The Theory of Valuations

O. F. G. Schilling0821815040, 9780821815045

The realization of the close connection between the theory of algebraic functions of one variable and the theory of algebraic numbers gave rise to the theory of valuations. The arithmetic approach of Dedekind and Weber to the theory of algebraic functions stimulated the question of whether there is an analogue to the power series expansions associated to a point of a Riemann surface. Hensel discovered such an analogue in his theory of p-adic numbers. He recognized that power series expansions can serve to clarify properties of systems of congruences which frequently occur in the allied theories of algebraic numbers and algebraic functions. In his book Theorie der algebraischen Zahlen he stated in 1908 the famous Reducibility Lemma on which a major part of the work on valuations is based. Thus, a powerful tool, already known to Newton in the discussion of plane curves, became available to algebra proper. The basic ideas of Hensel were developed further in the now classical papers of Chevalley, Krull, and Ostrowski.