Valuation Theory

Abstract

The notion of a valuation of a field arises when one attempts to assign magnitudes to the elements of a field. The classical case is that of the absolute value |α| in the field of real numbers or in the field of rational numbers. Of basic importance for the study of arithmetic properties of the rational and more generally of number fields (finite algebraic extensions of the rationals) are the p-adic valuations of the field of rational numbers. For a given prime p the valuation φ _p (α) of the rational number α indicates the power of p which divides the rational number α. Valuations play a fundamental role also in the study of algebraic function fields. For these it is necessary to generalize the notion somewhat so that it becomes equivalent to the notion of a place, which was first introduced by Dedekind and Weber in giving a purely algebraic definition of Riemann surfaces for algebraic functions. Valuation theory forms a solid link between algebra and analysis. On the one hand, it permits a precise study of algebraic functions and, on the other hand, it leads to the introduction of analytic notions (convergence, integration) in the study of arithmetic questions.