Abstract Algebra: Applications to Galois Theory, Algebraic Geometry, Representation Theory and Cryptography

Celine Carstensen-Opitz; Benjamin Fine; Anja Moldenhauer; Gerhard Rosenberger

Book

Abstract Algebra

Applications to Galois Theory, Algebraic Geometry, Representation Theory and Cryptography

Celine Carstensen-Opitz , Benjamin Fine , Anja Moldenhauer and Gerhard Rosenberger

Language: English

Published/Copyright: 2019

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This book is in the series

De Gruyter Textbook

About this book

A new approach to conveying abstract algebra, the area that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras, that is essential to various scientific disciplines such as particle physics and cryptology. It provides a well written account of the theoretical foundations and it also includes a chapter on cryptography. End of chapter problems help readers with accessing the subjects.

Covering topics not found in competing works: Free groups, module theory, extensions of rings
New chapter about Groups, Algebras and Representations
New section about the hot topic Combinatorial and geometric group theory

Author / Editor information

Celine Carstensen-Opitz, Technical University of Dortmund, Germany; Benjamin Fine, Fairfield University, Connecticut, USA; Gerhard Rosenberger, University of Hamburg, Germany; Anja Moldenhauer, University of Hamburg, Germany.

Topics

Mathematics Algebra and Number Theory

Publicly Available

Frontmatter
I

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Preface
V

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Preface to the second edition
VII

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Contents
IX

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1. Groups, rings and fields
1

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2. Maximal and prime ideals
21

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3. Prime elements and unique factorization domains
29

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4. Polynomials and polynomial rings
53

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5. Field extensions
67

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6. Field extensions and compass and straightedge constructions
81

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7. Kronecker’s theorem and algebraic closures
93

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8. Splitting fields and normal extensions
119

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9. Groups, subgroups, and examples
125

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10. Normal subgroups, factor groups, and direct products
147

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11. Symmetric and alternating groups
169

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12. Solvable groups
179

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13. Groups actions and the Sylow theorems
189

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14. Free groups and group presentations
201

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15. Finite Galois extensions
227

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16. Separable field extensions
243

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17. Applications of Galois theory
257

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18. The theory of modules
275

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19. Finitely generated Abelian groups
293

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20. Integral and transcendental extensions
303

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21. The Hilbert basis theorem and the nullstellensatz
319

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22. Algebras and group representations
333

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23. Algebraic cryptography
365

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Bibliography
399

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Index
403

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Publishing information

Pages and Images/Illustrations in book

eBook published on:

September 2, 2019

eBook ISBN:

9783110603996

Edition:

2nd rev. and ext. edition

Pages and Images/Illustrations in book

Front matter:

14

Main content:

407

Illustrations:

15

Tables:

15

https://doi.org/10.1515/9783110603996

eBook ISBN: 9783110603996

Audience(s) for this book

Advanced Students, Mathematicians, Theoretical Physicists, Cryptologists; Academic Libraries

Abstract Algebra

Overview

About this book

Author / Editor information

Topics

Table of contents

Frontmatter

Preface

Preface to the second edition

Contents

1. Groups, rings and fields

2. Maximal and prime ideals

3. Prime elements and unique factorization domains

4. Polynomials and polynomial rings

5. Field extensions

6. Field extensions and compass and straightedge constructions

7. Kronecker’s theorem and algebraic closures

8. Splitting fields and normal extensions

9. Groups, subgroups, and examples

10. Normal subgroups, factor groups, and direct products

11. Symmetric and alternating groups

12. Solvable groups

13. Groups actions and the Sylow theorems

14. Free groups and group presentations

15. Finite Galois extensions

16. Separable field extensions

17. Applications of Galois theory

18. The theory of modules

19. Finitely generated Abelian groups

20. Integral and transcendental extensions

21. The Hilbert basis theorem and the nullstellensatz

22. Algebras and group representations

23. Algebraic cryptography

Bibliography

Index

Bibliographic data