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This book is designed as a text for a first-year graduate algebra course. The choice of topics is guided by the underlying theme of modules as a basic unifying concept in mathematics. The book emphasizes proofs with a maximum of insight and a minimum of computation in order to promote understanding. However, extensive material on computation (for example, computation of canonical forms) is provided.
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This book is designed as a text for a first-year graduate algebra course. The choice of topics is guided by the underlying theme of modules as a basic unifying concept in mathematics. The book emphasizes proofs with a maximum of insight and a minimum of computation in order to promote understanding. However, extensive material on computation (for example, computation of canonical forms) is provided.
Dieser Download kann aus rechtlichen Gründen nur mit Rechnungsadresse in A, B, BG, CY, CZ, D, DK, EW, E, FIN, F, GR, HR, H, IRL, I, LT, L, LR, M, NL, PL, P, R, S, SLO, SK ausgeliefert werden.
Produktdetails
- Produktdetails
- Verlag: Springer US
- Seitenzahl: 526
- Erscheinungstermin: 6. Dezember 2012
- Englisch
- ISBN-13: 9781461209232
- Artikelnr.: 44044456
- Verlag: Springer US
- Seitenzahl: 526
- Erscheinungstermin: 6. Dezember 2012
- Englisch
- ISBN-13: 9781461209232
- Artikelnr.: 44044456
- Herstellerkennzeichnung Die Herstellerinformationen sind derzeit nicht verfügbar.
1 Groups.- 1.1 Definitions and Examples.- 1.2 Subgroups and Cosets.- 1.3 Normal Subgroups, Isomorphism Theorems, and Automorphism Groups.- 1.4 Permutation Representations and the Sylow Theorems.- 1.5 The Symmetric Group and Symmetry Groups.- 1.6 Direct and Semidirect Products.- 1.7 Groups of Low Order.- 1.8 Exercises.- 2 Rings.- 2.1 Definitions and Examples.- 2.2 Ideals, Quotient Rings, and Isomorphism Theorems.- 2.3 Quotient Fields and Localization.- 2.4 Polynomial Rings.- 2.5 Principal Ideal Domains and Euclidean Domains.- 2.6 Unique Factorization Domains.- 2.7 Exercises.- 3 Modules and Vector Spaces.- 3.1 Definitions and Examples.- 3.2 Submodules and Quotient Modules.- 3.3 Direct Sums, Exact Sequences, and Horn.- 3.4 Free Modules.- 3.5 Projective Modules.- 3.6 Free Modules over a PID.- 3.7 Finitely Generated Modules over PIDs.- 3.8 Complemented Submodules.- 3.9 Exercises.- 4 Linear Algebra.- 4.1 Matrix Algebra.- 4.2 Determinants and Linear Equations.- 4.3 Matrix Representation of Homomorphisms.- 4.4 Canonical Form Theory.- 4.5 Computational Examples.- 4.6 Inner Product Spaces and Normal Linear Transformations.- 4.7 Exercises.- 5 Matrices over PIDs.- 5.1 Equivalence and Similarity.- 5.2 Hermite Normal Form.- 5.3 Smith Normal Form.- 5.4 Computational Examples.- 5.5 A Rank Criterion for Similarity.- 5.6 Exercises.- 6 Bilinear and Quadratic Forms.- 6.1 Duality.- 6.2 Bilinear and Sesquilinear Forms.- 6.3 Quadratic Forms.- 6.4 Exercises.- 7 Topics in Module Theory.- 7.1 Simple and Semisimple Rings and Modules.- 7.2 Multilinear Algebra.- 7.3 Exercises.- 8 Group Representations.- 8.1 Examples and General Results.- 8.2 Representations of Abelian Groups.- 8.3 Decomposition of the Regular Representation.- 8.4 Characters.- 8.5 Induced Representations.- 8.6 Permutation Representations.- 8.7 Concluding Remarks.- 8.8 Exercises.- Index of Notation.- Index of Terminology.
1 Groups.- 1.1 Definitions and Examples.- 1.2 Subgroups and Cosets.- 1.3 Normal Subgroups, Isomorphism Theorems, and Automorphism Groups.- 1.4 Permutation Representations and the Sylow Theorems.- 1.5 The Symmetric Group and Symmetry Groups.- 1.6 Direct and Semidirect Products.- 1.7 Groups of Low Order.- 1.8 Exercises.- 2 Rings.- 2.1 Definitions and Examples.- 2.2 Ideals, Quotient Rings, and Isomorphism Theorems.- 2.3 Quotient Fields and Localization.- 2.4 Polynomial Rings.- 2.5 Principal Ideal Domains and Euclidean Domains.- 2.6 Unique Factorization Domains.- 2.7 Exercises.- 3 Modules and Vector Spaces.- 3.1 Definitions and Examples.- 3.2 Submodules and Quotient Modules.- 3.3 Direct Sums, Exact Sequences, and Horn.- 3.4 Free Modules.- 3.5 Projective Modules.- 3.6 Free Modules over a PID.- 3.7 Finitely Generated Modules over PIDs.- 3.8 Complemented Submodules.- 3.9 Exercises.- 4 Linear Algebra.- 4.1 Matrix Algebra.- 4.2 Determinants and Linear Equations.- 4.3 Matrix Representation of Homomorphisms.- 4.4 Canonical Form Theory.- 4.5 Computational Examples.- 4.6 Inner Product Spaces and Normal Linear Transformations.- 4.7 Exercises.- 5 Matrices over PIDs.- 5.1 Equivalence and Similarity.- 5.2 Hermite Normal Form.- 5.3 Smith Normal Form.- 5.4 Computational Examples.- 5.5 A Rank Criterion for Similarity.- 5.6 Exercises.- 6 Bilinear and Quadratic Forms.- 6.1 Duality.- 6.2 Bilinear and Sesquilinear Forms.- 6.3 Quadratic Forms.- 6.4 Exercises.- 7 Topics in Module Theory.- 7.1 Simple and Semisimple Rings and Modules.- 7.2 Multilinear Algebra.- 7.3 Exercises.- 8 Group Representations.- 8.1 Examples and General Results.- 8.2 Representations of Abelian Groups.- 8.3 Decomposition of the Regular Representation.- 8.4 Characters.- 8.5 Induced Representations.- 8.6 Permutation Representations.- 8.7 Concluding Remarks.- 8.8 Exercises.- Index of Notation.- Index of Terminology.







