Algebraic Numbers and Algebraic Functions

Part 1 Fields with valuations: absolute values; the topology defined by an absolute value; complete fields; valuations, valuation rings and places; the representation by power series; ordered groups; general valuations. Part 2 Extensions: generalities on extensions; extensions of complete fields; extensions of incomplete fields; Dedekind domains and the string approximation theorem; extensions of Dedekind domains; different and discriminant. Part 3 Global fields: algebraic number fields; the product formula; the unit theorem; the class number. Part 4 Function fields: divisors on a function field; principal divisors and the divisor class group; Riemann's theorem and the speciality index; the genus; derivations and differentials; the Riemann-Roch theorem and its consequences; elliptic function fields; Abelian integrals and the Abel-Jacobi theorem. Part 5 Algebraic function fields in two variables: valuations on function fields of two variables.