Solving norm equations in relative number fields using $S$-units

In this paper, we are interested in solving the so-called norm equation ${\mathcal N}_{L/K} (x)=a$, 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 ${\mathbb Z}_L$), 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.