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図書
図書
1. Introduction to abstract algebra : from rings, numbers, groups, and fields to polynomials and Galois theory
Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger
出版情報: Baltimore, Maryland : Johns Hopkins University Press, 2014  xiv, 566 p. ; 26 cm
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目次情報: 続きを見る
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：
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