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1.

図書
図書
1. The fundamental theorem of algebra (: hbk)
Benjamin Fine, Gerhard Rosenberger
出版情報: New York : Springer, c1997  xi, 208 p. ; 25 cm
シリーズ名: Undergraduate texts in mathematics
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2.

図書
図書
2. Algebraic theory of the Bianchi groups
Benjamin Fine
出版情報: New York : M. Dekker, c1989  vii, 249 p. ; 24 cm
シリーズ名: Monographs and textbooks in pure and applied mathematics ; 129
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3.

図書
図書
3. Abstract algebra : applications to Galois theory, algebraic geometry, representation theory and cryptography. 2nd ed
Celine Carstensen-Opitz ... [et al.]
出版情報: Berlin : De Gruyter, c2019  xiii, 407 p. ; 24 cm
シリーズ名: De Gruyter graduate
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目次情報: 続きを見る
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 Sn / 11.3：
The simplicity of An / 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 ¿ / 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：
4.

図書
図書
4. 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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