Introduction to abstract algebra : from rings, numbers, groups, and fields to polynomials and Galois theory

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Introduction to abstract algebra : from rings, numbers, groups, and fields to polynomials and Galois theory / Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger

資料種別:

図書

出版情報:

Baltimore, Maryland : Johns Hopkins University Press, 2014

形態:

xiv, 566 p. ; 26 cm

著者名:

ISBN:

9781421411767 [1421411768]

書誌ID:

BB15836643

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:

Includes bibliographical references (p. 557-560) and index

主題:

Algebra, Abstract

言語:

英語

目次情報:

Preface / 0：

Abstract Algebra and Algebraic Reasoning / 1：

Abstract Algebra / 1.1：

Algebraic Structures / 1.2：

The\Algebraic Method / 1.3：

The\Standard Number Systems / 1.4：

The\Integers and Induction / 1.5：

Exercises / 1.6：

Algebraic Preliminaries / 2：

Sets and Set Theory / 2.1：

Set Operations / 2.1.1：

Functions / 2.2：

Equivalence Relations and Factor Sets / 2.3：

Sizes of Sets / 2.4：

Binary Operations / 2.5：

The\Algebra of Sets / 2.5.1：

Algebraic Structures and Isomorphisms / 2.6：

Groups / 2.7：

Rings and the Integers / 2.8：

Rings and the Ring of Integers / 3.1：

Some Basic Properties of Rings and Subrings / 3.2：

Examples of Rings / 3.3：

The\Modular Rings / 3.3.1：

Noncommutative Rings / 3.3.2：

Rings Without Identities / 3.3.3：

Rings of Subsets / 3.3.4：

Direct Sums of Rings / 3.3.5：

Summary of Examples / 3.3.6：

Ring Homomorphisms and Isomorphisms / 3.4：

Integral Domains and Ordering / 3.5：

Mathematical Induction and the Uniqueness of Z / 3.6：

Number Theory and Unique Factorization / 3.7：

Elementary Number Theory / 4.1：

Divisibility and Primes / 4.2：

Greatest Common Divisors / 4.3：

The\Fundamental Theorem of Arithmetic / 4.4：

Congruences and Modular Arithmetic / 4.5：

Unique Factorization Domains / 4.6：

Fields / 4.7：

Fields and Division Rings / 5.1：

Construction and Uniqueness of the Rationals / 5.2：

Fields of Fractions / 5.2.1：

The\Real Number System / 5.3：

The\Completeness of R (Optional) / 5.3.1：

Characterization of R (Optional) / 5.3.2：

The\Construction of R (Optional) / 5.3.3：

The\p-adic Numbers (Optional) / 5.3.4：

The\Field of Complex Numbers / 5.4：

Geometric Interpretation / 5.4.1：

Polar Form and Euler's Identity / 5.4.2：

DeMoivre's Theorem for Powers and Roots / 5.4.3：

Basic Group Theory / 5.5：

Groups, Subgroups and Isomorphisms / 6.1：

Examples of Groups / 6.2：

Permutations and the Symmetric Group / 6.2.1：

Subgroups and Lagrange's Theorem / 6.2.2：

Generators and Cyclic Groups / 6.4：

Factor Groups and the Group Isomorphism Theorems / 6.5：

Normal Subgroups / 7.1：

Factor Groups / 7.2：

Examples of Factor Groups / 7.2.1：

The\Group Isomorphism Theorems / 7.3：

Direct Products and Abelian Groups / 7.4：

Direct Products of Groups / 8.1：

Direct Products of Two Groups / 8.1.1：

Direct Products of Any Finite Number of Groups / 8.1.2：

Abelian Groups / 8.2：

Finite Abelian Groups / 8.2.1：

Free Abelian Groups / 8.2.2：

The\Basis Theorem for Finitely Generated Abelian Groups / 8.2.3：

Symmetric and Alternating Groups / 8.3：

Symmetric Groups and Cycle Structure / 9.1：

The\Alternating Groups / 9.1.1：

Conjugation in Sn / 9.1.2：

The\Simplicity of An / 9.2：

Group Actions and Topics in Group Theory / 9.3：

Group Actions / 10.1：

Conjugacy Classes and the Class Equation / 10.2：

The\Sylow Theorems / 10.3：

Some Applications of the Sylow Theorems / 10.3.1：

Groups of Small Order / 10.4：

Solvability and Solvable Groups / 10.5：

Solvable Groups / 10.5.1：

The\Derived Series / 10.5.2：

Composition Series and the Jordan-Holder Theorem / 10.6：

Topics in Ring Theory / 10.7：

Ideals in Rings / 11.1：

Factor Rings and the Ring Isomorphism Theorem / 11.2：

Prime and Maximal Ideals / 11.3：

Prime Ideals and Integral Domains / 11.3.1：

Maximal Ideals and Fields / 11.3.2：

Principal Ideal Domains and Unique Factorization / 11.4：

Polynomials and Polynomial Rings / 11.5：

Polynomial Rings over a Field / 12.1：

Unique Factorization of Polynomials / 12.2.1：

Euclidean Domains / 12.2.2：

F[x] as a Principal Ideal Domain / 12.2.3：

Polynomial Rings over Integral Domains / 12.2.4：

Zeros of Polynomials / 12.3：

Real and Complex Polynomials / 12.3.1：

The\Fundamental Theorem of Algebra / 12.3.2：

The\Rational Roots Theorem / 12.3.3：

Solvability by Radicals / 12.3.4：

Algebraic and Transcendental Numbers / 12.3.5：

Unique Factorization in Z[x] / 12.4：

Algebraic Linear Algebra / 12.5：

Linear Algebra / 13.1：

Vector Analysis in R3 / 13.1.1：

Matrices and Matrix Algebra / 13.1.2：

Systems of Linear Equations / 13.1.3：

Determinants / 13.1.4：

Vector Spaces over a Field / 13.2：

Euclidean n-Space / 13.2.1：

Vector Spaces / 13.2.2：

Subspaces / 13.2.3：

Bases and Dimension / 13.2.4：

Testing for Bases in Fn / 13.2.5：

Dimension and Subspaces / 13.3：

Algebras / 13.4：

Inner Product Spaces / 13.5：

Banach and Hilbert Spaces / 13.5.1：

The\Gram-Schmidt Process and Orthonormal Bases / 13.5.2：

The\Closest Vector Theorem / 13.5.3：

Least-Squares Approximation / 13.5.4：

Linear Transformations and Matrices / 13.6：

Matrix of a Linear Transformation / 13.6.1：

Linear Operators and Linear Functionals / 13.6.2：

Fields and Field Extensions / 13.7：

Abstract Algebra and Galois Theory / 14.1：

Field Extensions / 14.2：

Algebraic Field Extensions / 14.3：

F-automorphisms, Conjugates and Algebraic Closures / 14.4：

Adjoining Roots to Fields / 14.5：

Splitting Fields and Algebraic Closures / 14.6：

Automorphisms and Fixed Fields / 14.7：

Finite Fields / 14.8：

Transcendental Extensions / 14.9：

A\Survey of Galois Theory / 14.10：

An\Overview of Galois Theory / 15.1：

Galois Extensions / 15.2：

Automorphisms and the Galois Group / 15.3：

The\Fundamental Theorem of Galois Theory / 15.4：

A\Proof of the Fundamental Theorem of Algebra / 15.5：

Some Applications of Galois Theory / 15.6：

The\Insolvability of the Quintic / 15.6.1：

Some Ruler and Compass Constructions / 15.6.2：

Algebraic Extensions of R / 15.6.3：

Bibliography / 15.7：

Index

Preface / 0：

Abstract Algebra and Algebraic Reasoning / 1：

Abstract Algebra / 1.1：

Algebraic Structures / 1.2：

The\Algebraic Method / 1.3：

The\Standard Number Systems / 1.4：

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