Rings and fields

Preliminaries

Diophantine equations: Euclidean domains

Construction of projective planes: splitting fields and finite fields

Error codes: primitive elements and subfields

Construction of primitive polynomials: cyclotomic polynomials and factorization

Ruler and compass construction: irreducibility and constructibility

Pappus' theorem and Desargues' theorem in projective planes: Wedderburn's theorem

Solution of polynomials by radicals: Galois groups

Introduction to groups

Crytography: elliptic curves and factorization.