Abstract Algebra : : Applications to Galois Theory, Algebraic Geometry, Representation Theory and Cryptography / / Celine Carstensen-Opitz, Benjamin Fine, Gerhard Rosenberger, Anja Moldenhauer.

• Superior document: Title is part of eBook package: De Gruyter DG Plus eBook-Package 2019
• VerfasserIn: Carstensen-Opitz, Celine,; Fine, Benjamin,; Moldenhauer, Anja,; Rosenberger, Gerhard,
• Place / Publishing House: Berlin ;, Boston : : De Gruyter, , [2019]; ©2019
• Year of Publication: 2019
• Edition: 2nd rev. and ext. edition
• Language: English
• Series: De Gruyter Textbook
• Physical Description: 1 online resource (XIV, 407 p.)

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.

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