Abstract algebra : applications to Galois theory, algebraic geometry, representation theory and cryptography

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Abstract algebra : applications to Galois theory, algebraic geometry, representation theory and cryptography / Celine Carstensen-Opitz ... [et al.]

版:

2nd ed

資料種別:

図書

出版情報:

Berlin : De Gruyter, c2019

形態:

xiii, 407 p. ; 24 cm

シリーズ名:

De Gruyter graduate <BB04858644>

著者名:

ISBN:

9783110603934 [3110603934]

書誌ID:

BB29010743

子書誌情報

フルテキスト

所蔵情報

他の版・巻

書誌詳細

注記:

Other authors: Benjamin Fine, Anja Moldenhauer, Gerhard Rosenberger
Includes bibliographical references (p. [400]-401) and index

主題:

Algebra, Abstract; Crytography; Galois theory; Geometry, Algebraic

言語:

英語

目次情報:

Preface

Preface to the second edition

Groups, rings and fields / 1：

Abstract algebra / 1.1：

Rings / 1.2：

Integral domains and fields / 1.3：

Subrings and ideals / 1.4：

Factor rings and ring homomorphisms / 1.5：

Fields of fractions / 1.6：

Characteristic and prime rings / 1.7：

Groups / 1.8：

Exercises / 1.9：

Maximal and prime ideals / 2：

Prime ideals and integral domains / 2.1：

Maximal ideals and fields / 2.3：

The existence of maximal ideals / 2.4：

Principal ideals and principal ideal domains / 2.5：

Prime elements and unique factorization domains / 2.6：

The fundamental theorem of arithmetic / 3.1：

Prime elements, units and irreducibles / 3.2：

Unique factorization domains / 3.3：

Principal ideal domains and unique factorization / 3.4：

Euclidean domains / 3.5：

Overview of integral domains / 3.6：

Polynomials and polynomial rings / 3.7：

Polynomial rings over fields / 4.1：

Polynomial rings over integral domains / 4.3：

Polynomial rings over unique factorization domains / 4.4：

Field extensions / 4.5：

Extension fields and finite extensions / 5.1：

Finite and algebraic extensions / 5.2：

Minimal polynomials and simple extensions / 5.3：

Algebraic closures / 5.4：

Algebraic and transcendental numbers / 5.5：

Field extensions and compass and straightedge constructions / 5.6：

Geometric constructions / 6.1：

Constructible numbers and field extensions / 6.2：

Four classical construction problems / 6.3：

Squaring the circle / 6.3.1：

The doubling of the cube / 6.3.2：

The trisection of an angle / 6.3.3：

Construction of a regular n-gon / 6.3.4：

Kronecker's theorem and algebraic closures / 6.4：

Kronecker's theorem / 7.1：

Algebraic closures and algebraically closed fields / 7.2：

The fundamental theorem of algebra / 7.3：

Splitting fields / 7.3.1：

Permutations and symmetric polynomials / 7.3.2：

The fundamental theorem of symmetric polynomials / 7.4：

Skew field extensions of C and Frobenius's theorem / 7.6：

Splitting fields and normal extensions / 7.7：

Normal extensions / 8.1：

Groups, subgroups, and examples / 8.3：

Groups, subgroups, and Isomorphisms / 9.1：

Examples of groups / 9.2：

Permutation groups / 9.3：

Cosets and Lagrange's theorem / 9.4：

Generators and cyclic groups / 9.5：

Normal subgroups, factor groups, and direct products / 9.6：

Normal subgroups and factor groups / 10.1：

The group isomorphism theorems / 10.2：

Direct products of groups / 10.3：

Finite Abelian groups / 10.4：

Some properties of finite groups / 10.5：

Automorphisms of a group / 10.6：

Symmetric and alternating groups / 10.7：

Symmetric groups and cycle decomposition / 11.1：

Parity and the alternating groups / 11.2：

Conjugation in S_n / 11.3：

The simplicity of A_n / 11.4：

Solvable groups / 11.5：

Solvability and solvable groups / 12.1：

The derived series / 12.2：

Composition series and the Jordan-Holder theorem / 12.4：

Groups actions and the Sylow theorems / 12.5：

Group actions / 13.1：

Conjugacy classes and the class equation / 13.2：

The Sylow theorems / 13.3：

Some applications of the Sylowtheorems / 13.4：

Free groups and group presentations / 13.5：

Group presentations and combinatorial group theory / 14.1：

Free groups / 14.2：

Group presentations / 14.3：

The modular group / 14.3.1：

Presentations of subgroups / 14.4：

Geometric interpretation / 14.5：

Presentations of factor groups / 14.6：

Group presentations and decision problems / 14.7：

Group amalgams: free products and direct products / 14.8：

Finite Galois extensions / 14.9：

Galois theory and the solvability of polynomial equations / 15.1：

Automorphism groups of field extensions / 15.2：

The fundamental theorem of Galois theory / 15.3：

Separable field extensions / 15.5：

Separability of fields and polynomials / 16.1：

Perfect fields / 16.2：

Finite fields / 16.3：

Separable extensions / 16.4：

Separability and Galois extensions / 16.5：

The primitive element theorem / 16.6：

Applications of Galois theory / 16.7：

Field extensions by radicals / 17.1：

Cyclotomic extensions / 17.3：

Solvability and Galois extensions / 17.4：

The insolvability of the quintic polynomial / 17.5：

Constructibility of regular n-gons / 17.6：

The theory of modules / 17.7：

Modules over rings / 18.1：

Annihilators and torsion / 18.2：

Direct products and direct sums of modules / 18.3：

Free modules / 18.4：

Modules over principal ideal domains / 18.5：

The fundamental theorem for finitely generated modules / 18.6：

Finitely generated Abelian groups / 18.7：

The fundamental theorem: p-primary components / 19.1：

The fundamental theorem: elementary divisors / 19.3：

Integral and transcendental extensions / 19.4：

The ring of algebraic integers / 20.1：

Integral ring extensions / 20.2：

Transcendental field extensions / 20.3：

The transcendence of e and Â¿ / 20.4：

The Hilbert basis theorem and the nullstellensatz / 20.5：

Algebraic geometry / 21.1：

Algebraic varieties and radicals / 21.2：

The Hilbert basis theorem / 21.3：

The Hilbert nullstellensatz / 21.4：

Applications and consequences of Hilbert's theorems / 21.5：

Dimensions / 21.6：

Algebras and group representations / 21.7：

Group representations / 22.1：

Representations and modules / 22.2：

Semisimple algebras and Wedderburn's theorem / 22.3：

Ordinary representations, characters and character theory / 22.4：

Burnside's theorem / 22.5：

Algebraic cryptography / 22.6：

Basic cryptography / 23.1：

Encryption and number theory / 23.2：

Public key cryptography / 23.3：

The Diffie-Hellman protocol / 23.3.1：

The RSA algorithm / 23.3.2：

The El-Gamal protocol / 23.3.3：

Elliptic curves and elliptic curve methods / 23.3.4：

Noncommutative-group-based cryptography / 23.4：

Free group cryptosystems / 23.4.1：

Ko-Lee and Anshel-Anshel-Goldfeld methods / 23.5：

The Ko-Lee protocol / 23.5.1：

The Anshel-Anshel-Goldfeld protocol / 23.5.2：

Platform groups and braid group cryptography / 23.6：

Bibliography / 23.7：

Index

Preface

Preface to the second edition

Groups, rings and fields / 1：

Abstract algebra / 1.1：

Rings / 1.2：

Integral domains and fields / 1.3：

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