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Dummit D.S., Foote R.M. — Abstract Algebra

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Название: Abstract Algebra

Авторы: Dummit D.S., Foote R.M.

Аннотация:

Widely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the reader's understanding. In this way, readers gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 657

Добавлена в каталог: 08.11.2005

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Предметный указатель

Frobenius reciprocity      642
Function field      559
Function field, elliptic      566
Fundamental theorem of algebra      457 530
Fundamental Theorem of Arithmetic      6 287
Fundamental Theorem of Finitely Generated Abelian Groups      159ff 199 379
Fundamental Theorem of Finitely Generated Modules over a P.I.D.      373 375 377
Fundamental Theorem of Galois Theory      481ff 487
Fundamental Theorem on Symmetric Functions      523 536ff
F[x]-module      317ff 361ff
G-invariant      574
Galois closure      508
Galois conjugates      487
Galois extension      475 485 486 487
Galois group      475
Galois group of       477 521
Galois group of       537 558
Galois group of       495
Galois group of       503
Galois group of $\mathbb{F}_{p^n}$       479 499
Galois group of $\mathbb{Q} (2^{1/8},i)$       490ff
Galois group of $\mathbb{Q} (2^{1/8},i)$ , over quadratic subfields      495
Galois group of $\mathbb{Q} (\sqrt{(2+\sqrt{2})(3+\sqrt{3})})$       498
Galois group of $\mathbb{Q} (\sqrt{2+\sqrt{2}})$       496
Galois group of $\mathbb{Q} (\sqrt{2})$       476
Galois group of $\mathbb{Q} (\sqrt{2},\sqrt{3})$       476 521
Galois group of $\mathbb{Q} (\sqrt{D_1},\sqrt{D_2})$       496
Galois group of $\mathbb{Q} (\zeta_5)$       511
Galois group of $\mathbb{Q} (\zeta_n)$       510
Galois group of $\mathbb{Q} (\zeta_n+\zeta_n^{-1})$       517
Galois group of $\mathbb{Q} (\zeta_p)$       511
Galois group of $\mathbb{Q} (\zeta_{13})$       512
Galois group of a biquadratic      532
Galois group of a cubic      527
Galois group of a quartic      529 532
Galois group of composite extension      505 506 507 513
Galois group of cyclotomic field      510 513
Galois group of finite field      479 499
Galois groups of polynomials      476 520ff 558
Galois groups over Q      553ff
Galois groups, infinite      558ff 565
Gauss sums      550
Gauss — Jordan elimination      349ff
Gaussian integers      272 277 287 309
Gauss’ Lemma      298
General linear group      33 236 340
General polynomial      522
Generalized character      636
Generalized eigenspace      413
Generators      26 53 60ff 220
Generators for free module      330
Generators of subgroup      60
Generators of submodule      327
GL(V)      340 345 571
Greatest common divisor      4 5 273 279 285
Group      17
Group action, left versus right      130
Group of units in a ring      228
Group ring      236ff 571
Group, alternating      108ff
Group, cyclic      53ff
Group, dihedral      24
Group, divisible      65 86
Group, general linear      33
Group, modular      72 87 101
Group, quasidihedral      71 87 101
Group, quaternion      34
Group, semidihedral      see “Quasidihedral group”
Group, special linear      47
Group, symmetric      28
Groups of medium order      203ff
Groups of order $3^3\dot7\dot13\dot409$       214 216 636ff
Groups of order       146
Groups of order       175 185
Groups of order 12      145 184
Groups of order 30      145 183
Groups of order 6      146
Groups of order pq      136 144 183
Groups, G-stable      574
Groups, simple of order 168      209ff 216
Groups, table of small order      169
Hall subgroup      102 202 626
Hall’s theorem      198 626
Hamilton Quaternions      226 230 231 237 257
Harmonic analysis      610
Heptadecagon, construction of      518ff
Hilbert’s basis theorem      289
Hilbert’s Nullstellensatz      312
Hilbert’s Specialization Theorem      562
Hilbert’s Theorem      90 497
Hilbert’s Theorem, additive form      497
Holder Program      104 585
Holomorph      180 186
Homogeneous      536
Homomorphism of modules      322ff
Homomorphism, group      35
Homomorphism, ring      239
Hypernilpotent group      193
Ideal      242
Ideal, generated by set      250
Idempotent      269 333 349 592 604 606
Identity, group      18
Identity, ring      225
Image of a map      2
Image of linear transformation, computing      354
Indecomposable module      578 580
Index, of subgroup      90
Induced character      628ff 630 635
Induced homomorphism      101
Induced module      628ff 630
Induced representation      630ff
Infinite Galois groups      558ff
Injective      2
Injective module      585ff 590
Inner automorphism      135
Inner product, of characters      605
Inseparable degree of a polynomial      463
Inseparable degree of field extension      563
Inseparable extension      463 558ff
Inseparable polynomial      458
Insolvability of the quintic      542 552
Integral domain      229 235 255 256 264 265 282
Internal direct product      173
Intertwine      578
Invariant Factor Decomposition Algorithm      391 393
Invariant factors      159 374
Invariant factors for a matrix      388 414
Invariant subspace      319
Inverse      18
Inverses of matrices, computing      352 365
Irreducibility criteria      301ff
Irreducibility, of cyclotomic polynomial      304 466
Irreducible character      601 605 607
Irreducible element      282ff
Irreducible module      333 578 583
Irreducible polynomial      301
Irreducible polynomials of degree n over $\mathbb{F}_p$       500 502
Isomorphism of modules      322
Isomorphism of rings      239
Isomorphism Theorems for groups      96ff
Isomorphism Theorems for modules      326
Isomorphism Theorems for rings      234 246
Isomorphism, group      35
Isotypic component      605 612
Jacobson radical      258
Join      65
Jordan block      404
Jordan canonical form      402ff 404 405 407ff
Jordan — Holder theorem      103
k-stage Euclidean Domains      289

Kernel of action      42 50 133
Kernel of homomorphism      39 75
Kernel of linear transformation, computing      354
Kernel of module homomorphism      322
Kernel, ring homomorphism      239
Klein 4-group      68
Kronecker — Weber Theorem      514 536
Krull topology      565
Kummer extensions      540
Kummer generators for cyclic extensions      550
Kummer theory      540
Lagrange resolvent      539
Lagrange’s Theorem      44 89
Lattice for quotient group      99ff
Lattice of subfields      488
Lattice of subfields of $\mathbb{Q}(2^{1/8},i)$       495
Lattice of subfields of $\mathbb{Q}(\sqrt[3]{2}, p)$       481
Lattice of subfields of $\mathbb{Q}(\sqrt{2}, \sqrt{3})$       481
Lattice of subfields of $\mathbb{Q}(\zeta_3)$       512
Lattice of subgroups      65ff
Lattice of subgroups of       111
Lattice of subgroups of       69 100
Lattice of subgroups of $D_{16}$       70
Lattice of subgroups of $QD_{16}$       72
Lattice of subgroups of       69 99
Lattice of subgroups of       68
Lattice of subgroups of $\mathbb{Z}/12\mathbb{Z}$       68
Lattice of subgroups of $\mathbb{Z}/2\mathbb{Z}$       66
Lattice of subgroups of $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ ,      68
Lattice of subgroups of $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$       72
Lattice of subgroups of $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$       72
Lattice of subgroups of $\mathbb{Z}/4\mathbb{Z}$       66
Lattice of subgroups of $\mathbb{Z}/6\mathbb{Z}$       67
Lattice of subgroups of $\mathbb{Z}/8\mathbb{Z}$       67
Lattice of subgroups of $\mathbb{Z}/p^n\mathbb{Z}$       67
Lattice of subgroups of Klein 4-group      68
Lattice of subgroups of modular group      72
Lattice of subgroups of quasidihedral group      493
Laurent series      238
Least common multiple      4 278 288
Least residue      9
Left regular representation      121
Length, of cycle      29
Lexicographic ordering      536
Linear equations, solving      351
Linear functionals      357
Linear independence of characters      482
Linear representation      571
Linear transformation      323 335
Linearly independent      336 353 370
Local ring      260
localization      266 267
Lower central series      196
Luroth’s theorem      560
Maschke’s Theorem      580 586
Matrix representation      571
Matrix ring      235
Matrix, of a linear transformation      341ff
Maximal ideal      253ff 256 258 280
Maximal real subfield, of cyclotomic field      517
Maximal subgroup      64 118 190
Maximal subgroups, of solvable groups      202
Minimal normal subgroup      202
Minimal polynomial      385
Minimal polynomial of a field element      433
Minimum condition      591
Minor      364
Mobius inversion formula      502
Modular arithmetic      9
Modular group      72 87 101 188
Modular representations      577
Module      315
Module over $\mathbb{Z}$       317 367
Module over F[x]      317ff 367
Module over group ring      574
Monic      234
Monomial      234 293
Monster simple group      600 615
Multilinear      361
Multiple root, of polynomial      306 307
Multiple roots      458
Multiplicativity of extension degrees      435
Multiplicity, of root      458
n-cycle      29 117
n-cycle, holomorph of      188
Natural projection      83 243 325
Newton’s Formulas      533
Nilpotence class      192
Nilpotent element      231 249
Nilpotent group      192
Nilpotent ideal      250
Nilpotent matrix      413
Nilradical      249 258 267
Noetherian modules      369 380
Noetherian ring      288
Nonsingular      340 343
Norm      232 271 496
Norm of a character      608
Norm of element in a field      496
Normal extension      449 564 565
Normal subgroup      82
Normalize      94
normalizer      49
Null space      340
Odd permutation      108
Orbit      44 116 612
Order of a set      1
Order of an element      21
Order of permutation      31
Ordered basis      336
Outer automorphism      138
p-extensions      510
p-group      140 190ff
p-groups, characters of group of order       621
p-groups, representations of      585 599
P.I.D.      see “Principal Ideal Domain”
Partition      3 127 162
Pell’s equation      288
Perfect fields      462 463
Perfect group      176
Periods, in cyclotomic fields      512 518
Permutation      3 28
Permutation character      602 608 609 633
Permutation groups      117
Permutation matrix      157
Permutation representation      41 113 115
Pivotal element      350
Polynomial ring      234ff 291ff
Polynomials with , as Galois group      555
Power series      238 257 289
Power series of matrices      414
Power, of ideal      250
Preimage      2
Presentation      27 220
Primary component      144 333 334 376
Primary decomposition theorem      162 376
Primary ideal      260 294
Prime element      282ff
Prime factorization      283
Prime ideal      254 267 280
Prime spectrum      260
Prime subfield      266 423
primitive      118
Primitive element      429
Primitive Element Theorem      509
Primitive roots of unity      452 465
Principal character      601
Principal ideal      250 273
Principal ideal domain      279ff 285 294 328 333 334 367ff 586 589
Principal Ideal Domain, that is not Euclidean      280

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