Study of Absolute Values

Abstract

The notion of an absolute value of a field K is a generalization of the notion of ordinary absolute value of the field C of complex numbers. A real valued function defined on a field K into non-negative real numbers is called absolute value of K if (x) = 0 , x = 0; (xy) = (x) (y) and (x + y) (x) + (y) 8x; y 2 K: In this thesis, we study absolute values and its basic properties and some significant results like Ostrowski's Theorem, Approximation Theorem and Independence Theorem. We also discuss Archimedean and non-Archimedean absolute values, completion of fields with respect to absolute values. A non-Archimedean absolute value gives rise to what is called (additive) valuation. A detailed exposition of discrete valuations is brought out. We also study Hensel's Lemma and some of its applications.