totally ordered, 444 algebra, 284, 543 abelian Lie algebra, 163 central simple, 209 absolute Galois group, 480 crossed product, 328 ACC, 128, 282, 300 cyclic, 328 Accessory Irrationalities, 199 division, 209, 331 action of group, 5, 152 enveloping, 548 r-transitive, 76 finitely generated, 604 transitive, 6, 187 generated by n elements, 558 acyclic, 262 graded, 550 sheaf, 381 not-necessarily-associative, 161 additive category, 340 algebra map, 543 additive functor, 340, 465 algebraic additive notation, 130 closure, 341 Adelard of Bath, 4 element, 79 Adian, S. I., 125 extension, 79 adjacency, 127 integer, 193, 449, 455 adjacency matrix, 17 conjugate, 196 adjacent (graph), 16 number field, 449 adjoining a unit, 39 numbers, 340 adjoining to field, 79 algebraically adjoint closed, 341 functors, 666 dependent, 345 linear transformation, 431, 437 algorithm matrix, 584 Buchberger, 646 Adjoint Functor Theorem, 401 Euclidean, 17

537 538 Index

into disjoint cycles, 118 basis almost all, 319 dependence, 349 almost split, 302 free abelian group, 328 alternating free algebra, 556 bilinear form, 418 free group, 82 group, 141 free module, 329, 481 multilinear, 563 free monoid, 92 space, 418 ideal, 283 alternating sum, 26 standard, 253 Amitsur, S. A., 209, 328, 560 vector space annihilator, 133, 508 finite-dimensional, 252 element, 379 infinite-dimensional, 319 module, 381 Basis Theorem antanairesis, 16 finite abelian groups, 367, 499 anti-isomorphism, 293 Hilbert, 286 Apollonius, 4 Bass, H., 300, 415, 498 Archimedes, 4 Bautista, R., 160 Arf invariant, 429 Beltrami, E., 594 Arf, C., 429 Besche, H. U., 4 Artin, E., 58, 149 biadditive, 509 Artin, M., 429 bidegree, 334 artinian ring, 128, 286 Bifet, E., 225 ascending central series, 44 bifunctor, 521 ascending chain condition, 282 bijection, 241 associated polynomial function, 593 bilinear form, 417 associated prime ideal, 620 alternating, 418 associates, 52 nondegenerate, 420 associativity, 29, 128 skew, 418 generalized, 131, 553 symmetric, 417 Atiyah, M., 415 negative definite, 426 atlas, 525 positive definite, 426 augmentation, 145, 338 bilinear function, 417, 509 augmentation ideal, 145, 338 bimodule, 470 Auslander, M., 160, 223, 302, 490, 496, 518, binary operation, 29 522 Binomial Theorem Auslander–Buchsbaum Theorem, 520, 522 commutative ring, 32 automorphism exterior algebra, 569 field, 180 birational map, 627 group, 155, 228 Bkouche, R., 488 inner, 8 block in G-set, 77 outer, 9 Block, R. E., 166 Axiom of Choice, 313 Boole, G., 129 Boolean group, 129 (B,N) pair, 81 Boolean ring, 33, 41 b-adic digits, 23 Boone, W. W., 125 Baer sum, 244, 302 boundaries, 262 Baer Theorem, 537 bouquet, 110 Baer, R., 97, 235, 425, 494 bracelet, 24 bar resolution, 316 bracket, 162 normalized, 318 Brauer group, 218 Barr, M., 87, 401 relative, 220 Barratt, M. G., 271 Brauer, R., 159, 190, 216, 219 Barratt–Whitehead Theorem, 271 Brauer–Thrall conjectures, 159 base b, 23 Bruck, R. H., 72 base of topology, 675 Bruck–Ryser Theorem, 72 basepoint, 104, 463 Brunelleschi, F., 68 basic subgroup, 521 Buchberger’s algorithm, 646 Index 539

Buchberger’s Theorem, 643 universal, 314 Buchberger, B., 629, 640 central simple algebra, 209 Buchsbaum, D. A., 223, 349, 518, 522 centralizer Burnside Basis Theorem, 48 group element, 8 Burnside ring, 199 subgroup, 8 Burnside’s Lemma, 20, 183 subset of algebra, 212 Burnside’s problem, 125 chain, 314 Burnside’s Theorem, 168, 202 chain map, 259 Burnside, W., 20, 125 over f, 275 change of rings, 475 ∞ C -function, 35 character, 173, 203 cancellation law afforded by, 173 domain, 34 degree, 173 group, 130 generalized, 179 Cardano, G., 5 induced, 187 Carmichael, R., 35 irreducible, 173 Carnap, R., 461 kernel, 184 Cartan, E., 66, 161, 166 linear, 173 Cartan, H., 380, 475, 538 regular, 176 Cartan–Eilenberg Theorem, 538 restriction, 191 cartesian product, 235 table, 180 castle problem, 8 trivial, 176 Casus Irreducibilis, 189 character group, 532 categorical statement, 355 characteristic of field, 60 category, 443 characteristic polynomial, 390 G-category, 408 characteristic subgroup, 32 abelian, 346 chessboard, 23 additive, 340 Chevalley, C., 65, 161, 332 cocomplete, 385 Ch’in Chiu-shao, 8 cogenerator, 398 Chinese Remainder Theorem composition, 443 Z, 25 enough injectives, 378 k[x], 89 enough projectives, 378 circle operation, 280 exact, 349 circle group, 129 generator, 385 circuit, 17, 107 morphism, 443 Claborn, L., 471 noetherian, 391 class objects, 443 function, 176 opposite, 465 group, 471 pre-additive, 446 number, 471 small, 525 sums, 155 virtually small, 405 class equation, 11 Cauchy sequence, 654 class group, 540 Cauchy’s Theorem, 10 class of nilpotent group, 44 Cauchy, A.-L., 7 Classification Theorem of Finite Simple Cayley graph, 18 Groups, 66, 176, 227 Cayley Theorem, 2 Clifford algebra, 572 Cayley, A., 2, 3, 140 Clifford, W. K., 572 Cayley–Hamilton Theorem, 392 coboundary, 241 Cech,ˇ E., 380 cocomplete, 385 center cocycle identity, 238 group, 155 codiagonal, 302, 341 Lie algebra, 168 coefficients, 41 matrix ring, 268, 281 cofactor, 584 ring, 277 cofinal subset, 318 centerless, 155 cofinite, 41, 596 central extension, 314 cogenerator, 398 540 Index

Cohen, I. S., 317, 450 covering, 111 coherent sheaf, 525 de Rham, 574 Cohn, P. M., 474 differentiations, 257 cohomological dimension, 323 dimension, 103 cohomology direct sum, 354 sheaf, 379 modulus, 129 cohomology group, 241 pointed, 104 cohomology groups of G, 309 quotient, 104, 260 coinduced module, 326 simplicial, 103 cokernel, 297 simply connected, 108 additive category, 343 subcomplex, 260 Cole, F., 55 zero, 257 colimit (see direct limit), 658 component, 109 colon ideal, 603 composite integer, 11 coloring, 21 composite of functions, 239 Columbus, 4 composition factors, 195 column space of matrix, 270 composition series, 195, 302 commensurable, 13 length, 195 common divisor, 10 composition, category, 443 in Z, 10 compositum, 209 several polynomials, 103 congruence mod I, 55 two polynomials, 66 congruence class, 244 commutative, 128 congruent mod m, 19 commutative diagram, 305 congruent matrices, 419 commutative ring, 32 conjugacy class, 8, 157 Dedekind, 467 conjugate domain, 34 algebraic integers, 196 DVR, 421 elements in field extension, 459 euclidean ring, 98 group elements, 154 factorial, 104 intermediate fields, 207 field, 37 conjugate subgroups, 8 integers in number field, 449 conjugation Jacobson, 610 Grassmann algebra, 567 local groups, 154 regular, 514 quaternions, 276 PID, 101 connected, 105 polynomial ring, 42 connecting homomorphism, 265 several variables, 45 constant reduced, 598 g-sheaf, 360 UFD, 105 presheaf, 369 valuation ring, 444 sheaf, 370 commutator, 35 constant function, 236 subgroup, 35 constant functor, 462 compact, 674 constant polynomial, 44 companion matrix, 385 constant term, 44 Comparison Theorem, 273 content, 109 complement, 40, 325 continuous, 675 of subgroup, 231 contracting homotopy, 265 complete factorization, 120 contraction of ideal, 449 complete graph, 17 contragredient, 198 completely decomposable, 425 contravariant functor, 464 completely reducible, 170 convolution, 274, 282 completion, 655 coordinate field, 625 complex, 257 coordinate list, 253 acyclic, 262 coordinate ring, 597 component, 109 Copernicus, 4 connected, 105 coproduct Index 541

family of objects, 452 several variables, 631 two objects, 447 representation, 169 corestriction, 320 several variables, 630 Corner, A. L. S., 425 degree-lexicographic order, 634 Correspondence Theorem deleted resolution, 273 groups, 165 derivation, 587 modules, 298 group, 248 rings, 279 Lie algebra, 162 coset principal, 248 ideal, 55 ring, 161 subgroup, 144 derivative, 46 cosyzygy, 489 derived series covariant functor, 461 groups, 37 covering complex, 111 Lie algebra, 165 fiber, 111 Derry, D., 425 intermediate, 116 Desargues, G., 68 lifting, 111 Descartes, R., 3, 7 projection, 111 descending central series, 43 regular, 117 Lie algebra, 165 sheets, 111 determinant, 576 universal, 116 diagonal, 341 covering space, 359 diagonal map, 302 Cramer’s Rule, 586 diagonalizable, 394, 401 crossed homomorphism, 248 diagram, 305 crossed product algebra, 328 commutative, 305 Cubic Formula, 5 diagram chasing, 308 cubic polynomial, 44, 188 Dickson, L. E., 58, 63, 122, 222, 327 Culler, M., 126 dicyclic group, 96 cycle Dieudonné, J., 558 homology, 262 differential form, 574 permutation, 117 differentiations, 257 cycle structure, 120 dihedral group, 136 cyclic infinite, 34, 123 group, 141 dilatation, 59 module, 296 dimension, 255, 322 cyclic algebra, 328 dimension shifting, 273 cyclotomic field, 461 Diophantus, 4, 445 cyclotomic polynomial, 93 direct image, 374 direct limit, 658 DCC, 128, 286, 301 direct product De Morgan laws, 41 commutative rings, 54 De Morgan, A., 41 groups, 167 de Rham complex, 168, 574 modules, 323, 451 de Rham, G., 168, 574 rings, 275 Dean, R. A., 39 direct sum decomposable, 425 additive category, 341 Dedekind ring, 467, 535 complexes, 354 Dedekind Theorem, 204 matrices, 384 Dedekind, R., 174, 204, 446 modules, 323, 324, 451 degree external, 324, 326 G-set, 5 internal, 326 character, 173 vector spaces, 259, 268 euclidean ring, 98 direct summand, 325 extension field, 78 direct system, 657 graded map, 550 transformation, 662 homogeneous element, 550 directed graph, 18 polynomial, 42 connected, 18 542 Index

directed set, 659 elementary cancellation, 84 Dirichlet, J. P. G. L., 368, 446, 466, 471 elementary divisors discrete, 678 finite abelian group, 373 discrete valuation, 332, 444 matrix, 397 discrete valuation ring, 421 elementary matrix, 410 discriminant, 223 elimination ideal, 648 bilinear form, 420 empty word, 83 of OE , 467 endomorphism of cubic, 224 abelian group, 274 of quartic, 230 module, 294 disjoint permutations, 117 ring, 274 disjoint union, 452 Engel’s Theorem, 165 distributivity, 29 Engel, F., 165 divides enlargement of coset, 62, 165, 298 commutative ring, 36 enough injectives, 378 in Z, 9 enough projectives, 378 divisible module, 496 enveloping algebra, 548 division algebra, 209 epimorphism (epic), 345 Division Algorithm equal subsets, 236 k[x], 62 equality of functions, 118 k[x1,...,xn], 637 equalizer in Z, 10 condition, 369 division ring, 275 equivalence characteristic p, 331 inverse, 356 divisor equivalence class, 244 in Z, 9 equivalence relation, 243 Dlab, V., 160 equivalent domain categories, 356 commutative ring, 34 extensions, 241, 295 DVR, 421 filtration, 302 morphism, 443 height sequences, 423 of function, 236 matrices, 406 PID, 101 normal series, 197 regular local ring, 516 representations, 171 UFD, 105 series, groups, 197 Double Centralizer Theorem, 212 Eratosthenes, 4 double coset, 81 etymology double cover, 370 K-theory, 415 dual basis, 269 abelian, 219 dual space, 260, 269 abelian category, 346 duals in category, 450 acyclic complex, 262 DVR, 421 adjoint functors, 392, 666 Dye, R. L., 429 affine, 594 Dynkin diagrams, 167 affine space, 627 Dynkin, E., 167 alternating group, 141 artinian, 149 Eckmann, B., 310, 314 automorphism, 155 edge, 16 canonical form, 386 trivial, 17 coherent ring, 525 Eick, B., 4 commutative diagram, 305 eigenvalue, 388 cubic, 44 eigenvector, 388 cycle, 117 Eilenberg, S., 310, 358, 417, 441, 475, 491, dihedral group, 136 538 domain, 34 Eisenstein Criterion, 95 exact sequence, 575 Eisenstein integers, 32 Ext, 295 Eisenstein, G., 95 exterior algebra, 562 Index 543

factor set, 237 factored, 310 field, 37 short, 306 flat, 529 splice, 310 free group, 92 triangle, 268 functor, 461 Exchange Lemma, 256 homology, 225 exponent homomorphism, 47 group, 376 ideal, 446 module, 381 isomorphism, 47 extension kernel, 50 central, 314 Latin square, 71 universal, 314 left exact, 469 groups, 227 nilpotent, 165 modules, 295, 306 polyhedron, 136 of ideal, 449 power, 130 extension field, 78 profinite, 477 algebraic, 79 pure subgroup, 364 degree, 78 quadratic, 44 finite, 78 quasicyclic, 503 Galois, 207, 475 quaternions, 276 inseparable, 182 quotient group, 162 normal, 190 radical, 598 pure, 187 rational canonical form, 386 purely transcendental, 345 regular representation, 203 radical, 187 ring, 29 separable, 182 symplectic, 424 simple, 214 syzygy, 486, 492 extension sheaf, 375 Tor, 307 exterior algebra, 562 torsion subgroup, 359 exterior derivative, 574 variety, 594 exterior power, 562 vector, 248 Euclid, 4 FAC, 384, 525 Euclid’s Lemma, 69, 98, 101 factor groups, 192 integers, 12 factor modules, 302 Euclidean Algorithm I factor set, 237 integers, 17 factorial ring (see UFD), 104 Euclidean Algorithm II faithful G-set, 6 integers, 18 faithful module, 292 Euclidean Algorithm, k[x], 70 family of supports, 380 euclidean ring, 98 Fano plane, 69 Eudoxus, 4 Fano, G., 69 Euler φ-function, 142 Feit, W., 34, 219 Euler Theorem, 148 Feit–Thompson Theorem, 219 Euler, L., 19, 446 Fermat Little Theorem, 22 Euler–Poincaré characteristic, 110, 262 Fermat prime, 96 evaluation homomorphism, 49 Fermat’s Theorem, 148 even permutation, 124 Fermat, P., 445 exact Ferrari, Lodovici, 5 abelian category, 347 FFR, 500 category, 349 fiber, 111 functor, 355, 469 Fibonacci, 4, 590 left, 467 field, 37 right, 517 algebraic closure, 341 hexagon, 325 algebraically closed, 341 sequence, 305 finite, 186 almost split, 302 fraction, 38 complexes, 260 Galois, 88 544 Index

perfect, 401 group, 82 prime, 59 module, 329, 481 rational functions, 44 monoid, 92 15-puzzle, 124, 126 resolution, 255 filtration, 302, 333 free product, 118 length, 302 freeness property, 330 refinement, 302 Freudenthal, H., 310 filtrations Freyd, P., 355 equivalent, 302 Frobenius finite automorphism, 186 extension, 78 complement, 204 order (module), 379 group, 204 topology, 479, 679 kernel, 205 finite index topology, 675 Reciprocity, 191 finite-dimensional, 251 Theorem finitely generated Frobenius kernels, 206 algebra, 604 real division algebras, 215 ideal, 283 Frobenius group, 46 module, 296 Frobenius, F. G., 20, 24, 174, 187, 198, 203, finitely generated group, 93 215, 327, 374 finitely presented group, 93 full functor, 355 finitely presented module, 488 full subcategory, 349 Finney, Jr., R. L., 488 fully invariant, 32 First Isomorphism Theorem function, 236 abelian category, 358 bijection, 241 commutative rings, 58 constant, 236 complexes, 260 identity, 236 groups, 163 inclusion, 237 modules, 297 injective, 238 vector spaces, 269 polynomial, 44 Fitting subgroup, 49 rational, 45 Fitting, H., 49 restriction, 239 Five Lemma, 309 surjective, 238 fixed field, 202 functor fixed points, 253 additive, 340, 465 fixed-point-free, 203 constant, 462 fixes, 117, 180 contravariant, 464 flabby sheaf, 381 contravariant Hom, 464 flat dimension, 491 covariant, 461 flat module, 529 covariant Hom, 461 flat resolution, 491 exact, 355, 469 forgetful functor, 462 forgetful, 462 formal power series full, 355 one variable, 41 identity, 461 several variables, 515 left exact, 467, 468 Formanek, E., 560 representable, 351, 528 four-group, 137 right exact, 517 fraction field, 38 two variables, 521 fractional ideal, 469, 539 fundamental group, 87, 463 Fraenkel,A.A.H.,442 Fundamental Theorem Frattini Argument, 38 Arithmetic, 198 Frattini subgroup, 47 finite abelian groups Frattini Theorem, 47 elementary divisors, 374 free invariant factors, 376 abelian group, 328 finitely generated abelian groups algebra, 556 elementary divisors, 374 commutative algebra, 558, 671 invariant factors, 377 Index 545

Galois Theory, 211, 479 global dimension, left, 490 modules global section, 362 elementary divisors, 382 gluing, 369 invariant factors, 382 Godement resolution, 382 symmetric functions, 208 Godement, R., 382 symmetric polynomials, 208, 639 Going Down Theorem, 453 Going Up Theorem, 453 G-category, 408 Goldman, O., 604 G-domain, 606 Goodwillie, T. G., 590 G-ideal, 608 Gordan, P., 285 g-map, 365 graded algebra, 550 G-set, 5 graded map, 550 block, 77 graph, 16 degree, 5 adjacency matrix, 17 faithful, 6 adjacent, 16 primitive, 77 automorphism, 16 g-sheaf, 360 Cayley, 18 constant, 360 complete, 17 stalk, 360 connected, 16 zero, 361 directed, 18 Gabriel, P., 160, 166, 386 edge, 16 Galligo, A., 487 isomorphism, 16 Galois extension, 207, 475 labeled, 18 Galois field, 88 vertex, 16 Galois group, 181, 475 Grassmann algebra, 566 absolute, 480 Grassmann, H. G., 68, 566 Galois Theorem, 86 greatest common divisor Galois, E., 8, 146 domain, 97 Gaschütz, W., 254 in Z, 10 Gauss Theorem several polynomials, 103 R[x]UFD,110 two polynomials, 66 cyclotomic polynomial, 96 Green,J.A.,314 Gauss’s Lemma, 111 Griess, R., 168 Gauss, C. F., 215 Griffith, P. A., 518 Gaussian elimination, 409 Gröbner, W., 640 Gaussian equivalent, 410 Gröbner basis, 640 Gaussian integers, 32 Gromov, M., 126 gcd, 10 Grothendieck group, 404, 406, 408 Gelfond, A., 347 Jordan–Hölder, 412 Gelfond-Schneider Theorem, 347 reduced, 407 general linear group, 50, 128 Grothendieck, A., 336, 402, 412, 415, 441, general polynomial, 84 592 generalized associativity, 131, 553 group generalized character, 179 p-group, 11 generalized matrix, 142 abelian, 128 generalized quaternions, 82, 254 additive notation, 130 generate affine, 139 dependence, 349 algebra, 274 generator alternating, 141 category, 403 axioms, 128, 138 cyclic group, 141 Boolean, 129 of ModR, 385 circle group, 129 generators and relations, 92, 403 conjugacy class, 157 algebra, 556 cyclic, 141 Gerard of Cremona, 4 dicyclic, 96 germ, 364 dihedral, 136 germs, 371 finitely generated, 93 546 Index

finitely presented, 93 abelian group, 422 four-group, 137 prime ideal, 502 free, 82 height (rational function), 353 free abelian, 328 height sequence, 422 Frobenius, 46, 204 Heisenberg group, 46 fundamental, 87 Heller, A., 415 Galois, 181 Herbrand quotient, 325 general linear, 128 Herbrand, J., 325 generalized quaternions, 82 hereditary ring, 474 hamiltonian, 156 Hermite, C., 122 Heisenberg, 46 hermitian, 437 infinite dihedral, 34 Herstein,I.N.,208 Mathieu, 75 higher center, 44 maximal condition, 35 Higman, D. G., 159 minimal generating set, 48 Higman, G., 125, 215 modular, 123, 173 Hilbert, D., 29, 209, 232, 285 nilpotent, 44 Basis Theorem, 286 normalizer condition, 46 Nullstellensatz, 600, 612 perfect, 36, 78 Theorem 90, 217, 327 polycyclic, 34 Theorem on Syzygies, 500 Prüfer, 503 Hipparchus, 4 projective unimodular, 51 Hirsch length = Hirsch number, 41 quasicyclic, 503 Hirsch,K.A.,41 quaternions, 156 Hirzebruch,F.E.P.,415 quotient, 162 Hochschild, G. P., 336 simple, 173 Hölder, O., 198 solvable, 34, 192 Hom functor special linear, 50, 140 contravariant, 464 special unitary, 437 covariant, 461 special unitary group, 235 homogeneous element, 550 stochastic, 139 homogeneous ideal, 550 symmetric, 117, 128 homology, 262 topological, 461, 678 homology groups of G, 309 torsion, 359 homomorphism torsion-free, 359 R-homomorphism, 291 unitary, 437 algebra, 543 unitriangular, 30 commutative ring, 47 group algebra, 274 graded algebra, 550 group object, 460 group, 150 group of units, 37 conjugation, 154 Gruenberg, K. A., 481 natural map, 162 Grushko Theorem, 120 Lie algebra, 164 Grushko, I. A., 120 ring, 279 Gutenberg, 4 homotopic, 265 Hall subgroup, 39 homotopic paths, 105 Hall Theorem, 38, 40 homotopy Hall, P., 38, 245 contracting, 265 Hamel basis, 321 Hopf’s formula, 314 Hamel, G. K. W., 321 Hopf, H., 310, 314 Hamilton, W. R., 156, 276, 327, 392 Hopkins, C., 139 hamiltonian, 156 Hopkins–Levitzki Theorem, 139 Haron, A. E. P., 355 Houston, E., 218 Hasse, H., 219, 429 Hume, J., 3 Hasse–Minkowski Theorem, 429 Hurewicz, W., 305, 310 Hausdorff, F., 676 hyperbolic plane, 424 height hypersurface, 596 Index 547

IBN, 483 infinite order, 133, 379 ideal, 50, 278 infinite-dimensional, 251 augmentation, 145, 338 inflation, 321 basis of, 283 initial object, 459 colon, 603 injections commutative ring, 50 coproduct, 447, 452 elimination, 648 direct sum of modules, 327 finitely generated, 283 injective, 238 fractional, 539 dimension, 488 generated by subset, 53 limit (see direct limit), 658 homogeneous, 550 module, 492 invertible, 469, 539 object in category, 348 left, 278 injective resolution, 256 Lie algebra, 164 injectively equivalent, 489 maximal, 74 inner automorphism, 8, 155 minimal, 129 inner product, 417 minimal left, 287 matrix, 419 monomial, 645 space, 417 nilpotent, 614 inseparable order, 379 extension, 182 primary, 617 polynomial, 182 prime, 75 integers, 9 principal, 51 integers mod m, 31 proper, 50 integers, algebraic number field, 449 radical, 598 integral right, 278 basis, 461 two-sided, 278 closure, 448 ideal generated by X, 280 element, 446 idempotent, 132, 177 extension, 446 identity integral closure, 604 function, 236 integral domain (see domain), 34 functor, 461 integrally closed, 448 group element, 128 intermediate covering complex, 116 morphism, 443 intermediate field, 207 image Invariance of Dimension, 255, 256 abelian category, 346 invariant (of group), 152 function, 236 invariant basis number, 483 linear transformation, 260 invariant factors module homomorphism, 296 finite abelian group, 376 inclusion, 237 matrix, 386 increasing p ≤ n list, 565 invariant subspace, 295 indecomposable, 333, 425 inverse Independence of Characters, 203 commutative ring, 36 independent list, 252 function, 241 maximal, 257 Galois problem, 232 indeterminate, 43 group element, 128 index of subgroup, 147 image, 61 induced limit, 653 character, 187 right, 282 class function, 189 system, 651 module, 187 inverse image, 375 representation, 187 inverse image (simplicial map), 111 induced map, 461, 464 invertible ideal, 469, 539 homology, 264 invertible matrix, 585 induced module, 326 invertible sheaf, 526 induced topology, 676 irreducible induction (transfinite), 345 character, 173 548 Index

element, 67 group homomorphism, 153 module (see simple module), 299 Lie homomorphism, 164 representation, 156, 170 linear transformation, 260 variety, 614 module homomorphism, 296 irredundant, 620 ring homomorphism, 50, 279 union, 616 Killing, W. K. J., 66, 161, 166 Isaacs, I. M., 343 Klein, F., 68 isometry, 135, 429 Kronecker delta, 30 isomorphic Kronecker product, 520 commutative rings, 47 Kronecker Theorem, 83 groups, 150 Kronecker, L., 24, 374 modules, 291 Krull dimension, 502 stably, 411 Krull Theorem, 609 isomorphism Krull, W., 159, 318, 479, 504 R-isomorphism, 291 Krull–Schmidt Theorem, 159 category, 445 Kulikov,L.Yu.,521 complexes, 259 Kummer, E., 446 groups, 150 Kurosh,A.G.,425,448 modules, 291 rings, 47 labeled graph, 18 vector spaces, 259 Lady, E. L., 427 Ivanov, S. V., 125 Lagrange Theorem, 146 Iwasawa Theorem, 79 Lagrange, J.-L., 7, 146 Iwasawa, K., 79 Lam, C., 72 Lamé, G., 446 Jacobi identity, 49 Lambek, J., 533 Lie algebra, 163 Landau, E., 14, 139 Jacobi,C.G.J.,49 Laplace expansion, 583 Jacobson radical, 130 Laplace, P.-S., 583 Jacobson ring, 610 Lasker, E., 620 Jacobson semisimple, 130 Latin square, 71, 157, 466 Jacobson, N., 164, 610 orthogonal, 71 Janusz,G.J.,150,222 lattice, 210 Jategaonkar, A. V., 490 Laurent polynomials, 281, 443 Jónnson, B., 425 Laurent, P. A., 281 Jordan canonical form, 397 law of inertia, 427 Jordan, C., 24, 55, 63, 198 Law of Substitution, 128, 237 Jordan, P., 166 laws of exponents, 132 Jordan–Dickson Theorem, 63 Lazard, M., 666 Jordan–Hölder category, 412 leading coefficient, 42 Jordan–Hölder Theorem least common multiple Grothendieck group, 412 commutative ring, 72 groups, 198 in Z, 14 modules, 303 least criminal, 40 Jordan–Moore Theorem, 55 Least Integer Axiom, 9 juxtaposition, 83 left adjoint, 393 left derived functors, 275 k-algebra, 543 left exact functor, 467 k-linear combination, 250 left hereditary ring, 535 k-map, 343 left noetherian ring, 284 Kaplansky Theorem, 535 left quasi-regular, 131 Kaplansky, I., 52, 223, 282, 411, 434, 501, length 522, 560 composition series, 195 Kepler, J., 68 cycle, 117 kernel filtration, 302 additive category, 343 module, 303 character, 184 normal series, 192 Index 549

word, 83 Luther, M., 4 Leonardo da Pisa (Fibonacci), 4 Lying Over, 450 Leray spectral sequence, 380 Lyndon, R. C., 336 Leray Theorem, 381 Lyndon–Hochschild–Serre, 336 Leray, J., 380, 381 Levi, F., 97 m-adic topology, 676 Levitzki, J., 139, 560 Mac Lane, S., 310, 441, 461, 553 lexicographic order, 631 Mann, A., 12, 97 Lichtenbaum, S., 416 mapping problem, universal, 449 Lie algebra, 162 Maschke’s Theorem, 140, 337 Lie’s Theorem, 165 Maschke, H., 140, 337 Lie, M. S., 161 Mathieu, E., 75 lifting, 228, 483 Matlis, E., 491 limit (see inverse limit), 653 matrix Lindemann, F., 347 elementary, 410 linear generalized, 142 fractional transformation, 353 linear transformation, 263 functional, 473 nilpotent, 401 polynomial, 44 nonsingular, 128 representation, 170 permutation, 170 transformation, 259 scalar, 158, 268 nonsingular, 259 strictly triangular, 269 linear character, 173 unitriangular, 30 linear combination maximal in Z, 10 normal subgroup, 14 module, 296 tree, 108 vector space, 250 maximal condition linearly dependent list, 252 groups, 35 linearly independent infinite set, 319 maximal element linearly independent list, 252 poset, 314 list, 250 maximal ideal, 74 coordinate, 253 maximal independent list, 257 increasing p ≤ n, 565 maximum condition, 128, 283 linearly dependent, 252 Mayer, W., 271 linearly independent, 252 Mayer–Vietoris Theorem, 271 local homeomorphism, 359 McKay, J. H., 12 local ring McLain,D.H.,12 regular, 514 Merkurjev, A. S., 416 localization metric space, 673 map, 428, 435 Milnor, J. W., 125, 416 module, 435 minimal ring, 428 left ideal, 129, 287 locally closed, 375 polynomial locally connected, 379 matrix, 393 locally constant, 370 prime ideal, 318 locally free sheaf, 525 prime over ideal, 504 locally isomorphic, 422 minimal generating set, 48 Lodovici Ferrari, 7 minimal normal subgroup, 37 long exact sequence, 267 minimal polynomial Lo´ s, J., 454 algebraic element, 80 lower central series, 43 minimum condition, 128, 287 lowest terms Minkowski, H., 429, 471 in Q, 12 minor, 581 in k[x], 69 Mitchell, B., 355, 386 Lubkin, S., 355 Möbius, A. F., 68, 86 Lüroth, J., 355 modular group, 123, 173 Lüroth’s Theorem, 355 modular law, 300 550 Index

module, 288 Nakayama, T., 131 bimodule, 470 natural cogenerator, 398 isomorphism, 523 cyclic, 296 transformation, 523 divisible, 496 natural map, 57 faithful, 292 groups, 162 finitely generated, 296 modules, 297 finitely presented, 488 rings, 279 flat, 529 vector spaces, 269 free, 329, 481 natural numbers, 9, 141 generator, 385 Navarro, G., 369 injective, 492 N/C Lemma, 9 left, 288 Neumann, B. H., 215 primary, 381 Neumann, H., 215 projective, 484 Niccolò Fontana (Tartaglia), 4 quotient, 297 Nielsen, J., 97 right, 289 nilpotent simple, 299 element, 598 small, 385 ideal, 132 torsion, 380 Lie algebra, 165 torsion-free, 359, 380 matrix, 401 trivial, 136 nilpotent group, 44 modulus, 129 class, 44 Molien, T., 154, 338 nilpotent ideal, 614 monic polynomial, 42 nilradical, 608 several variables, 631 Nobeling, G., 537 monkey, 27 Noether, E., 163, 215, 219, 284, 620 monoid, 133 noetherian, 128, 284, 301 W +(Ω), 632 category, 391 free, 92 nondegenerate, 420 monomial ideal, 645 quadratic form, 429 monomial order, 630 nonderogatory, 394 degree-lexicographic order, 634 nongenerator, 47 lexicographic order, 631 nonsingular monomorphism (monic), 344 linear transformation, 259 Monster, 168 matrix, 128 Moore Theorem, 88 nontrivial subgroup, 139 Moore, E. H., 55, 88 norm, 216, 456 Moore, J., 358, 491 algebraic integer, 196 Morita equivalent, 388 euclidean ring, 98 Morita, K., 388 normal morphism, 443 basis, 463 epic, 345 extension, 190 identity, 443 series, 192 monic, 344 factor groups, 192 Motzkin, T. S., 101 length, 192 moves, 117 refinement, 197 multilinear function, 552 subgroup, 153 alternating, 563 generated by X, 158 multiplication by r, 291 Normal Basis Theorem, 466 multiplication table, 150 normal series multiplicative subset, 428 refinement, 34 multiplicity, 72 normal subgroup Munshi, R., 613 maximal, 14 minimal, 37 Nagata, M., 223 normalized bar resolution, 318 Nakayama’s Lemma, 131 normalizer, 8 Index 551

normalizer condition, 46 (p)-primary module, 381 not-necessarily-associative algebra, 161 pairwise disjoint, 245 Novikov, P. S., 124, 125 Papp, Z., 498 nullhomotopic, 264 Pappus, 4 Nullstellensatz, 600, 612 parallelogram law, 248 weak, 599, 612 parity, 19, 124 number field Parker,E.T.,72 algebraic, 449 partially ordered set, 209 cyclotomic, 463 chain, 314 quadratic, 455 directed set, 659 discrete, 652 O’Brien, E. A., 4 well-ordered, 316 objects of category, 443 partition, 55, 245 obstruction, 292 partition of n, 377 odd permutation, 124, 126 Pascal, B., 68 Ol’shanskii, A. Yu., 508 path Ol shanskii, A. Yu., 125 circuit, 17, 107 one-to-one reduced, 17, 107 (injective function), 238 path class, 106 one-to-one correspondence Peirce decomposition, 134 (bijection), 241 Peirce, C. S., 134 onto function perfect field, 56, 401 (surjective function), 238 perfect group, 36, 78 Open(X), 353 periodic cohomology, 315 opposite category, 465 permutation, 116 opposite ring, 292 adjacency, 127 orbit, 6 complete factorization, 120 orbit space, 6 cycle, 117 order disjoint, 117 group, 135 even, 124 group element, 133 odd, 124, 126 power series, 46 parity, 124 order ideal, 300, 379 signum, 125 order-reversing, 210 transposition, 117 ordered abelian group, 444 permutation matrix, 16, 170 totally ordered, 444 PI-algebra, 560 ordered pair, 235 PID, 101 orthogonal Pigeonhole Principle, 261 basis, 425 Plücker, J., 68 complement, 421 Poincaré, H., 42, 150, 224, 225 direct sum, 424 pointed complex, 104 group, 431 pointed spaces, 463 matrix, 158 pointwise operations, 35 orthogonal Latin squares, 71 Pólya, G., 23 orthogonality relations, 182 polycyclic group, 34 orthonormal basis, 425 polynomial, 42 outer automorphism, 9, 155 n variables, 45 commuting variables, 559 φ-function, 142 cyclotomic, 93 p-complement, 39 function, 593 p-adic topology, 675 general, 84 p-adic integers, 655 irreducible, 67 p-adic numbers, 655 monic, 42 p-complement, 39 noncommuting variables, 556 p-group, 11, 31 reduced, 224 p-primary, 481 separable, 182 p-primary abelian group, 362 skew, 275 552 Index

zero, 42 categorical polynomial function, 44, 593 family of objects, 452 polynomial identity, 560 two objects, 450 Poncelet, J. V., 68 direct Pontrjagin duality, 501 groups, 167 Pontrjagin, L. S., 333 modules, 323, 451 poset, 209, 314 rings, 275 positive definite, 426 product topology, 678 positive word, 91 profinite completion, 656 power series, 41, 515 profinite group, 680 powers, 130 projection (covering complex), 111 pre-additive category, 446 projections Premet, A., 166 direct sum of modules, 327 presentation, 92 product, 450, 452 preserves projective finite direct sums, 342 dimension, 486 preserves multiplications, 276 general linear group, 73 presheaf, 671 hyperplane, 69 constant, 369 limit (see inverse limit), 653 direct image, 374 line, 69 inverse image, 375 module, 484 map, 353 object in category, 348 of sections, 363 plane, 166 primary component, 362, 381 space, 69 primary decomposition, 362 subspace, 69 commutative rings, 481, 620 unimodular group, 51 irredundant, 620 projective resolution, 255 primary ideal, 617 projective space, 68, 69 belongs to prime ideal, 618 projective unimodular group, 402 Prime Avoidance, 509 projectively equivalent, 487 prime element, 105 projectivity, 72 prime factorization proper in Z, 11 class, 442 polynomial, 72 divisor, 106 prime field, 59 ideal, 50 prime ideal, 75 subgroup, 139 associated, 620 submodule, 295 belongs to primary ideal, 618 subring, 32 minimal, 318 subset, 237 minimal over ideal, 504 subspace, 249 primitive Prüfer, H., 365, 503 element, 66 Prüfer group, 503 theorem, 214 Prüfer topology, 676 polynomial, 108 pullback, 455 associated, 109 pure ring, 158 extension, 187 root of unity, 92 subgroup, 364 primitive G-set, 77 submodule, 370 primitive element, 85 purely inseparable, 164 principal purely transcendental, 345 kG-module, 136 pushout, 456 character (see trivial character), 176 Pythagorean triple, 15, 623 ideal, 51 primitive, 15 ideal domain, 101 Pythagorus, 4 principal derivation, 248 Principal Ideal Theorem, 504 Qin Jiushao, 8 product quadratic field, 455 Index 553

quadratic form, 428 commutative ring, 598 equivalence, 429 mod {g1,...,gm}, 636 nondegenerate, 429 polynomial, 224 quadratic polynomial, 44 reduced path, 17, 107 Quartic Formula, 7 reduced word, 84 quartic polynomial, 44, 189 reduction, 84, 636 resolvent cubic, 7 Ree, R., 66 quasi-isomorphic, 426 Rees, D., 498, 504 quasicyclic group, 503 refinement, 34, 197, 302 quasiordered set, 445 reflexive relation, 243 quaternions, 156 regular character, 176 division ring, 276 regular covering complex, 117 generalized, 82, 254 regular G-set, 203 Quillen, D., 349, 416, 487 regular local ring, 514 quintic polynomial, 44 regular map, 626 quotient regular on module, 498 (Division Algorithm) regular representation, 169 k[x], 63 Reisz Representation Theorem, 422 (Division Algorithm) in Z, 10 Reisz, M., 422 complex, 104, 260 Reiten, I., 160, 302 group, 162 relation, 243 Lie algebra, 164 relative Brauer group, 220 module, 297 relatively prime space, 258 k[x], 69 quotient ring, 57, 278 in Z, 12 integers, 12 r-chart, 525 UFD, 107 r-cycle, 117 remainder, 10 R-homomorphism, 291 k[x], 63 R-isomorphism, 291 k[x1,...,xn], 637 R-linear combination, 296 mod G, 637 R-map, 291 repeated roots, 74 R-module, 288 representable functor, 351, 528 R-sequence, 507 representation r-transitive, 75 character, 173 sharply r-transitive, 76 completely reducible, 170 Rabinowitz trick, 600 group, 135 radical extension, 187 irreducible, 156, 170 radical ideal, 598 linear, 170 Rado, R., 369 regular, 169 rank, 419 representation of ring, 292 free abelian group, 329 representation on cosets, 2 free module, 482 representative of coset, 144 linear transformation, 269 residually finite, 90 matrix, 270 residue field, 511 rank (free group), 89 resolution rational canonical form, 386 bar, 316 rational curve, 625 deleted, 273 rational functions, 44 flat, 491 rational map, 626 free, 255 Razmyslov, Yu. P., 560 injective, 256 real projective space, 68 projective, 255 realizes the operators, 232 resolvent cubic, 7, 229 Recorde, R., 3 restriction, 239 reduced cohomology, 320 abelian group, 502 representation, 191 basis, 648 restriction sheaf, 375 554 Index

resultant, 225 dual, 500 retract, 325 Schanuel, S., 223, 351, 415 retraction, 102, 325 Schering, E., 24, 374 right R-module, 289 Schmidt, O. Yu., 159 right adjoint, 393 Schneider, T., 347 right derived functors, 285, 288 Schottenfels Theorem, 64 right exact functor, 518 Schottenfels, I. M., 64, 402 ring, 29, 273 Schreier Refinement Theorem, 34 artinian, 128, 286 groups, 197 Boolean, 33, 41 modules, 302 commutative, 32 Schreier transversal, 98 Dedekind, 535 Schreier, O., 97 division ring, 275 Schur’s Lemma, 146, 200 quaternions, 276 Schur, I., 245 endomorphism ring, 274 Scipio del Ferro, 4 group algebra, 274 Second Isomorphism Theorem hereditary, 474 groups, 164 Jacobson, 610 modules, 297 left hereditary, 535 secondary matrices, 417 left noetherian, 128, 284 section, 362 opposite, 292 global, 362 polynomial, 42 zero, 362 self-injective, 499 Seidenberg, A., 450 semisimple, 150, 335 self-adjoint, 436 simple, 144 self-injective, 499 skew polynomial, 42 self-normalizing, 27 unique factorization domain, 522, 541 semidirect product, 230 von Neumann regular, 493 semigroup, 133 zero, 31 semisimple ring extension, 446 Jacobson, 130 Ringel, C., 160 ring, 150 Roiter, A. V., 160 semisimple module, 334 root semisimple ring, 335 multiplicity, 72 separable polynomial, 64 element, 182 root of unity, 92, 129 extension, 182 primitive, 92 polynomial, 182 Rosset, S., 42, 560 series Rotman, J. J., 427, 488 composition, 302 roulette wheel, 23 factor modules, 302 Ruffini, P., 7 Serre, J.-P., 97, 223, 336, 384, 441, 487, Russell paradox, 442 525, 592 Russell, B. A. W., 442 Serre–Auslander–Buchsbaum Theorem, 518 Ryser, H. J., 72 sesquilinear, 436 set, 442 S-polynomial, 641 sgn, 125 Salmerón, L., 160 Shafarevich, I., 232 Sarges, H., 286 Shapiro’s Lemma, 324 S¸asiada, E., 454 Shapiro, A., 324 saturated, 444 sharply r-transitive, 76 scalar sheaf matrix, 158, 268 abelian groups, 370 multiplication, 247 acyclic, 381 module, 288 coherent, 525 transformation, 268 cohomology, 379 scalar-closed, 508 constant, 370 Schanuel’s Lemma, 489 direct image, 374 Index 555

double cover, 370 subspace, 250 extension, 375 Smith normal form, 411 extension by zero, 375 Smith, H. J. S., 411 flabby, 381 solution germs, 371 linear system, 249 inverse image, 375 universal mapping problem, 449 locally free, 525 solution space, 144, 249 map, 371 solvable restriction, 375 by radicals, 188 sheet, 359 group, 192 skyscraper, 370 Lie algebra, 165 space, 360 solvable group, 34 structure, 361 spans, 250 sheafification, 372 infinite-dimensional space, 319 g-sheaf, 365 Spec(R) presheaf, 366 topological space, 615 sheet, 359 special linear group, 50, 140 sheets, 111 special unitary group, 235, 437 Shelah, S, 309 Specker, E., 537 short exact sequence, 306 spectral sequence, 334 almost split, 302 splice, 310 split, 307 split extension shuffle, 571 groups, 230 signature, 427 modules, 295 signum, 125 split short exact sequence, 307 similar matrices, 154, 267 splits Simmons, G. J., 86 polynomial, 72, 84 simple splitting field extension, 214 central simple algebra, 211 group, 173 polynomial, 84 Lie algebra, 164 squarefree integer, 15 module, 299, 334 stabilizer, 6 ring, 144 stabilizes an extension, 246 transcendental extension, 353 stably isomorphic, 411 simple components, 149 stalk, 671 simplex, 103 g-sheaf, 360 dimension, 103 Stallings, J. R., 120, 324 simplicial map, 104 standard basis, 253 inverse image, 111 standard polynomial, 560 simply connected, 108 Stasheff, J., 553 Singer, R., 95 Steinberg, R., 66 single-valued, 237 Steinitz Theorem, 214, 484 skeletal subcategory, 384 Steinitz, E., 214 skeleton, 103 Stevin, S., 3 skew field, 275 Stickelberger, L., 24, 374 skew polynomial ring, 42 Strade, H., 166 skew polynomials, 275 string, 373 Skolem, T., 215 strongly indecomposable, 426 Skolem–Noether Theorem, 215 structure constants, 328 skyscraper sheaf, 370 structure sheaf, 361, 525 slender, 454 subalgebra, Lie, 163 small category, 525 subbase of topology, 675 small class (= set), 442 subcategory, 349, 446 small module, 385 full, 349 Small, L., 288, 332, 474, 535 skeletal, 384 smallest subcomplex, 260 element in partially ordered set, 316 inverse image, 111 556 Index

subfield, 38 algebra, 559 generated by X, 59 bilinear form, 417 prime field, 59 function, 208 subgroup, 139 group, 117 basic, 521 space, 417 center, 155 symmetric difference, 33, 129 centralizer of element, 8 symmetric functions centralizer of subgroup, 8 elementary, 84, 180 characteristic, 32 symmetric group, 128, 242 commutator, 35 symmetric relation, 243 conjugate, 8 symmetry, 135 cyclic, 141 symplectic Fitting, 49 basis, 424 Frattini, 47 group, 431 fully invariant, 32 syzygy, 486 generated by X, 143 Hall, 39, 245 tangent half-angle formula, 624 index, 147 target, 236, 443, 463 nontrivial, 139 Tarry, G., 72 normal, 153 Tarski monsters, 508 generated by X, 158 Tarski, A., 508 normalizer, 8 Tartaglia, 4 proper, 139 tensor algebra, 556 pure, 364 tensor product, 510 self-normalizing, 27 sheaves, 525 subnormal, 46, 192 terminal object, 459 Sylow, 25 Thales of Miletus, 4 torsion, 359 Theatetus, 4 submatrix, 581 Third Isomorphism Theorem submodule, 295 groups, 165 cyclic, 296 modules, 298 generated by X, 296 Thompson, J. G., 34, 46, 203, 207, 219 proper, 295 Thrall, R. M., 159 torsion, 379 Three Subgroups Lemma, 50 subnormal subgroup, 46, 192 T.I. set, 208 subpresheaf, 371 Tietze, H. F. F., 107 subquotient, 333 top element, 670 subring, 32, 277 topological group, 461, 678 subring generated by X, 280 topological space subsheaf, 371 metric space, 673 subspace, 249 topology, 675 invariant, 295 p-adic, 675 proper, 249 base, 675 smallest, 250 compact, 674 spanned by X, 250 discrete, 678 subword, 83 finite index, 675 superalgebra, 572 generated by S, 675 support, 323 Hausdorff, 676 surjective, 238 induced, 676 Suslin, A. A., 416, 487 product, 678 Suzuki, M., 66 Prüfer, 676 Swan, R. G., 325, 411 subbase, 675 Sylow subgroup, 25 torsion Sylow Theorem, 26, 27 group, 359 Sylow, L., 24 module, 380 Sylvester, J. J., 426 subgroup, 359 symmetric submodule, 379 Index 557

torsion-free, 359, 380 universal covering complex, 116 totally ordered abelian group, 444 universal mapping problem, 449 trace, 172, 222, 456 solution, 449 trace form, 456 upper bound, 210, 314 Trace Theorem, 222 upper central series, 44 transcendence basis, 349 transcendence degree, 351 valuation ring, 444 transcendental element, 79 discrete, 421 transcendental extension, 353 van der Waerden trick, 87 transfer, 314 van der Waerden, B. L., 215 transfinite induction, 345 van Kampen, E. R., 109 transformation of direct system, 662 Vandermonde matrix, 589 transgression, 321 Vandermonde, A.-T., 589 transition functions variety, 594 locally free sheaf, 526 affine, 594 transition matrix, 264 irreducible, 614 transitive vector bundle, 361 r-transitive, 75 vector space, 247 group action, 6 vertices, 103 sharply r-transitive, 76 Viète, F., 3, 6 transitive relation, 243 Vietoris, L., 271 transpose, 248 virtually small, 405 transposition, 117 Vogtmann, K., 126 transvection von Dyck, W., 82 2 × 2 matrix, 53 von Neumann regular, 493 n × n matrix, 53 von Neumann, J., 493 linear transformation, 59 vonStaudt,K.G.C.,68 transversal, 97 Schreier, 98 Watts, C. E., 396–398, 663 tree, 17, 107 weak dimension, 492, 493 maximal, 108 Wedderburn Theorem triangulated space, 224 finite division rings, 146, 215 trivial character, 176 Wedderburn, J. M., 146, 327 trivial edge, 17 Wedderburn–Artin Theorem, 149, 154 trivial module, 136 wedge of p factors, 562 twin primes, 16 Weierstrass, K., 347 type weight, 630 abelian group, 423 well-defined, 237 type T(X | R), 93 well-ordered, 316 type (pure extension field), 187 Weyl, H., 166 Whitehead’s problem, 309 UFD, 105 Whitehead, J. H. C., 271 Ulm, H., 372 Widman, J., 3 unimodular matrix, 50 Wielandt, H., 27 unique factorization domain, 105 Wiles, A. J., 441, 445, 593 unique factorization, k[x], 71 Williams, K. S., 102 unit, 36 Wilson’s Theorem, 149 noncommutative ring, 133 Wilson, J., 149 unit (adjoint functors), 393 Wilson, R. L., 166 unitary Witt, E., 146 group, 437 Wolf, J. A., 125 matrix, 437 word transformation, 437 empty, 83 unitriangular, 30 length, 83 universal positive, 91 central extension, 314 reduced, 84 Coefficients Theorem, 307 word on X,83 558 Index

yoke, 492 Yoneda Lemma, 350 Yoneda, N., 291, 302, 350, 528

Zaks, A., 278 Zariski closure, 602 topology on kn, 596 on Spec(R), 615 Zariski topology, 524 Zariski, O., 524, 596 Zassenhaus Lemma, 195 modules, 302 Zassenhaus, H., 195, 245 Zermelo, E. E. F., 442 zero complex, 257 zero divisor, 34 zero g-sheaf, 361 zero object, 459 zero of polynomial, 593 zero polynomial, 42 zero ring, 31 zero section, 362 zero-divisor, 288 on module, 498 ZFC, 442 Zorn’s Lemma, 314 Zorn, M., 314 This book is the second part of the new edition of Advanced Modern Algebra XLI ½VWX TEVX TYFPMWLIH EW +VEHYEXI 7XYHMIW MR Mathematics, Volume 165). Compared to the previous edition, the QEXIVMEPLEWFIIRWMKRM½GERXP]VISVKERM^IHERHQER]WIGXMSRWLEZI FIIRVI[VMXXIR8LIFSSOTVIWIRXWQER]XSTMGWQIRXMSRIHMRXLI ½VWX TEVX MR KVIEXIV HITXL ERH MR QSVI HIXEMP8LI ½ZI GLETXIVW SJ XLI FSSO EVI HIZSXIH XS KVSYT XLISV]VITVIWIRXEXMSR XLISV] homological algebra, categories, and commutative algebra, respec- XMZIP]8LIFSSOGERFIYWIHEWEXI\XJSVEWIGSRHEFWXVEGXEPKIFVEKVEHYEXIGSYVWI EWEWSYVGISJEHHMXMSREPQEXIVMEPXSE½VWXEFWXVEGXEPKIFVEKVEHYEXIGSYVWISVJSV WIPJWXYH]

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