564–566Abelian category, 330
Abelian p-extension, 509–514
Abhyankar, S., 595
Absolute value, 558–562
archimedean, 561
for a finite dimensional extension field, 585–588, 637
for F(x), 566
non-archimedean, 561
for 564–566
trivial, 559
Abstract dependence relation, 122–125
Adjoint functor, 48
Adjoint relative to a sesquilinear form, 209
Adjugant, 49
Adjunction, 49
Affine algebraic variety, 427–433
Albert, A. A., 502
affine, 519
central, 215
exterior, 141
filtered, 147
finite dimensional central simple, 215–226
free, 44
modules for and representations of, 211
polynomial, 43
separable, 374
similar, 227
structure theory of, 210–215
symmetric, 141
tensor, 139
Algebraically closed field, 427, 464
Algebraic closure, 464
existence of, 466
isomorphism of, 468
separable, 490
Algebraic dependence, 514–515
Algebraic integer, 284
Amalgamated sum, 90
Amitsur, S. A., 393
Amitsur’s complex, 334
Annihilator of a module, 186
Anti-automorphisms of End ΔV, 210
Anti-derivation, 379
Anti-hermitian form, 209
Anti-representation, 195
Approximation theorem, 562–564
Archimedean ordered, 654
Arf invariant, 244
Arrow, 10
Artin, E., 184, 312, 483, 501, 651, 660–662, 651, 670, 671
Artin, E., Nesbitt, C. J., and Thrall, R. M., 483
Artin, E., and Schreier, O., 501, 509, 650, 674–677
Artin, E., and Whaples, G., 563
Artin, M., 526
Artinian module, 101
commutative, 425–427
Artin-Rees lemma, 441–442
Asano, K., 306
Ascending chain condition, 101
Associated prime ideal, 437
Ax, J., 631
Axiom of choice, 2
Baer sum, 353
Balanced product, 126
Banaschewski’s lemma, 307
Barsotti, J., 544
Base, 123
Beauville, A., 523
Bergman, G., 195
Bézoutiant, 663
Bialgebra, 145
Bifunctor, 20
Bimodules, 133–137
invertible, 182
Blichfeldt-Brauer theorem, 308–309
Boolean algebras, 75–76
Boundary, 332
Bourbaki, N., 197
Brauer, R., 245, 247, 305–312, 313–314, 495
Brauer group, 226–228, 475–484, 495
of local field, 608–611
Brauer-Suzuki theorem, 324
Burnside, W., 215, 246, 247, 284, 524
Burnside’s theorem, 213, 284–286
Cardinal numbers, 5
addition of, 3
arithmetic of, 3–4
multiplication of, 4
Cartan, H., and Eilenberg, S., 326, 387
Cartier, P., 544
Castelnuovo-Zariski theorem, 523
Category (definition), 9–10
abelian, 330
additive, 328
discrete, 12
dual, 13
equivalent, 27
isomorphic, 26
of objects over or below, 14
products of, 13
small, 12
Cayley, A., 184
Character, 269
of abelian groups, 280–281
of direct products, 279–280
generalized, 306
irreducible, 269
linear, 280
realizable over a subfield, 315
table, 277–278
unit, 271
Character group, 281
Characteristic series, 109
Characteristic subgroup, 105
Chevalley, C., 244
Chevalley, C., and Eilenberg, S., 326
Chief series, 109
Ci-field, 669
Circle composition, 194
Claborn, L., 648
Classes, 6
Class function, 269
induced, 306
Class group, 626
Clemens, C. H., 523
Clifford, W. K, 244
Clifford algebra, 228
historical note on, 245
main involution in, 239
structure of, 234
Clifford group, 238
vector representation of, 238
(A. H.) Clifford’s theorem, 255, 299
Coalgebra, 144–145
dual, 146
Coboundary, 333
Cocycle, 333
Codomain, 9
Cogenerator, 177
Cohomology group, 333
Cohomology of algebras, 370–375
Cohomology of groups, 355–363
Colliot-Thélène, J. L., 523
Completely ramified extension field, 598
Completely reducible module, 117–122
Complete reducibility of modules for semisimple artinian rings, 208
Complex, 331
chain (positive), 333
cochain (negative), 333
over a module, 339
under a module, 341
Composition series, 108
Congruence, 61
of identities, 91
Connecting homomorphism, 336
Content, 630
Contracting homomorphism, 357
Contravariant (hom functor), 38
Coproduct, 34
Coresolution, 341
injective, 341
Correspondence, 53
diagonal, 53
inverse, 53
Corro, D. R, 323
Countably infinite set, 3
Co variant (hom functor), 38
Crossed homomorphism, 360
principal, 361
Crossed product, 475
Cross section, 287
Curtis, C., and Reiner, I., 325
Cycle, 332
Cyclic algebras, 484–485
Cyclic p-extensions, 513
Dedekind, R., and Weber, H., 557
Dedekind domain, 623
characterizations, 625–631
integral extension of, 631–634
Degree
of a central simple algebra, 222
of a word, 78
Dense, 197
Density theorems, 197–201
for completely reducible modules, 197
for primitive rings, 199
Demeyer and Ingraham, 462
Dependence relation, 122
higher 147
into a module, 373
Derivation Galois theory (for purely
inseparable extensions of exponent one), 541, 542
Derived functor, 342–346
Diagonalization, 145
Descending chain condition, 101
Dickson, L. E., 254, 472, 476,
Dimensionality (of completely reducible module), 125
Dirac, P. A. M., 245
Directed set, 59
Direct limit, 70
of Ω-algebras, 72
Direct product (internal), 116
Direct sum (internal), 110
of complexes, 332
of quadratic forms, 665
of representations, 253
Discriminant (ideal), 640
Discriminant of quadratic form, 232, 613
Disjoint representations, 302
Divisor (ideal), 622
Domain, 9
integrally closed, 412
Double centralizer, 257–258
Dual basis lemma, 152–153
Eckmann, B., 326
Eckmann, B., and Schopf, A., 160
Eilenberg, S., and MacLane, S., 8, 326, 368
Eisenstein polynomial, 596
Eklof, P, 77
Elementary group, 305
Enveloping algebra, 213
Epic, 15
in Grp and R-mod, 16
in Ring, 17
Equivalence of categories, 27
Equivalence of categories mod-R and mod-Mn(R), 29–31, 170
Equivalence relation, 54
Essential extension, 161
Essential monomorphism, 161
Even Clifford algebra, 236
Even Clifford group, 238
Exact functor, 98
Exact sequence, 96
Exchange axiom, 123
Exchange property, 116
Exponent
of central simple algebra, 497
of group, 270
of purely inseparable field extension, 495
Ext, 346–353
Extension
of base field, 220
of complete fields, 569–573, 582–583
of groups, 363–369
of homomorphisms to places, 580–585
of modules, 348–350
and extension of groups, 364
normalized, 481
Faithful functor, 22
Faithful module, 174
Faithful representation, 191
Feit, W., and Thompson, J., 246
Field composite, 550
Filter, 76
principal, 78
proper, 76
Finitely generated, 103
Finite sets, 3
Finite topology, 469
First isomorphism theorem (for Ω-algebras), 63
Fitting’s lemma, 113–114
Five lemma, 100
Flat module, 153–155
characterization by Tor, 354
Flatness of localization, 399
Formally real fields, 651–655
Free algebra, 44
Free field composites, 551
Free group, 89
Free module, 43
Free Ω-algebra, 78–80
Free product, 85
of groups, 85
Free resolution of 
, 359
Free ring, 86
Freudenthal, H., 600
Frobenius automorphism, 506, 599
Frobenius groups, 317–325
Frobenius kernel, 320
Frobenius reciprocity theorem, 297
Fully invariant series, 109
Fully invariant subgroup, 105
Functor, 19
abelianizing, 20
additive, 97
contravariant, 19
diagonal, 21
duality, 21
faithful, 22
forgetful, 20
full, 22
identity, 20
injection, 20
power, 20
projection, 21
representable, 37–40
Fundamental theorem of homomorphisms (of Ω-algebras), 62
Galois extension field, 474
Galois group of simple purely transcendental extension, 522
Galois theory, 471–475, 486–489, 541–544
Gauss’ lemma, 594
Generator (for a category of modules), 173, 175
Gerstenhaber, M., 544
Global dimension, 377–378
G-module, 355
regular, 356
Gödel-Bernays (GB) system, 6
“Going down” theorem, 413
“Going up” theorem, 412
Goldschmidt, D. M., and Isaacs, I. M., 306, 325
Graded homomorphism, 384
Graded module, 384
Graded ring associated with an ideal, 448
Graves, J. J., 184
Griffiths, P. A., 523
Grothendieck, A., 9
Group algebra, 251–252
center of, 260
Group determinant, 282
Group of fractional ideals, 624
Groupoid, 12
Group with operators, 56
Hajja, M., 524
Hartshorne, R., 523
Hasse invariant, 613
HausdorfT, F., 583
Height (of a prime ideal), 456
Hensel, K., 557
Hermitian form, 209
Herstein, I. N., 245
Higher derivation, 540
Hilbert basis theorem, 420–421
Hilbert Nullstellensatz, 429, 564–565, 583, 584
Hilbert polynomial (for a graded module), 443–448
Hilbert-Serre theorems, 445–447
Hilbert’s seventeenth problem, 651
Hilbert’s syzygy theorem, 385
Hom functors, 37–40
contravariant, 38
covariant, 38
Homogeneous component, 121
Homogeneous rational function, 517
Homological dimension, 375–378
Homology
functor, 333
group, 362
module, 333
Homomorphism of Ω-algebras, 60
fundamental theorem, 62
Homotopy, 337–339
Hopf, H., 326
Huppert, B., 369
Hurwitz’s theorem, 665
I-adic topology, 455–461
Ideal, 65
fractional, 620
integral, 620
invertible, 621
maximal, 68
nil, 195
principal, 620
Idealizer, 199
Indecomposable submodule, 438
Induced representation, 288
Inductive limit. See Direct limit
Inertial group, 304
Ingraham. See Demeyer and Ingraham Infinite Galois theory, 486–489
fundamental theorem of, 488
Initial object, 36
Injective hull (envelope), 163
Injective module, 156–164
Integral dependence, 408–412
Integrally closed domain, 412
I ntertwining number, 296
Inverse limit, 72
of Ω-algebras, 73
Irredundant primary decomposition, 436
Iskovkikh, V. A., 523
Isolated primary component, 434
] sometric isomorphism, 568
Isomorphism (in a category), 12
Isomorphism of H2(G, F*) with Br(E/F), 481
Isomorphisms of simple artinian rings, 206
Isomorphism theorems (for Ω-algebras), 63, 65
Jacobi, C. G. J., 323
Jacobian matrix, 538
Jacobson–Bourbaki correspondence, 468–471
Jacobson radical of a ring, 192–197
characterizations of, 196
of artinian ring, 202
Janusz, G. J., 649
Jordan-Hölder theorem, 108
Kan, D., 48
Kaplansky, I., 462
Kaplansky, I., and Kneser, M., 659–660
Keisler, L., 662
fc-fold transitive permutation group, 278
Kolchin’s theorem, 215
Koszul complex, 383
Krasner’s lemma, 584
Krull dimension, 450–455
Krull intersection theorem, 442
Krull–Schmidt theorem, 115
Krull’s principal ideal theorem, 452
Kummer extension, 498–501
Lam, T. Y., 677
-factor group, 104
-group, 104
indecomposable, 117
strongly indecomposable, 117
-subgroup, 104
Landau, D., 659
Lang, S., 669
Lasker, E., 388
Lasker–Macaulay–Noether theory, 389
Lasker–Noether decomposition theorem, 433
Lattice, 56
complete, 57
of congruences, 66–70
Laurent series (formal), 425, 597
Lexicographic order, 555
Lie algebra of endomorphisms, 531
Lifting of idempotents, 405, 457
Linear disjointness, 525
Local fields, 595–599
Brauer group of, 608–611
quadratic forms over, 591–598, 611–618
Local-global relations, 400–403
Localization of modules, 397–400
of rings, 393–397
Locally compact totally disconnected abelian groups, 602
Local ring, 111
regular, 454
Lower central series, 90
Lukasiewicz notation, 55
Luroth’s theorem, 522
“Lying over” theorem, 411
Macaulay, F. S., 388
MacLane, S., 530
Magnus, W., 90
Malcev, A. I., 375
Manin, J., 523
Maschke, H., 313
Maschke’s theorem, 253
Maximum condition, 101
Maximum spectrum, 403
McCrimmon, K., 196
McKenna, K., 662
M-group, 311
Minimum condition, 101
Module, 12
artinian, 101
completely reducible, 119, 208
over a Dedekind domain, 643–649
with differentiation, 332
divisible, 158
faithful, 174
flat, 153–155
free, 43
graded, 384
indecomposable, 111
induced, 288
injective, 156–164
irreducible, 117, 205, 208–209
noetherian, 101
projective, 148–155
strongly indecomposable, 112
Molien, T., 184
Monic, 15
in Grp and R-mod, 16
in Ring, 17
Morita, K., 9
Morita context (pre-equivalence data), 164–171
ring of, 171
Morita similar (rings), 179
Morita theorems, 167–168, 178, 182
Morphism, 9
epic, 15
monic, 15
zero, 328
Motzkin, T., 674
Multiplication ring, 203
Multiplicity (of irreducible component of a representation), 260
Mumford, D., 523
Nakayama’s lemma, 415
Natural isomorphism, 23
Natural transformation, 23–25
Newton–Puiseaux series, 595
Nilpotent ideal, 191
Nil radical, 391
characterization of (Krull’s theorem), 392
Noether, E., 184, 327, 388, 422, 436, 472
Noetherian module, 103
Noetherian ring, 420–425
Noether normalization theorem, 518
Non-degenerate (form), 209
Normal algebraic extension field, 489
structure of, 492–494
Normal closure, 490
Normal primary decomposition, 437
Normal series, 105
Normic polynomial, 670
Ojanguren, M., and Sridharan, M. R., 544
Ω-algebra, 53–58
Ordered abelian group, 575
isolated subgroup of, 583
Ordered field, 651
Order homomorphism, 574
Ordinal number, 5
Orthogonal idempotents, 110
Orthogonality relations, 274, 276
Orthogonal transformation, 237
Ostrowski’s theorem, 571–573
Ostwald, W., 324
Otobe, Y., 600
p-adic absolute value of Q, 558
Partially ordered set, 57
Partial order, 54
Peirce decomposition, 197
Perfect closure, 494
Pfister quadratic form, 667
Pfister theory of quadratic forms, 663–669
Place, 579
p-Lie algebra, 531
Poincare series, 445
Pontrjagin, L., 600
Power series (formal), 422–424
Pre-equivalence data (Morita context), 164–171
Primary decomposition (of modules and ideals), 433–440
Primary ideal, 434
Primary submodule, 434
Prime avoidance lemma, 391
Prime ideal, 389
associated with primary submodule, 434
Primitive ideal, 192
Primitive ring, 187
commutative, 189
right, 195
Produced representation, 299
Product
in a category, 33
n-ary, 54
nullary, 55
in an Ω-algebra, 54–55
Projections (determined by a direct decomposition of modules), 111
Projective class group, 419–420
Projective limit. See Inverse limit Projective linear group, 256
Projective module, 148–155
characterization by Ext, 348
rank of, 414–419
Projective representation, 256, 369
Projective resolution of a sequence, 343
Puiseaux series, 595
Pullback diagram, 36
Purely inseparable element, 491
Purely inseparable extension field, 491
Purely transcendental extension, 516
Pushout diagram, 37
Quadratic form, 228–229
over local fields, 611–618
multiplicative, 665
strongly multiplicative, 666
Quadratic reciprocity law, 323
Quasi-algebraically closed field, 670
Quasi-elementary group, 307
Quasi-regular element, 194
Quasi-regular ideal, 194
Quaternion algebra, 232, 611–615
Quillen, D., 152
Quotient algebra, 61
Quotient object, 18
Rabin, M., 77
Radical (Jacobson) of a ring, 192–197, 392
of artinian ring, 202
nil, 392
Ramification index, 588, 637–638
Ramified extension field, 599
Real closed field, 632–633, 653–654
characterization, 674
Real closure of an ordered field, 655–657
Reduced orthogonal group, 240
Reduced product, 87
Reduced word, 87
Rees ring, 438
Regular extension field, 549
Regular ring (in the sense of von Neumann), 196
Reid, J. D., 495
Relation, 54
Representation of a group, 247
absolutely irreducible, 263
completely reducible, 253
conjugate, 255
contragredient, 251
disjoint, 302
equivalence of, 251
factor, 252
induced, 288
irreducible, 253
irreducible constituents of, 255
matrix, 248
permutation, 248
produced, 299
regular, 248
similarity of, 248
subrepresentation of, 252
tensor product, 249
unit, 251
Representation of a ring
completely reducible, 187
faithful, 186
irreducible, 186
regular, 191
Representations of dihedral group, 261–262
Representations of S„, 265–269
Resolution (of a module), 339–341
free, 340
Koszul, 383
projective, 339
standard resolution of an algebra, 371–372
standard resolution of , 357
Restricted Lie algebra (p-Lie algebra), 531
Restriction of scalars, 291
Retraction, 15
Rim, D. S., 593
Ring of algebraic integers, 284
Ring of endomorphisms, 96, 185
R-sequence, 454
Saltman, D., 523
Sansuc, J. J., 523
Schanuel’s lemma, 155
Schreier refinement theorem, 106
Schur index, 316
Schur relations, 273
Schur’s lemma, 118
Second isomorphism theorem (for Ω-algebras), 65
Section, 15
Semi-direct product, 367
Semi-linear isomorphism, 205
Semi-linear map, 205
Semi-primitivity of polynomial rings (Amitsur’s theorem), 393
Separability, 525–530
Separability and inseparability degree, 495
Separable algebra, 374
Separable algebraic closure, 490
Separable algebraic extension, 491
Separable extension field, 529
Separable splitting field (of central simple algebra), 495–498
Separably generated extension field, 527
Separating transcendency base, 527
Sesquilinear form, 209
adjoint relative to, 210
Set, I
directed, 59
Shoda’s theorem, 304
Simple components of semi-simple artinian ring, 204
Skolem–Noether theorem, 222
Snake lemma, 337
Spectrum (prime), 403–408
Spin group, 240
Spinorial norm, 240
Split algebra, 220
Splitting field
of a central simple algebra, 220–221
of group, 264
of a set of polynomials, 467
Springer, T., 655
Sridharan, M. R., 544
Stone, M. H., 403
Strongly indecomposable group, 117
Strongly indecomposable module, 112, 114
Sturm, J. F., 650
Sturm’s theorem, 655
Subcategory, 11
Subdirect decomposition of Ω-algebras, 69
Subdirectly irreducible Ω-algebras, 69
Subdirect product of Ω-algebras, 66–70
Subobject, 18
Suslin, A., 152
Swinnerton-Dyer., 523
Symmetric algebra, 141–142
System of parameters, 454
Tableau (Young), 265
Taketa’s theorem, 311–312
Tamagawa, T., 630
Tarski, A., 52
Tarski’s theorem, 650
Teichmiiller, O., 327
Tensor algebra, 140
Tensor product
of algebras, 143,
of fields, 44
of molecules, 125–133
of quadratic forms, 644
of representations, 249
Tensor product functor, 129–130
Terminal object, 36
Thrall, R. M., 483
Tor, 353–355
Totally disconnected locally compact division rings, 599–608
structure theorem for, 607
Totally positive elements, 657–660
Trace bilinear form, 641
Transcendency base, 516
separating, 527
Transcendency degree, 516
Ultrafilter, 76
Ultraproducts, 75–78
of division rings, 77
of Ω-algebras, 77
Unipotent, 215
Unirational, 523
Units of polynomial ring (McCoy’s theorem), 393
Universal enveloping algebra of a Lie algebra, 142–143
Universal map, 42
Universl object, 42
Universals, 40–45
Unramified extension field, 598
Unramified primes, 640
Valence, 78
Valuation, 578–579
canonical, 578
definition, 578
discrete, 590
exponential, 576
rank one, 583
Valuation ring, 577
discrete, 591
Value group, 575
van Dantzig, D., 600
van der Waerden, B. L., 87
Varieties (of Ω-algebras), 81–86
coproducts in, 84
free algebras for, 82
internal characterization of, 92
von Neumann, J., 265
Weber, H., 557
Wedderburn, J. H. M., 184, 327, 485
Wedderburn-Artin theorems, 171–173, 203–204
matrix formulation, 210
Wedderburn “principal” theorem, 374
Wedderburn theorem (on finite division rings), 225
Weil, A., 549
Well-ordered sets, 3
Weyl, H., 245
Weyl algebra, 192
Whitehead, A. N., 52
Whitehead, J. H. C., 374
Whitney, H., 125
Witt index, 241
Witt vectors, 501–509
and abelian p-extension, 509–514
Wronskian, 540
Yoneda’s lemma, 39
Young, A., 265
Zariski, O., and Samuel, P., 462, 618
Zariski topology, 403, 430–432
Zassenhaus, H., 367
Zassenhaus’ lemma, 106
Zermelo–Fraenkel axiomatization of set theory, 1, 3
Zero (in a category), 36
Zero divisor of a module, 433
Zorn’s lemma, 2