Chapter 4: Groups, Rings, and Fields

Alexander I. Kheyfits

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Chapter 4: Groups, Rings, and Fields

Alexander I. Kheyfits

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This chapter is in the book Discrete Mathematics With Cryptographic Applications

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Kheyfits, Alexander I.. "Chapter 4: Groups, Rings, and Fields". Discrete Mathematics With Cryptographic Applications: A Self-Teaching Introduction, Berlin, Boston: Mercury Learning and Information, 2021, pp. 67-78. https://doi.org/10.1515/9781683927624-005

Kheyfits, A. (2021). Chapter 4: Groups, Rings, and Fields. In Discrete Mathematics With Cryptographic Applications: A Self-Teaching Introduction (pp. 67-78). Berlin, Boston: Mercury Learning and Information. https://doi.org/10.1515/9781683927624-005

Kheyfits, A. 2021. Chapter 4: Groups, Rings, and Fields. Discrete Mathematics With Cryptographic Applications: A Self-Teaching Introduction. Berlin, Boston: Mercury Learning and Information, pp. 67-78. https://doi.org/10.1515/9781683927624-005

Kheyfits, Alexander I.. "Chapter 4: Groups, Rings, and Fields" In Discrete Mathematics With Cryptographic Applications: A Self-Teaching Introduction, 67-78. Berlin, Boston: Mercury Learning and Information, 2021. https://doi.org/10.1515/9781683927624-005

Kheyfits A. Chapter 4: Groups, Rings, and Fields. In: Discrete Mathematics With Cryptographic Applications: A Self-Teaching Introduction. Berlin, Boston: Mercury Learning and Information; 2021. p.67-78. https://doi.org/10.1515/9781683927624-005

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Chapters in this book

Frontmatter i
Contents v
Preface xi
Chapter 1: A Brief Survey of Elementary Functions 1
Chapter 2: Propositional Algebra 27
Chapter 3: Naïve and Formal (Axiomatic) Set Theory 51
Chapter 4: Groups, Rings, and Fields 67
Chapter 5: Predicates and Quantifiers—Algebraic Theory 79
Chapter 6: Binary Relations and Relational Databases 93
Chapter 7: Combinatorics 117
Chapter 8: Elements of Number Theory 139
Chapter 9: Boolean Functions 163
Chapter 10: Hashing Functions and Cryptographic Maps 171
Chapter 11: Generating Polynomials and Inversion Formulas 179
Chapter 12: Systems of Representatives 187
Chapter 13: Boolean Algebras 193
Chapter 14: Combinatorial Circuits 201
Chapter 15: Complete Systems of Boolean Functions and Bases 211
Chapter 16: Introductory Graph Theory, Euler’s Formula, and Unbreakable Ciphers 225
Chapter 17: Trees and Digraphs 245
Chapter 18: Computations and Algorithms 263
Chapter 19: Finite Automata 271
Chapter 20: Introduction to Game Theory 297
Chapter 21: Information Theory and Coding 309
Chapter 22: Probability Theory with a Finite Sample Space and the Birthday Problem 319
Chapter 23: Turing Machines, P and NP Classes, and Other Models of Computation 341
Chapter 24: Answers and Solutions to Selected Exercises 351
Bibliography 361
Index 365

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https://doi.org/10.1515/9781683927624-005

Chapters in this book

Frontmatter i
Contents v
Preface xi
Chapter 1: A Brief Survey of Elementary Functions 1
Chapter 2: Propositional Algebra 27
Chapter 3: Naïve and Formal (Axiomatic) Set Theory 51
Chapter 4: Groups, Rings, and Fields 67
Chapter 5: Predicates and Quantifiers—Algebraic Theory 79
Chapter 6: Binary Relations and Relational Databases 93
Chapter 7: Combinatorics 117
Chapter 8: Elements of Number Theory 139
Chapter 9: Boolean Functions 163
Chapter 10: Hashing Functions and Cryptographic Maps 171
Chapter 11: Generating Polynomials and Inversion Formulas 179
Chapter 12: Systems of Representatives 187
Chapter 13: Boolean Algebras 193
Chapter 14: Combinatorial Circuits 201
Chapter 15: Complete Systems of Boolean Functions and Bases 211
Chapter 16: Introductory Graph Theory, Euler’s Formula, and Unbreakable Ciphers 225
Chapter 17: Trees and Digraphs 245
Chapter 18: Computations and Algorithms 263
Chapter 19: Finite Automata 271
Chapter 20: Introduction to Game Theory 297
Chapter 21: Information Theory and Coding 309
Chapter 22: Probability Theory with a Finite Sample Space and the Birthday Problem 319
Chapter 23: Turing Machines, P and NP Classes, and Other Models of Computation 341
Chapter 24: Answers and Solutions to Selected Exercises 351
Bibliography 361
Index 365

Chapter 4: Groups, Rings, and Fields

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