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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
Determinant      248 435ff 450 488
Determinant computing      441
Determinant ideal      671
Diagonal subgroup      49 89
Diagonalizable matrices criterion      493 494
Dickson’s Lemma      334
Differential      723
Differential of a morphism      728
Dihedral group      23ff
Dihedral group, as Galois group      617ff
Dihedral group, characters of      881 885
Dihedral group, commutator subgroup of      171
Dihedral group, conjugacy classes in      132
Dimension of $\bigwedge^k(V)$      449
Dimension of $\mathcal S^k(V)$      446
Dimension of $\mathcal T^k(V)$      443
Dimension of a ring      750 754ff
Dimension of a tensor product      421
Dimension of a variety      681 729
Dimension of a vector space      408 411
Dimension shifting      802
Diophantine equations      14 245 276 278
Direct factor      455
Direct limit      268 358 741
Direct product infinite      157 357 414
Direct product of free modules      358
Direct product of groups      18 152ff 385 593
Direct product of injective modules      793
Direct product of injective resolutions      793
Direct product of modules      353 357 358 385
Direct product ofrings      231 233 265ff
Direct product, characters of      879
Direct sum of injective modules      403
Direct sum of modules      351ff 357 385
Direct sum of projective modules      392 403 793
Direct sum of projective resolutions      793
Direct sum of rings      232
Direct sum, infinite      158 357 414
Direct summand      373 385 451
Directed set      268
Dirichlet’s Theorem on Primes in Arithmetic Progressions      557
Discrete cohomology groups      808ff
Discrete G-module      808
Discrete valuation      232 238 272 755
Discrete valuation ring      232 272 755ff 762
Discriminant      610
Discriminant as resultant      621
Discriminant of $p^{th}$ cyclotomic polynomial      621
Discriminant of a cubic      612
Discriminant of a polynomial      610
Discriminant of a quadratic      611
Discriminant of aquartic      614
Distributive laws      34 223
divides      4 252 274
Divisibility of ideals      767
Divisible module      397
Divisible, group      66 86 167
Division algorithm      4 270 299
Division ring      224 225 834
Divisor      274
Domain      1
Double coset      117
Double dual      432 823 914
Doubling the Cube impossibility of      531ff
Doubly transitive      117 877
Dual basis      432
Dual group      167 815 823
Dual module      404 404
Dual numbers      729
Dual vector space      431
Echelon      425
Eigenspace      473
Eigenvalue      414 423 472
Eigenvector      414 423 472
Eisenstein’s Criterion      309ff 312
Elementary Abelian group      136 155 339 654
Elementary divisor      161ff 465ff
Elementary divisor, decomposition      161ff 464
Elementary divisor, decomposition algorithm      495
Elementary Jordan matrix      492
Elementary row and column operations      424 470ff 479ff
Elementary symmetric functions      607
Elimination ideal      328ff
Elimination theory      327ff
Elliptic function      600
Elliptic function field      653
Elliptic integral      14
Elliptic, curve      14
Embedded prime ideal      685
Embedding      83 359 569
Endomorphism      347
Endomorphism ring      347
Equivalence class      3 45 114
Equivalence of categories      734 916
Equivalence of short exact sequences      381
Equivalence relation      3 45 114
Equivalent extensions      381 787 824
Equivalent representations      846 869
Euclidean algorithm      5 271
Euclidean Domain      270ff 299
Euclidean Domain modules over      470 490
Euler $\varphi$-function      7 8 11 267 315 539ff 589
Euler’s theorem      13 96
Evaluation homomorphism      244 255 432ff
Exact sequence      378
Exact, functor      391 396
Exactness of tensor products      399
Exactness, of Horn      389ff 393ff
Exceptional characters      901
Exponent of a group      165ff 626
Exponential map      86
Exponential notation      20 22
Exponential of a matrix      503ff
Extension field      511ff
Extension of ideals      693 708ff
Extension of modules      378
Extension of scalars      359ff 363ff 369 373
Extension problem      104 378 776
Extension Theorem, for Isomorphisms of Fields      519 541
Extension, of a map      3 386 393
Exterior algebra      446
Exterior power      446
Exterior product      see “Wedge product”
External direct sum      353
External, direct product      172
F-algebra      see “Algebra”
Factor group      see “Quotient group”
Factor set      824ff
Factor through, homomorphism      100 365
Factorial variety      726
Factorization      283ff
Faithful, action      43 112ff.
Faithful, functor      914
Faithful, representation      840
Fano plane      210
Feit-Thompson Theorem      104 106 149 196 212
Fermat primes      601
Fermat’s Little Theorem      96
Fermat’s Theorem on sums of squares      291
Fiber      2 73ff 240ff
Fiber product of homomorphisms      407
Fiber sum of homomorphisms      407
Field      34 224 226 510ff
Field extension      511ff
Field generated by      511 516
Field norm      229
Field of fractions      260ff
Field of p-adic numbers      759
Field of rational functions      264 516 530 567 585 647ff 681 721
Finite covering      704
Finite dimensional      408 411
Finite extensions      512ff 521 526
Finite fields      34 301 529
Finite fields of four elements      516 653
Finite fields, algebraic closure of      588
Finite fields, existence and uniqueness of      549ff
Finite fields, Galois groups of      566 586
Finite fields, subfields of      588
Finite group      17
Finitely generated, field extension      524ff 646
Finitely generated, group      65 158 218ff
Finitely generated, ideal      251 317
Finitely generated, k-algebra      657
Finitely generated, module      351ff 458
Finitely presented, group      218ff
Finitely presented, module      795ff
First Order Diophantine Equation      276 278
First Orthogonality Relation      872
Fitting ideal      671
Fitting’s Lemma      668
Fixed field      560
Fixed point free      41 132
Fixed set      131 798
Fixed, element      558
Flat module      400ff 405ff 790 795
FORM      297
Formal Laurent series      238 265 756 759
Formal power series      238 258 265 668
Formally real fields      530
Fourier analysis      875ff
Fractional ideal      760ff
Fractional linear transformations      567 647
Frattini subgroup      198ff
Frattini’s Argument      193
Free generators      218
Free generators of a module      354
Free group      215ff
Free module      338 352 354ff 358 400
Free nilpotent group      221
Free rank      159 218 355 460 464
Free, abelian group      158 355
Frobenius automorphism      549 556 566 586 589 604
Frobenius group      168 638 643ff 896
Frobenius group as Galois group      638
Frobenius group, characters of      896
Frobenius kernel      896
Frobenius reciprocity      904
Full functor      914
Function      1
Function field      646 653
functor      391 396 398 913
Functor, contravariant      395 913
Functor, covariant      391 398 913
Fundamental matrix      506
Fundamental theorem of algebra      545 615ff
Fundamental Theorem of Arithmetic      6 289
Fundamental Theorem of Finitely Generated Abelian Groups      158ff 196 468
Fundamental Theorem of Finitely Generated Modules over a Dedekind Domain      769ff
Fundamental Theorem of Finitely Generated Modules over a P.I.D.      462 464 466
Fundamental Theorem of Galois Theory      574ff
Fundamental Theorem on Symmetric Functions      608
G-invariant      843
G-module      798
G-stable      843
Galois closure      594
Galois cohomology groups      809ff
Galois conjugates      573
Galois extension      562 572ff
Galois group      562ff 574ff
Galois group of $x^3-2$      564ff 568 576
Galois group of $x^4+1$      579
Galois group of $x^4-2x^3-2$      582
Galois group of $x^6-2x^3-2$      623 644
Galois group of $x^n-a$      636
Galois group of $x^p-x-a$      589
Galois group of $\mathbb F_{p^n}$      566 586
Galois group of $\mathbb Q(2^{1/8}, i)$ or $x^8-2$      577ff
Galois group of $\mathbb Q(2^{1/8}, i)$ over quadratic subfields      581
Galois group of $\mathbb Q(\sqrt{(2+\sqrt 2)(3+\sqrt 3)})$      584
Galois group of $\mathbb Q(\sqrt{(2+\sqrt 2)})$      582
Galois group of $\mathbb Q(\sqrt{2})$      563
Galois group of $\mathbb Q(\sqrt{2}, \sqrt 3)$      563ff 567 576
Galois group of $\mathbb Q(\sqrt{D_1}, \sqrt{D_2})$      582
Galois group of $\mathbb Q(\zeta_5)$      597
Galois group of $\mathbb Q(\zeta_n)$      596
Galois group of $\mathbb Q(\zeta_n+\zeta_n^{-1})$      601 603
Galois group of $\mathbb Q(\zeta_p)$      597
Galois group of $\mathbb Q(\zeta_{13})$      598ff
Galois group of a biquadratic      582
Galois group of a composite extension      592
Galois group of a cubic      612
Galois group of a cyclotomic field      599
Galois group of a general polynomial      609
Galois group of a quadratic      563
Galois group of a quartic      615 618
Galois groups infinite      651ff
Galois groups of polynomials      606ff
Galois groups over $\mathbb Q$      640ff
Galois theory      14 105 558ff
Gaschiitz’s Theorem      838
Gauss sum      637
Gauss — Jordan elimination      327 424ff
Gaussian integers      229ff 271 278 289ff 377
Gauss’ Lemma      303 530 819 824
General linear group      35 89 236 413 418
General polynomial      607 609 629 646
General polynomial division      320ff 331
Generalized associative law      18
Generalized character      898
Generalized eigenspace      501
Generalized quaternion group      178
Generating set      61ff
Generator      25ff 54 218ff
Generator of $S_n$      64 107ff
Generator of $S_p$      111
Generator of a cyclic group      57
Generator of a free module      354
Generator of a subgroup      61ff
Generator of a submodule      351
Generator of an ideal      251
Generic point      733
Germs of continuous functions      269
Global sections      740
Globally asymptotically stable      508
Going-down Theorem      694 728
Going-up Theorem      694 720
Graded ideal      443
Graded lexicographic ordering (grlex)      331
Graded, ordering      331
Graded, ring      443
Graph      210 669 687
Graph coloring      335ff
Greatest common divisor (g.c.d.)      4 252 274ff 287
Greatest common divisor of ideals      767
Grevlex monomial ordering      331
Groebner basis      315ff 319ff 664ff 702 712
Groebner basis in field extensions      672
Group      13 16ff
Group extensions      824ff
Group of $n^{th}$ roots of unity      see “Root of unity”
Group of units in a ring      226
Group ring      236ff 798 840
Group table      21
Groups of order $2p^2$      186
Groups of order $3^3$      7 13 409 212ff 898ff
Groups of order $p^2$      125 137
Groups of order $p^2q$      144
Groups of order $p^3$      179 183 198 199ff 886
Groups of order 12      144 182
Groups of order 147      185
1 2 3 4 5
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