Computing Local Artin Maps, and Solvability of Norm Equations

Let L = K(α) be an Abelian extension of degree n of a number field K, given by the minimal polynomial of α over K. We describe an algorithm for computing the local Artin map associated with the extension L / K at a finite or infinite prime v of K. We apply this algorithm to decide if a nonzero a K is a norm from L, assuming that L / Kis cyclic