附註:Includes bibliographical references (page 166) and index.
0. Preliminaries. Definition of rings and fields. Vector spaces. Bases. Equivalence relations. Axiom of choice -- 1. Diophantine equations: Euclidean domains. Euclidean domain of Gaussian integers. Euclidean domains as unique factorization domains -- 2. Construction of projective planes: splitting fields and finite fields. Existence and uniqueness of splitting fields and of finite fields of prime power order -- 3. Error codes: primitive elements and subfields. Existence of primitive elements in finite fields. Subfields of finite fields. Computation of minimum polynomials -- 4. Construction of primitive polynomials: cyclotomic polynomials and factorization. Basic properties of cyclotomic polynomials. Berlekamp's factorization algorithm -- 5. Ruler and compass constructions: irreducibility and constructibility. Product formula for the degree of composite extensions. Irreducibility criteria for polynomials over the rationals. The field of constructible real numbers -- 6. Pappus' theorem and Desargues' theorem in projective planes: Wedderburn's theorem. Proof of Wedderburn's theorem -- 7. Solution of polynomials by radicals: Galois groups. Basic definitions and results in Galois groups. Discriminants -- 8. Introduction to groups. Group axioms. Subgroup lattice. Class equation. Cauchy's theorem. Transitive permutation groups. Soluble groups -- 9. Cryptography: elliptic curves and factorization. Euler's function. Discrete logarithms. Elliptic curves. Pollard's method of factorizing integers. Elliptic curve factorization of integers.
