Items related to Algebra: An Approach via Module Theory
Synopsis
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.
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- PublisherSpringer
- Publication date2011
- ISBN 10 1461209242
- ISBN 13 9781461209249
- BindingPaperback
- LanguageEnglish
- Number of pages544
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