Abhyankar S.S. — Lectures on Algebra Volume 1 :: Электронная библиотека попечительского совета мехмата МГУ

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Abhyankar S.S. — Lectures on Algebra Volume 1

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Название: Lectures on Algebra Volume 1

Автор: Abhyankar S.S.

Аннотация:

This book is a timely survey of much of the algebra developed during the last several centuries including its applications to algebraic geometry and its potential use in geometric modeling. The present volume makes an ideal textbook for an abstract algebra course, while the forthcoming sequel, "Lectures on Algebra II", will serve as a textbook for a linear algebra course. The author's fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the applications of group theory to the calculation of Galois groups is evident in the second volume which contains more local rings and more algebraic geometry. Both books are based on the author's lectures at Purdue University over the last few years.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 746

Добавлена в каталог: 28.03.2010

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Предметный указатель

Abel      2
Abelian      3 600
Abhyankar's proof of Newton's Theorem      98
Absolute algebraic closure      22
Absolute value      53—54
Absolutely saturated chain of prime ideals      250 615
Abstract disjoint union      683
action      651—656
Action (conjugation)      653
Action (faithful)      678
Action (transitive)      656 678
Action of a group on a set      651
Action of prime power group on a set      655
Additive abelian group      3 600
Additive abelian groups being modules over the ring of integers, direct sum theory applies to them      205
Adjoint of a homomorphism      446
Adjoint of a matrix      90 607
Admissibility Lemma      452—465
Admissible (pair of homomorphisms)      445
Aesop's Fables      16
Affine algebraic variety      149 611
Affine And (Relevant) Projective Varieties Relations Theorem      552
Affine And (Relevant) Projective Varieties Relations Theorem as related to Ideals And Homogeneous Ideals      594
Affine coordinate ring      104 147—149
Affine model      156 633
Affine Normalization Theorem      531 630
Affine portion (or portion at finite distance)      68 550
Affine ring      133 610
Affine semimodel      156 633
Affine varieties      146—151
Affine variety set      146—147
Algebra      1 29 596—597
Algebra (over a ring)      340 621
Algebra-theoretic (also called multiplicative) direct sum      340—341 621
Algebraic (extension of) over      12 603
Algebraic closure      22 46—48 161 603
Algebraic closure, absolute or relative      22
Algebraic element      12 603
Algebraic function field      515 630
Algebraic function field over a domain      530
Algebraic geometry      31 98—99
Algebraic independence      12 602
Algebraic points      152 548
Algebraically blowing-up simple center      559—566
Algebraically closed      32 603
Almost Isomorphism Theorem      465—479
Alternating group      5 25 600 672
Alternating sum of lengths      408—414
Alternative function field      147—149
Alternative homogeneous function field      542—543
Analytic branch      70
Analytic geometry      31 63—65 98
Analytic hypersurface      66 70
Analytic independence      38
Analytic Independence Theorem      241 614
Annihilator      112 217 608
Annihilator of nonprimary submodule      170
Annihilators and colons      112
Annihilators and direct sums      205
Annihilators and radicals      112
Annihilators of finitely generated modules      299
Another Ord Valuation Lemma      299 618
Antitransitive (permutation) group      662 678
Approximate root      58—59
Archimedean      53
Archimedeanly comparable      53 677
Arithmetic genus      398
Artin      7
Artin — Rees lemma      220—225 613
Artin — Schreier extensions      641
Artinian modules and rings      129—132 610
Artinian ring iff noetherian ring of dimension zero      131
Artinian: DCC $\Leftrightarrow$ MNC      129
Ascending chain condition      113
Askwith      63 98
Assassinator = colorful (like annihilator) long form of ass      117 216—217 609
Assigned value group      41
Associate      13 602
Associated graded ring      272 616
Associated graded ring in connection with a Corollary about regular local ring      277
Associated graded ring in connection with its triviality      587
Associated graded rings and leading ideals      585
Associated graded rings with their generalized versions      586
Associated primes      115—116 609
Associative or associativity      3 599
Associator      117 216—217 609
Auslander — Buchsbaum      328 339
Autoequivalent homomorphisms      445
Automorphism      7—9 599—602
Auxiliary Theorem (about complete or normal models or semimodels)      533 592
Axiom of Choice      33 44—52 603—605
Backward shift      434
Basic Finite Field Theorem      519
Basic Perfect Field Theorem      519
Basic Primitive Element Theorem      520
Basic Roots of Unity Theorem      517
Basis      8—9 602
Basis of an additive abelian group      365
Beejganit      1 596
Bell      65
bezout      31 66 98—102
Bhaskara      371 596
Bhaskaracharya (Bhaskara + Acharya)      1
Bijection      3 584 598
Bijective      3 584 598
Bilinear map      94—95
Binary relation      33 37
Binomial coefficients      290
Binomial theorem      96—97
Bivariate      17
Block (in the sense of permutation groups)      664
Block Lemma      664
Block matrices      90 172—173
Blowing-up primary ideals      553
Blowing-up simple center (geometrically or algebraically)      555—566
Blowup of a line in three space      595
Blowup of equimultiple simple center does not increase multiplicity (proof in Third Monoidal Theorem)      555—559 561 571 632
Blowup of points and lines      167—168
Blowup of singularities      160—161
Blowup Theorem (relative to Prime Ideal or Simple Center)      560—561
Bombay      98
Bracketed colon      118—119 609
Bracketed colon operation      224 578—579
Brim-full      375 623
Burnside      661
Burnside's Lemma      671
Burnside's theorem      661 671
Calculus      63—64
Cancellative      6 209 600
Canonical      7—8 600—602
Canonical basis      435 446
Canonical epimorphism (induced by homogeneous generators)      435
Canonical homomorphism (induced by homogeneous elements)      435
Canonical monomorphism      484
Canonical valuation      382
Cardano      2
Cardinal      33 50
Cardinality      33
Cartesian product      32 603
Cartesian product of a family      203
Castelnuovo      98
Catenarian      268 616
Catenarian Condition Lemma      270
Catenarian Domain Corollary      416
Catenarian Domain Theorem      271
Catenarian Ring Theorem      271 616
Cauchy multiplication      10—11 35—36
Cauchy relative to      94 608
Cauchy sequence      53—55 86—88 608
Cauchy's theorem      656

Cayley      98
Cayley and Burnside      661
Cayley's theorem      661
Center of a group      654
Center of a monoidal transformation      558
Center of a quadratic transformation      557
Center of a quasilocal ring      156 632—633
Centralize      654
centralizer      654
Centralizer Lemma      661
Chain in a poset      33
Characteristic      4 21 603
Characteristic function      169
Characteristic matrix      644
Characteristic polynomial      644
Characteristic subgroup      667
Characterization Theorem for Real Ranks      380 623
Characterization Theorem for Segment-Full Losets and Segment-Completions      377
Characterizations of PIDs, PIRs, SPIRs, and UFDs      348—357 621
Chevalley      98 597
Chicago World's Fair      4 61
Chinese Remainder Lemma      369
Chinese remainder theorem      365 371 622
Chord      63
Chrystal      43
circle      63 107 149
Circular and Fermat cones      161
Class equation      655
Classical group      62
Classification Theorem      5 28
Closure and closedness corollaries (of Krull Intersection Theorem) for ideals as well as modules      222 227
Codimension      414—415 627
Codimension of Product Theorem      418 627
Coefficient field      555
Coefficient set      555
Cofactor      90 607
Cofinal      46
Cohen      299 597
Cohen — Macaulay Lemma      291
Cohen — Macaulay module      281 618
Cohen — Macaulay ring      281 618
Cohen — Macaulay Rings (Conditions for)      360
Cohen — Macaulay Theorem      296 619
Cohen's Noetherian Theorem      229
Cohn's Two By Two Theorem (or Cohn's Example)      503 629
Collection of objects      3
College Algebra (of Rings and Ideals)      597
Colon and nonzerodivisor ideal corollary      218
Colons of ideals or modules      108—109 608
column      61
Column rank      90
Comaximal ideals      131—133 610
Common divisor      17—18
Common multiple      21—22
Commutative      3 6 600
Commutative Monoids Lemma      366
Commutative triangle      442
Commutator      26
Commutator subgroup      26
Comparison lemma      314
Complement      4 598
Complementary prospectral variety      534
Completable      484 629
Complete field      53
Complete Intersection Theorem      307 620
Complete intersection, (noetherian ring being called a local complete intersection)      301 619
Complete intersection, ideal-theoretic      281 618
Complete intersection, local ring theoretic      300 619
Complete model      156—159 633
Complete Normality Lemma      277 618
Complete ordered abelian group      53
Complete quasilocal ring      85—89 607
Complete relative to      94 608
Complete semimodel      156 633
Complete set of conjugates      657
Complete system of orthogonal idempotents      343
Completely normal domain      277 618
Completing the power      74
Completing the square      1 74
Completing the square method      74
Completing unimodular rows      483—497 629
Completing Unimodular Rows Lemma      486
Completing Unimodular Rows Theorem      487
Completing unimodular tuples      483—497 629
complex numbers      3 52—59 599
Components (= homogeneous components) of an element in a graded ring      206
composition      3 598
Composition of valuations      381—384 624 687
Composition series      124—127
Compositum      415 628
Condition for inseparable element      646
Conditions for a common factor or a common root of two univariate polynomials      165
Conditions for a loset to be segment-full      378
Conditions for DVR      359 588 622
Conductor      310 620
conic      63
Conjugacy classes      653
Conjugacy Lemma      660
conjugate      653
Conjugation action      653
Conjugation rule      660
Content of a polynomial      81 606
Contracted and extended ideals      142
Convergent sequence      53 85—89 608
Coordinate ring      146—149
Coprime      19
Coprincipal prime ideal (nonmaximal)      376 681
Core      375 623 681
Core-full      375 623
Coset      5 599
Countable      32
Counting properly      31
Cramer's rule      162—165 174—177
Cramer's rule for modules      220
Cremona      98
Criterion for Extensions of Derivations      649
Criterion for Separable Algebraic (Field) Extensions      650
Criterion of Separability (Jacobian)      651
Cubic curve      108
Cubic discriminant      103
Curve      30—32 104 149—150
Cusp      71 160—161
Cuspidal cubics      160—161
CYCLE      24
Cyclic extension      15 636
Cyclic extensions of degree equal to characteristic      641
Cyclic extensions of degree prime to characteristic      637
Cyclic group      4 599 641
Cyclic permutation      24
Cylinder      30
Decimal expansion      55
Decomposition of ideals and varieties      260 267 545—547
Decomposition of ideals and varieties (projective version)      545—547
Dedekind      364 597
Dedekind completion      375
Dedekind cut      57 374
Dedekind domain      364 622
Dedekind Domain Characterization and its Properties      366 369 622
Dedekind map (closed or open)      375
Dedekind map (of a pair of losets)      375
Degree form      93
Degree of a curve or surface      31
Degree of a homogeneous component      206
Degree of a homogeneous element      206
Degree of a hypersurface      31
Degree of a polynomial      10—11
Degree of a variety      31
Degree of inseparability      642
Degree of separability      642
Dehomogenization      198

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