Solving norm equations in relative number fields using S-units

In this paper, we are interested in solving the so-called norm equation NL/K(x) = a, where L/K is a given arbitrary extension of number fields and a a given algebraic number of K. By considering S-units and relative class groups, we show that if there exists at least one solution (in L, but not necessarily in ZL), then there exists a solution for which we can describe precisely its prime ideal factorization. In fact, we prove that under some explicit conditions, the S-units that are norms are norms of S-units. This allows us to limit the search for rational solutions to a finite number of tests, and we give the corresponding algorithm. When a is an algebraic integer, we also study the existence of an integral solution, and we can adapt the algorithm to this case.