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: |