Dummit D.S., Foote R.M. — Abstract algebra :: Электронная библиотека попечительского совета мехмата МГУ

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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.

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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

Предметный указатель

Groups of order 168      207ff
Groups of order 30      143 182
Groups of order 4p      186
Groups of order 56      185
Groups of order 60      145ff 186
Groups of order 75      185
Groups of order pq      143 179 181
Groups, table of small order      167
Hall subgroup      101 200 829 890
Hall’s theorem      105 196 890
Hamilton Quaternions      224ff 231 237 249 299
Harmonic analysis      875
Heisenberg group      35 53 174 179 187
Hilbert’s Nullstellensatz      675 700ff
Hilbert’s Zahlbericht      815
Holder Program      103ff
Holomorph      179 186
Hom of direct products      404
Hom of direct sums      388 388 404
Homeomorphism      738
Homogeneous cochains      810
Homogeneous component of a graded ring      443
Homogeneous component of a polynomial      297
Homogeneous ideal      299
Homogeneous of degree m      621
Homogeneous polynomial      297
Homological algebra      391 655 776ff
Homology groups      777
Homomorphism of algebras      343 657
Homomorphism of complexes      777
Homomorphism of fields      253 512
Homomorphism of graded rings      443
Homomorphism of groups      36 73ff 215
Homomorphism of modules      345ff
Homomorphism of rings      239ff
Homomorphism of short exact sequences      381ff
Homomorphism of tensor algebras      450
Homotopic      792
Hubert’s Basis Theorem      316 334 657
Hubert’s Specialization Theorem      648
Hubert’s Theorem      90 583 814
Hubert’s Theorem, additive form      584 815
Hypernilpotent group      191
Hypersurface      659
icosahedron      see “Platonic solids”
Ideal      242ff
Ideal generated by set      251
Ideal quotient      333 691
Idempotent      267 856
Idempotent linear transformation      423
Identity matrix      236
Identity of a group      17
Identity of a ring      223
Image of a k-algebra homomorphism, computing      665ff
Image of a linear transformation, computing      429
Image of a map      2
Implicitization      678
Incidence relation      210
Indecomposable module      847
Independence of characters      569 872
Independent transcendentals      645
Index of a field extension      512
Index of a subgroup      90ff
Induced, character      892ff 898
Induced, module      363 803 811 812 893
Induced, representation      893
Inductive limit      see “Direct limit”
Inequivalent extensions      379ff
Inert prime      749 775
Infinite cyclic group      57 811
Infinite Galois groups      651ff
Inflation homomorphism      806
Inhomogeneous cochains      810
Injective envelope      see “Injective hull”
Injective hull      398 405 405
Injective map      2
Injective module      395ff 403ff 784
Injective resolution      786
Injectively equivalent      407
Inner automorphism      134
Inner product of characters      870ff
Inseparable degree of a field extension      650
Inseparable degree of a polynomial      550
Inseparable extension      551 566
Inseparable polynomial      546
Insolvability of the quintic.      625 629
INTEGER      1 695ff
Integers mod n      see “ $\mathbb Z/n\mathbb Z$ ”
Integral basis      698 775
Integral closure      229 691ff
Integral domain      228 235
Integral element      691
Integral extension      691ff
Integral group ring $(\mathbb ZG)$       237 798
Integral ideal      760
Integral Quaternions      229
Integrally closed      691ff
Internal direct sum      354
Internal, direct product      172
Intersection of ideals, computing      330ff
Intertwine      847
Invariant factor      159ff 464 774
Invariant factor decomposition      159ff 462ff
Invariant Factor Decomposition Algorithm      480
Invariant factor of a matrix      475 477
Invariant subspace      341 843
Inverse image      2
Inverse limit      268 358 652ff
Inverse of a fractional ideal      760
Inverse of a map      2
Inverse of an element in a group      17
Inverse of matrices      427 440
Invertible fractional ideal      760
Irreducibility of a cyclotomic polynomial      310
Irreducibility, criteria      307ff
Irreducible algebraic set      679
Irreducible character      866 870 873
Irreducible element      284
Irreducible element in $\mathbb Z[i]$       289ff
Irreducible ideal      683
Irreducible module      356 847
Irreducible polynomial      287 512ff 572
Irreducible polynomial of degree n over $\mathbb F_p$       301 586
Irreducible topological space      733
Isolated prime ideal      685
Isomorphism of algebras      343
Isomorphism of cyclic groups      56
Isomorphism of groups      37
Isomorphism of modules      345
Isomorphism of rings      239
Isomorphism of short exact sequences      381
Isomorphism of vector spaces      408
Isomorphism Theorems for groups      97ff
Isomorphism Theorems for modules      349
Isomorphism Theorems for rings      243 246
Isomorphism type      37
Isomorphism, classes      37
Isotypic component      869
Jacobson radical      259 750
Join      67 88
Jordan block      492
Jordan canonical form      457 472 492ff
Jordan-Holder Theorem      103ff
k-stage Euclidean Domains      294
k-tensors      442
Kernel of a k-algebra homomorphism      678
Kernel of a k-algebrahomomorphism, computing      665ff
Kernel of a linear transformation, computing      429
Kernel of ahomomorphism      40 75 239 345
Kernel, of agroup action      43 51 112ff

Klein 4-group (Viergruppe)      68 136 155
Kronecker product      421ff 431
Kronecker — Weber Theorem      600
Krull dimension      704 750ff 754
Krull topology      652
Krull’s Theorem      652
Kummer extensions      627 817
Kummer generators for cyclic extensions      636
Kummer theory      626 816 823
Lagrange resolvent      626
Lagrange’s Theorem      13 45 89ff 460
Lattice of subfields      574
Lattice of subfields of $\mathbb Q(2^{1/8}, i)$       581
Lattice of subfields of $\mathbb Q(\sqrt[3]{2}, \rho)$       568
Lattice of subfields of $\mathbb Q(\zeta_{13})$       598
Lattice of subgroups      66ff
Lattice of subgroups for quotient group      98ff
Lattice of subgroups of       111
Lattice of subgroups of       69 99
Lattice of subgroups of $D_{16}$       70
Lattice of subgroups of $QD_{16}$       72 580
Lattice of subgroups of       69 99
Lattice of subgroups of       69
Lattice of subgroups of $\mathbb Z/12\mathbb Z$       68
Lattice of subgroups of $\mathbb Z/2\mathbb Z$       67
Lattice of subgroups of $\mathbb Z/2\mathbb Z\times\mathbb Z/2\mathbb Z$       71
Lattice of subgroups of $\mathbb Z/2\mathbb Z\times\mathbb Z/2\mathbb Z$ (Klein 4-group)      68
Lattice of subgroups of $\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z$       72
Lattice of subgroups of $\mathbb Z/4\mathbb Z$       67
Lattice of subgroups of $\mathbb Z/6\mathbb Z$       68
Lattice of subgroups of $\mathbb Z/8\mathbb Z$       67
Lattice of subgroups of $\mathbb Z/n\mathbb Z$       67
Lattice of subgroups of $\mathbb Z/p^n\mathbb Z$       68
Lattice of subgroups of the modular group of order      16 72
Laurent series      see “Formal Laurent series”
Leading coefficient      234 295
Leading term      234 295 318
Leading term ideal of      318ff
Least common multiple (l.c.m.)      4 279 293
Least residue      9
Left derived functor      788
Left exact      391 395 402
Left group action      43
Left ideal      242 251 256
Left inverse in a ring      233
Left inverse of a map      2
Left module      337
Left multiplication      44 118ff 531
Left Principal Ideal Domain      302
Left regular representation      44 120
Left translation      44
Left zero divisor      233
Legendre symbol      818
Length of a cycle      30
Lexicographic monomial ordering      317ff 622
Lie groups      505 876
Lifts      386
Linear algebraic sets      659
Linear character      569
linear combination      5 275 280 408
Linear equations, solving      425ff
Linear functional      431
Linear representation      840
Linear transformation      34Qff 346 408
Linearly independent, characters      569 872
Linearly independent, vectors      409
Local homomorphism      723 744
Local ring      259 717 752ff 755
Local ring of an afflne variety      721ff
localization      106ff 795 796
Localization at a point in a variety      722
Localization at a prime      708ff 718
Localization of a module      714ff
Locally ringed spaces      745
Locus      659
Long Exact Sequence      778 789
Long Exact Sequence in Group Cohomology      802
Lower central series      193
Lueroth’s Theorem      647
MAP      1 215
Maschke’s Theorem      453 849
Matrix      34 235 415ff
Matrix of a composition      418
Matrix of a linear transformation      415ff
Matrix representation      840
Matrix ring      235ff 418
Matrix ring ideals of      249
Maximal ideal      253ff 280 512
Maximal order      232
Maximal real subfield of a cyclotomic field      603
Maximal spectrum      731
Maximal spectrum of $\mathbb Z[i]$       735
Maximal spectrum of $\mathbb Z[x]$       736
Maximal spectrum of k[x, y]      735
Maximal spectrum of k[x]      735
Maximal subgroup      65 117 131 188 198
Maximal subgroup of solvable groups      200
Middle linear map      see “Balanced map”
Minimal element      4
Minimal Grobner basis      325ff
Minimal normal subgroup      200
Minimal polynomial      474
Minimal polynomial of a field element      520
Minimal polynomial of a field element, computing      667
Minimal primary decomposition      683
Minimal prime ideal      298 688
Minimum condition      855
Minkowski’s Criterion      441
Minor      439
Mobius inversion formula      555 588
Modular arithmetic      9 224
Modular group of order      16 72 186
Modular representations      846
Module      337ff
Module of fractions      714
Module over $\mathbb Z$       339 456ff
Module over a Dedekind Domain      769ff
Module over a group ring      798ff 843ff
Module over a P.I.D.      456ff
Module over F[x]      340ff 456ff
Module sheaf of      748
Monic      234
Monomial      297
Monomial ideal      318 332 334
Monomial ordering      317
Monomial part      297
Monomial term      297
Monster simple group      865
Morphism      911
Morphism of affine algebraic sets      662
Morphism of affine schemes      743
Multidegree      297 318
Multilinear form      435
Multilinear map      372 435
Multiple      252 274
Multiple roots of a polynomial      312 545 547
Multiplicative field norm      230 582
Multiplicative function      7 267
Multiplicative subgroup of a field      314
Multiplicativity of extension degrees      523 529
Multiplicity of a root      313 545
Nakayama’s Lemma      751
Natural      83 167 432 911ff
Natural projection      83 243 348 916
Newton’s Formulas      618
Nilpotence class      190
Nilpotent group      190—198
Nilpotent ideal      251 258 674
Nilpotent matrix      502
Nilpotent, element      231 250 596 689

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