Chris Thron, Texas A&M University-Central Texas, Journalism, Media Studies & Communications, 6 Application: Introduction to Cryptography, 11 Abstract Groups: Definitions and Basic Properties, 12 Application: Further Topics in Cryptography, 13 Equivalence Relations and Equivalence Classes, 15 Application: Error-Detecting and Correcting Codes, 17 Exploration: relating poynomials and matrices, 20 Appendix: Induction proofs-patterns and examples. \newcommand{\gf}{\operatorname{GF}} Tom Judson's Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. In our math department at Texas A&M — Central Texas, most of the majors are certifying secondary teachers. \newcommand{\gt}{>} 1 Preliminaries1.1 A Short Note on Proofs1.2 Sets and Equivalence Relations2.1 Mathematical Induction2.2 The Division Algorithm, 3 Groups3.1 Integer Equivalence Classes and Symmetries3.2 Definitions and Examples3.3 Subgroups, 4 Cyclic Groups4.1 Cyclic Subgroups4.2 Multiplicative Group of Complex Numbers4.3 The Method of Repeated Squares, 5 Permutation Groups5.1 Definitions and Notation5.2 Dihedral Groups, 6 Cosets and Lagrange’s Theorem6.1 Cosets6.2 Lagrange’s Theorem6.3 Fermat’s and Euler’s Theorems, 7 Introduction to Cryptography7.1 Private Key Cryptography7.2 Public Key Cryptography, 8 Algebraic Coding Theory8.1 Error-Detecting and Correcting Codes8.2 Linear Codes8.3 Parity-Check and Generator Matrices8.4 Efficient Decoding, 9 Isomorphisms9.1 Definition and Examples9.2 Direct Products, 10 Normal Subgroups and Factor Groups10.1 Factor Groups and Normal Subgroups10.2 The Simplicity of the Alternating Group, 11 Homomorphisms11.1 Group Homomorphisms11.2 The Isomorphism Theorems, 12 Matrix Groups and Symmetry12.1 Matrix Groups12.2 Symmetry, 13 The Structure of Groups13.1 Finite Abelian Groups13.2 Solvable Groups, 14 Group Actions14.1 Groups Acting on Sets14.2 The Class Equation14.3 Burnside’s Counting Theorem, 15 The Sylow Theorems15.1 The Sylow Theorems15.2 Examples and Applications, 16 Rings16.1 Rings16.2 Integral Domains and Fields16.3 Ring Homomorphisms and Ideals16.4 Maximal and Prime Ideals16.5 An Application to Software Design, 17 Polynomials17.1 Polynomial Rings17.2 The Division Algorithm17.3 Irreducible Polynomials, 18 Integral Domains18.1 Fields of Fractions18.2 Factorization in Integral Domains, 19 Lattices and Boolean Algebras19.1 Lattices19.2 Boolean Algebras19.3 The Algebra of Electrical Circuits, 20 Vector Spaces20.1 Definitions and Examples20.2 Subspaces20.3 Linear Independence, 21 Fields21.1 Extension Fields21.2 Splitting Fields21.3 Geometric Constructions, 22 Finite Fields22.1 Structure of a Finite Field22.2 Polynomial Codes, 23 Galois Theory23.1 Field Automorphisms23.2 The Fundamental Theorem23.3 Applications, Mathematical Association of America Abstract Algebra: Theory and Applications is an open-source textbook that is designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. See the note about the various Editionsand changes. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory. Faced with this situation, we decided to create a book that our students could actually read for themselves. Excelente libro de Algebra, pero llegó en mal estado. In this way we have been able to dedicate class time to problem-solving and personal interaction rather than rehashing the same material in lecture format.

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