TOPICS IN THE THEORY OF ALGEBRAIC FUNCTION FIELDS

By ABIGAIL SMITH

CONTENTS ALGEBRAIC AND NUMERICAL ANTECEDENTS ALGEBRAIC AND TRANSCENDENTAL EXTENSIONS ABSOLUTE VALUES OVER Q RIEMANN SURFACES EXERCISES ALGEBRAIC FUNCTION FIELDS OF ONE VARIABLE THE FIELD OF CONSTANTS VALUATIONS, PLACES, AND VALUATION RINGS ABSOLUTE VALUES AND COMPLETIONS VALUATIONS IN RATIONAL FUNCTION FIELDS ARTIN’S APPROXIMATION THEOREM THE RIEMANN–ROCH THEOREM DIVISORS PRINCIPAL DIVISORS AND CLASS GROUPS REPARTITIONS OR ADELES DIFFERENTIALS THE RIEMANN–ROCH THEOREM AND ITS APPLICATIONS EXAMPLES FIELDS OF RATIONAL FUNCTIONS AND FUNCTION FIELDS OF GENUS 0 ELLIPTIC FUNCTION FIELDS AND FUNCTION FIELDS OF GENUS 1 QUADRATIC EXTENSIONS OF K(X) AND COMPUTATION OF THE GENUS EXTENSIONS AND GALOIS THEORY EXTENSIONS OF FUNCTION FIELDS GALOIS EXTENSIONS OF FUNCTION FIELDS DIVISORS IN AN EXTENSION COMPLETIONS AND GALOIS THEORY INTEGRAL BASES DIFFERENT AND DISCRIMINANT DEDEKIND DOMAINS DIFFERENT AND DISCRIMINANT IN DEDEKIND DOMAINS DISCRETE VALUATION RINGS AND COMPUTATION OF THE DIFFERENT RAMI?CATION IN ARTIN–SCHREIER AND KUMMER EXTENSIONS RAMI?CATION GROUPS 510 EXERCISES