Rings and Fields Book

0	Preliminaries	1
	Definition of rings and fields
	Vector spaces
	Bases
	Equivalence relations
	Axiom of choice
1	Diophantine equations: Euclidean domains	13
	Euclidean domain of Gaussian integers
	Euclidean domains as unique factorization domains
2	Construction of projective planes: splitting fields and finite fields	25
	Existence and uniqueness of splitting fields and of finite fields of prime power order
3	Error codes: primitive elements and subfields	49
	Existence of primitive elements in finite fields
	Subfields of finite fields
	Computation of minimum polynomials
4	Construction of primitive polynomials: cyclotomic polynomials and factorization	65
	Basic properties of cyclotomic polynomials
	Berlekamp's factorization algorithm
5	Ruler and compass constructions: irreducibility and constructibility	83
	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	93
	Proof of Wedderburn's theorem
7	Solution of polynomials by radicals: Galois groups	109
	Basic definitions and results in Galois groups
	Discriminants
8	Introduction to groups	135
	Group axioms
	Subgroup lattice
	Class equation
	Cauchy's theorem
	Transitive permutation groups
	Soluble groups
9	Crytography: elliptic curves and factorization	151
	Euler's function
	Discrete logarithms
	Elliptic curves
	Pollard's method of factorizing integers
	Elliptic curve factorization of integers
	Further reading	166
	Index	167