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1.

図書

図書
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:
2.

図書

図書
Larry C. Grove
出版情報: New York ; Tokyo : Academic Press, 1983  xiii, 299 p. ; 24 cm
シリーズ名: Pure and applied mathematics ; 110
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3.

図書

図書
C. L. Liu
出版情報: New York : McGraw-Hill, c1977  x, 294 p. ; 25 cm
シリーズ名: McGraw-Hill computer science series
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4.

図書

図書
John R. Durbin
出版情報: New York : Wiley, c1985  xiv, 346 p. ; 26 cm
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目次情報: 続きを見る
Introduction
Mappings and Operations / I:
Mappings / 1:
Composition / 2:
Invertible Mappings
Operations / 3:
Composition as an Operation / 4:
Introduction to Groups / II:
Definition and Examples / 5:
Permutations / 6:
Subgroups / 7:
Groups and Symmetry / 8:
Equivalence. Congruence. Divisibility / III:
Equivalence Relations / 9:
Congruence / 10:
The Division Algorithm
Integers Modulon / 11:
Greatest Common Divisors / 12:
The Euclidean Algorithm
Factorization / 13:
Euler's Phi-Function
Groups / IV:
Elementary Properties / 14:
Generators / 15:
Direct Products
Cosets / 16:
Lagrange's Theorem / 17:
Isomorphism / 18:
More on Isomorphism / 19:
Cayley's Theorem / 20:
Group Homomorphisms / V:
Homomorphisms of Groups / 21:
Kernels
Quotient Groups / 22:
The Fundamental Homomorphism Theorem / 23:
Introduction to Rings / VI:
Integral Domains / 24:
Subrings
Fields / 26:
Characteristic / 27:
The Familiar Number Systems / VII:
Ordered Integral Domains / 28:
The Integers / 29:
Field of Quotients / 30:
The Field of Rational Number
Ordered Fields / 31:
The Field of Real Numbers
The Field of Complex Numbers / 32:
Complex Roots of Unity / 33:
Polynomials / VIII:
Definition and Elementary Properties / 34:
Factorization of Polynomials / 35:
Unique Factorization Domains / 37:
Quotient Rings / IX:
Homomorphisms of Rings / 38:
Ideals
Quotient Rings off[X] / 39:
Factorization and Ideals / 41:
Field Extensions / X:
Simple Extensions / 42:
Degrees of Extensions / 43:
Splitting Fields / 44:
Finite Fields / 45:
Galois Theory / XI:
Galois Groups / 46:
Separability and Normality / 47:
Fundamental Theorem of Galois Theory / 48:
Solvability by Radicals / 49:
Geometric Constructions / XII:
Three Famous Problems / 50:
Constructible Numbers / 51:
Impossible Constructions / 52:
Applications of Permutation Groups / XIII:
Groups Acting on Sets / 53:
Burnside's Counting Theorem / 54:
Sylow's Theorem / 55:
Symmetry / XIV:
Finite Symmetry Groups / 56:
Infinite Two-dimensional Symmetry Groups / 57:
On Crystallographic Groups / 58:
The Euclidean Group / 59:
Cryptography and Algebraic Coding / XV:
RSA Algorithm / 60:
Introduction to Algebraic Coding / 61:
Linear Codes / 62:
Standard Decoding / 63:
Error Probability / 64:
Lattices and Boolean Algebras / XVI:
Partially Ordered Sets / 65:
Lattices / 66:
Boolean Algebras / 67:
Finite Boolean Algebras / 68:
Switching / 69:
Sets / A:
Proofs / B:
Mathematical Induction / C:
Linear Algebra / D:
Solutions to Selected Problems / E:
Photo Credit List
Index of Notation
Index
Introduction
Mappings and Operations / I:
Mappings / 1:
5.

図書

図書
Garrett Birkhoff, Saunders MacLane
出版情報: New York : Macmillan, c1977  xi, 500 p. ; 25 cm
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6.

図書

図書
[by] B.L. van der Waerden ; in part based on lectures by E. Artin and E. Noether ; translated by Fred Blum and John R. Schulenberger
出版情報: New York : Ungar, [1970]  2 v. ; 25 cm
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7.

図書

図書
John B. Fraleigh
出版情報: Reading, Mass. : Addison-Wesley, 1968,c1982  xviii, 478 p. ; 24 cm
シリーズ名: Addison-Wesley series in mathematics
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Sets and Relations
Groups and Subgroups / I:
Introduction and Examples / 1:
Binary Operations / 2:
Isomorphic Binary Structures / 3:
Groups / 4:
Subgroups / 5:
Cyclic Groups / 6:
Generators and Cayley Digraphs / 7:
Permutations, Cosets, and Direct Products / II:
Groups of Permutations / 8:
Orbits, Cycles, and the Alternating Groups / 9:
Cosets and the Theorem of Lagrange / 10:
Direct Products and Finitely Generated Abelian Groups / 11:
Plane Isometries / 12:
Homomorphisms and Factor Groups / III:
Homomorphisms / 13:
Factor Groups / 14:
Factor-Group Computations and Simple Groups / 15:
Group Action on a Set / 16:
Applications of G-Sets to Counting / 17:
Rings and Fields / IV:
Integral Domains / 18:
Fermat's and Euler's Theorems / 20:
The Field of Quotients of an Integral Domain / 21:
Rings of Polynomials / 22:
Factorization of Polynomials over a Field / 23:
Noncommutative Examples / 24:
Ordered Rings and Fields / 25:
Ideals and Factor Rings / V:
Homomorphisms and Factor Rings / 26:
Prime and Maximal Ideas / 27:
Grouml;bner Bases for Ideals / 28:
Extension Fields / VI:
Introduction to Extension Fields / 29:
Vector Spaces / 30:
Algebraic Extensions / 31:
Geometric Constructions / 32:
Finite Fields / 33:
Advanced Group Theory / VII:
Isomorphism Theorems / 34:
Series of Groups / 35:
Sylow Theorems / 36:
Applications of the Sylow Theory / 37:
Free Abelian Groups / 38:
Free Groups / 39:
Group Presentations / 40:
Groups in Topology / VIII:
Simplicial Complexes and Homology Groups / 41:
Computations of Homology Groups / 42:
More Homology Computations and Applications / 43:
Homological Algebra / 44:
Factorization / IX:
Unique Factorization Domains / 45:
Euclidean Domains / 46:
Gaussian Integers and Multiplicative Norms / 47:
Automorphisms and Galois Theory / X:
Automorphisms of Fields / 48:
The Isomorphism Extension Theorem / 49:
Splitting Fields / 50:
Separable Extensions / 51:
Totally Inseparable Extensions / 52:
Galois Theory / 53:
Illustrations of Galois Theory / 54:
Cyclotomic Extensions / 55:
Insolvability of the Quintic / 56:
Appendix: Matrix Algebra
Notations
Answers to odd-numbered exercises not asking for definitions or proofs
Index
Sets and Relations
Groups and Subgroups / I:
Introduction and Examples / 1:
8.

図書

図書
Chow, Yutze
出版情報: New York : Gordon and Breach Science Publishers, c1976  1 v. ; 24 cm
シリーズ名: Modern abstract algebra / Yutze Chow ; Vol. 2
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9.

図書

図書
Matthew Hennessy
出版情報: Cambridge, Mass. : MIT Press, c1988  vi, 272 p. ; 24 cm
シリーズ名: MIT Press series in the foundations of computing
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10.

図書

図書
Chow, Yutze
出版情報: New York : Gordon and Breach Science Publishers, c1976  1 v. ; 24 cm
シリーズ名: Modern abstract algebra / Yutze Chow ; Vol. 1
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