Decomposition of primes in non-Galois extensions

In this thesis we consider the following question: Given a finite separable non-Galois extension F/K of a global field K, how a prime P of K decomposes in the field F. In the first part, we study the Galois extension M/K where M is the Galois closure of F/K and action of Galois group G of M/K over the set of primes of F lying over a prime P in K. We obtain a one to one correspondence between the double coset space of G with respect to certain subgroups of G (depending on P and F) and the set of primes of F lying over P. Under this correspondence ramification indices and inertia degrees are explicitly determined. Then we investigate the case where G is a finite group of Lie type and F is the intermediate field corresponding to a parabolic subgroup of G. We obtain that the number of primes of F lying over an unrami ed place with given residue degree can be given as polynomials in a power of the characteristic of the variety G. This polynomials depend on the length function on the certain subgroups of the Weyl group of G