Solving norm equations in relative number fields using 𝑆-units

Source abstract

In this paper, we are interested in solving the so-called norm equation N L / K ( x ) = a {\mathcal N}_{L/K} (x)=a , where L / K L/K is a given arbitrary extension of number fields and a a a given algebraic number of K K . By considering S S -units and relative class groups, we show that if there exists at least one solution (in L L , but not necessarily in Z L {\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 S -units that are norms are norms of S 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 a is an algebraic integer, we also study the existence of an integral solution, and we can adapt the algorithm to this case.