On the degree of Kummer extensions for commutative algebraic groups

Let A be a commutative algebraic group over a field K, and let G be a finitely generated subgroup of the K-rational points of A. The purpose of this thesis is to study the degrees of the Kummer extensions relative to A,K and G. For a positive integer N, this is the smallest extension of the N-torsion field of A over K that contains the N-division points of G. Finite explicit procedures to compute such degrees are given if A is the multiplicative group and K is a p-adic field, and if A is any product of one-dimensional algebraic tori and K is a number field. A finite explicit procedure to compute all the degrees for all positive integers N at once is given when A is the multiplicative group and K is a multiquadratic or quartic cyclic number field. Finally, an effective bound on the failure of maximality for such degrees is given if A is an abelian variety with complex multiplication and K is a number field. <br>