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E-raamat: Advanced Modern Algebra

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Advanced Modern AlgebraSuurem pilt
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Part One--this volume--can be used as a text for the first year of graduate algebra, says Rotman, but it and the forthcoming Part Two can also serve more advanced graduate students wishing to learn topics on their own. They comprise a reference as well, he says, containing many of the standard theorems and definitions that users of algebra need to know. Part One contains material for two courses. The first course covers classical formulas, classical number theory, commutative rings, groups, and Galois theory. The second course covers modules, Zorn's lemma, advanced linear algebra, categories of modules, multilinear algebra, and more commutative algebra. Annotation ©2016 Ringgold, Inc., Portland, OR (protoview.com)
Preface to Third Edition: Part 1 xi
Acknowledgments xiv
Part A Course I
Chapter A-1 Classical Formulas
3(6)
Cubics
4(2)
Quartics
6(3)
Chapter A-2 Classical Number Theory
9(20)
Divisibility
9(7)
Euclidean Algorithms
16(3)
Congruence
19(10)
Chapter A-3 Commutative Rings
29(86)
Polynomials
41(6)
Homomorphisms
47(8)
Quotient Rings
55(7)
From Arithmetic to Polynomials
62(12)
Maximal Ideals and Prime Ideals
74(9)
Finite Fields
83(6)
Irreducibility
89(8)
Euclidean Rings and Principal Ideal Domains
97(7)
Unique Factorization Domains
104(11)
Chapter A-4 Groups
115(64)
Permutations
116(7)
Even and Odd
123(4)
Groups
127(12)
Lagrange's Theorem
139(11)
Homomorphisms
150(9)
Quotient Groups
159(14)
Simple Groups
173(6)
Chapter A-5 Galois Theory
179(56)
Insolvability of the Quintic
179(8)
Classical Formulas and Solvability by Radicals
187(3)
Translation into Group Theory
190(10)
Fundamental Theorem of Galois Theory
200(23)
Calculations of Galois Groups
223(12)
Chapter A-6 Appendix: Set Theory
235(12)
Equivalence Relations
243(4)
Chapter A-7 Appendix: Linear Algebra
247(26)
Vector Spaces
247(12)
Linear Transformations and Matrices
259(14)
Part B Course II
Chapter B-1 Modules
273(40)
Noncommutative Rings
273(9)
Chain Conditions on Rings
282(6)
Left and Right Modules
288(12)
Chain Conditions on Modules
300(5)
Exact Sequences
305(8)
Chapter B-2 Zorn's Lemma
313(46)
Zorn, Choice, and Well-Ordering
313(6)
Zorn and Linear Algebra
319(4)
Zorn and Free Abelian Groups
323(11)
Semisimple Modules and Rings
334(5)
Algebraic Closure
339(6)
Transcendence
345(8)
Luroth's Theorem
353(6)
Chapter B-3 Advanced Linear Algebra
359(82)
Torsion and Torsion-free
359(3)
Basis Theorem
362(9)
Fundamental Theorem
371(1)
Elementary Divisors
371(3)
Invariant Factors
374(4)
From Abelian Groups to Modules
378(5)
Rational Canonical Forms
383(5)
Eigenvalues
388(7)
Jordan Canonical Forms
395(7)
Smith Normal Forms
402(15)
Inner Product Spaces
417(12)
Orthogonal and Symplectic Groups
429(7)
Hermitian Forms and Unitary Groups
436(5)
Chapter B-4 Categories of Modules
441(102)
Categories
441(20)
Functors
461(14)
Galois Theory for Infinite Extensions
475(6)
Free and Projective Modules
481(11)
Injective Modules
492(9)
Divisible Abelian Groups
501(8)
Tensor Products
509(13)
Adjoint Isomorphisms
522(7)
Flat Modules
529(14)
Chapter B-5 Multilinear Algebra
543(48)
Algebras and Graded Algebras
543(9)
Tensor Algebra
552(9)
Exterior Algebra
561(5)
Grassmann Algebras
566(7)
Exterior Algebra and Differential Forms
573(2)
Determinants
575(16)
Chapter B-6 Commutative Algebra II
591(60)
Old-Fashioned Algebraic Geometry
591(2)
Affine Varieties and Ideals
593(6)
Nullstellensatz
599(5)
Nullstellensatz Redux
604(10)
Irreducible Varieties
614(9)
Affine Morphisms
623(5)
Algorithms in k[ x1, . . ., xn]
628(1)
Monomial Orders
629(7)
Division Algorithm
636(3)
Grobner Bases
639(12)
Chapter B-7 Appendix: Categorical Limits
651(22)
Inverse Limits
651(6)
Direct Limits
657(2)
Directed Index Sets
659(7)
Adjoint Functors
666(7)
Chapter B-8 Appendix: Topological Spaces
673(8)
Topological Groups
678(3)
Bibliography 681(6)
Special Notation 687(6)
Index 693
Joseph J. Rotman, University of Illinois at Urbana-Champaign, IL, USA.
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