Algebraic Numbers and Algebraic Functions

Abstract

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.