On ramifications in extensions of rational function fields

Abstract

Let K (x) be a rational function field, which is a finite separable extension of the rational function field K (z). In the first part of the thesis, we have studied the number of ramified places of K (x) in K (x) =K (z). Then we have given a formula for the ramification index and the different exponent in the extension F (x) over a function field F, where x satisfies an equation f (x) = z for some z 2 F and separable polynomial f (x) 2 K [x]. In fact, this generalizes the well-known formulas for Kummer and Artin- Schreier extensions.