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



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Название: Abstract algebra

Авторы: Dummit D.S., Foote R.M.

Аннотация:

The principal change from the second edition is the addition of Grobner bases to this edition. The basic theory is introduced in a new Section 9.6. Applications to solving systems of polynomial equations (elimination theory) appear at the end of this section, rounding it out as a self-contained foundation in the topic. Additional applications and examples are then woven into the treatment of affine algebraic sets and Jt-algebra homo-morphisms in Chapter 15. Although the theory in the latter chapter remains independent of Grobner bases, the new applications, examples and computational techniques significantly enhance the development, and we recommend that Section 9.6 be read either as a segue to or in parallel with Chapter 15. A wealth of exercises involving Grobner bases, both computational and theoretical in nature, have been added in Section 9.6 and Chapter 15. Preliminary exercises on Grobner bases can (and should, as an aid to understanding the algorithms) be done by hand, but more extensive computations, and in particular most of the use of Grobner bases in the exercises in Chapter 15, will likely require computer assisted computation.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 3rd edition

Год издания: 2004

Количество страниц: 932

Добавлена в каталог: 07.02.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Nilradical      250 258 673 674
Noetherian ring      316 458 656ff 793
Noetherian, module      458 469
Noether’s Normalization Lemma      699ff
Noncommutative polynomial algebra      302 443
Nonflnitely generated ideal      298 657
Nongenerator      199
Nonpivotal      425
Nonprincipal ideal      252 273 298
Nonsimple field extension      595
Nonsingular curve      775
Nonsingular model      726
Nonsingular variety      725
Nonsingular, linear transformation      413
Nonsingular, matrix      417
Nonsingular, point      725 742 763
Norm      232 270 299
Norm of a character      872
Norm of an element in a field      582 585
Normal basis      815
Normal complement      385
Normal extension      537 650
Normal ring      691
Normal subgroup      82ff
Normal variety      726
Normalization      691 726
Normalize      82 94
Normalized factor set      825
Normalized section      825
Normalized, cocycle      827
normalizer      50ff 123ff 134 147 206ff
Null space      413
Nullity      413
Number fields      696
Object      911
Opposite algebra      834
Orbit      45 115ff 877
Order of a set      1
Order of an element in a group      20 55 57 90
Order of conductor      232
Order of zero or pole      756 763
Order, of a permutation      32
Ordered basis      409
Orthogonal characters      872
Orthogonal idempotents      377 856 870
Orthogonality relations      872
Outer automorphism group      137
p-adic integers      269 652 758ff
p-adic Laurent series      759
p-adic valuation      759
p-extensions      596 638
p-group      139 188
p-group, characters of      886
p-group, representations of      854 864
p-primary component      142 358 465
P.I.D.      see “Principal Ideal Domain”
Parabolic subgroup      212
Partition of n      126 162
Partition, of a set      3
Pell’s equation      230
Perfect field      549
Perfect group      174
Periods in cyclotomic fields      598 602 604
Permutation      3 29 42
Permutation character      866 877 895
Permutation even      108ff
Permutation group      116 120
Permutation matrix      157
Permutation module      803
Permutation odd      108ff
Permutation representation      43 112ff 203ff 840 844 852 877
Permutation sign of      108ff 436ff
Pivotal element      425
Platonic solids, symmetries of      28 45 92 111 148
Pole      756
Polynomial      234
Polynomial map      299 662
Polynomial ring      234ff 295ff
Polynomials with Sn as Galois group      642ff
Pontriagin dual group      787
Positive norm      270
Postage Stamp Problem      278
Power of an ideal      247
Power series of matrices      502ff
Power set      232
Preimage      2
Presentation      26ff 39 218ff 380
Primary component      see “p-primary component”
Primary Decomposition Theorem for ideals      681ff 716ff
Primary Decomposition Theorem for modules      357 465 772
Primary Decomposition Theorem, for abelian groups      161
Primary ideal      260 298 748
Prime      6
Prime element in a ring      284
Prime factorization      6
Prime factorization for ideals      765ff
Prime ideal      255ff 280 674
Prime ideal algorithm for determining      710ff
Prime spectrum      731ff
Prime subfield      264 511 558
Primes associated to a module      670
Primes associated to an ideal      670
Primitive central idempotent      856 870
Primitive element      517 594
Primitive Element Theorem      595
Primitive idempotent      856
Primitive permutation group      117
Primitive roots of unity      539ff
Principal character      866
Principal crossed homomorphisms      814
Principal fractional ideal      760
Principal ideal      251
Principal Ideal Domain (P.I.D.)      279ff 284 459
Principal Ideal Domain characterization of      281 289 294
Principal Ideal Domain that is not Euclidean      282
Principal open set      687 738
Product of ideals      247 250
Product of subgroups      93ff
Proflnite      809 813
Projection      83 423 453
Projection homomorphism      153ff
Projections of algebraic sets      679
Projective limit      see “Inverse limit”
Projective module      390ff 400 403ff 761 773 786
Projective plane      210
Projective resolution      779
Projectively equivalent      407
Public Key Code      279
Pullback of a homomorphism      407
Purely inseparable      649
Purely transcendental      646
Pushout of a homomorphism      407
Pythagoras’ equation rational solutions      584
Quadratic integer rings      229ff 248 271 278 286 293ff 698 749
Quadratic integer rings that are Euclidean      278
Quadratic integer rings that are P.I.D.s      278
Quadratic reciprocity law      819
Quadratic residue symbol      818
Quadratic, equation      522 533
Quadratic, extensions      522 533
Quadratic, field      227 698
Quadratic, subfield of cyclic quartic fields, criterion      638
Quadratic, subfleld of $\mathbb Q(\zeta_p)$      621 637
Quartic equations, formulas for roots      634ff
Quasicompact      688 738 746
Quasidihedral group      71ff 186
Quasidihedral group, as Galois group      579
Quaternion group      36
Quaternion group, as Galois group      584
Quaternion group, characters of      882
Quaternion group, generalized      178
Quaternion group, representations of      845 852
Quaternion ring      224 229 258
Quintic, insolvability      625 629
Quotient field      260ff
Quotient, computations in k-algebras      672
Quotient, group      15 73ff 76 574
Quotient, module      348
Quotient, ring      2410:
Quotient, vector space      408 412
Radical extension      625ff
Radical ideal      258 673 689
Radical of a zero-dimensional ideal      706ff
Radical of an ideal      258 673ff 701
Radical of an ideal computing      701
Radicals      625
Ramified prime      749 775
RANGE      2
Rank of a free module      338 354 356 358 459
Rank of a group      165 218 355
Rank of a linear transformation      413
Rank of amodule      460 468 469 471 719 773
Rational canonical form      457 472ff
Rational canonical form, computing      481ff
Rational functions      see “Field of rational functions”
Rational group ring      237
Rational numbers      1 260
Rational valued characters      879
Real numbers      1
Real numbers modulo      1 21 86
Reciprocity      229 621
Recognition theorem      171 180
Reduced Grobner basis      326ff
Reduced row echelon form      425
Reduced word      216ff
Reducible character      866
Reducible element      284
Reducible module      847
Reduction homomorphism      245 296 300 586
Reduction mod n      10 243 296 640
Reduction of polynomials mod p      586 589
Reflexive      3
Regular at a point      721
Regular local ring      725 755
Regular map      662 722
Regular representation      844 862ff
Relations      25ff 218ff 380
Relations matrix      470
Relative Brauer group      836
Relative degree of a field extension      512
Relative integral basis      775
Relatively prime      4 282
Remainder      5 270 320ff
Replacement Theorem      410 645
Representation      840ff
Representation permutation      43 112ff 203ff 840 844 852 877
Representative      3 9 77
Residue class      8
Resolvent cubic      614 623
Resolvent polynomials      642
Restricted direct product      158
Restriction maps      269 740
Restriction of scalars      359
Restricttion homomorphism      269 805 807
Resultant      619ff
Reverse of a polynomial      312
Right derived functor      785
Right Euclidean Domain      302
Right exact      400 402
Right group action      43 128 844 852
Right ideal      242 251
Right inverse in a ring      233
Right inverse of a map      2
Right module      337
Right regular representation      132
Right zero divisor      233
Ring      223
Ring of algebraic integers      695ff
Ring of continuous functions      225 227 259
Ring of dual numbers      729
Ring of fractions      260ff 708
Ring of integers      229
Ring of sets      232
Root      310 521
Root extension      627
Root of a polynomial      307ff 512
Root of unity      22 66 86 539ff 552
Row equivalent      425
Row rank      418 427 434
Row reduced      424
Ruler and compass constructions      534
Saturated      710
Saturation of an ideal      710ff
scalar      408
Scalar matrix      236
Scalar transformations      348
Schanuel’s Lemma      407
Scheme      745
Schur multiplier      838
Schur’s lemma      356 853 856
Schur’s Theorem      829
Second dual      see “Double dual”
Second Orthogonality Relation      872
Section      384 740
Semidihedral group      see “Quasidihedral group”
Semidirect product      175ff 383 385 821 829
Semisimple      855
Separable      551
Separable degree of a field extension      650
Separable degree of a polynomial      550
Separable extension      551 572 594ff
Separable polynomial      546 562 572
Separating transcendence base      650
Shapiro’s Lemma      804
Short exact sequence      379
Short exact sequence of complexes      778
Short Five Lemma      383
Similar central simple algebras      835
Similar matrices      419 476 493ff
Similar representations      846
Similar, linear transformations      419 476
Similarity      40
Simple algebra      832
Simple extensions      517 586 594
Simple group      91—92ff 149ff 201ff 212
Simple group classification of      103 212
Simple group of order      168 207ff
Simple group sporadic      104 865
Simple module      see “Irreducible module”
Simple radical extension      625
Simple ring      253 863
Simple tensor      360
Simultaneous Resolution      783
Singular point      725
Skew field      see “Division ring”
Skew-symmetrization      452
Smith normal form      479
Smooth      725 742
Snake lemma      792
Solution of cubic equations      630
Solution of quartic equations      634ff
Solvability of a quintic, criterion      630 639
Solvability of groups of odd order      see “Feit — Thompson Theorem”
Solvable by radicals      627ff
Solvable extensions      625ff
Solvable group      105 149 196ff 628 886 890
Solvable length      195ff
Solving algebraic equations      327ff
Solving linear equations      425ff
span      62 351 408 427
1 2 3 4 5
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