On Computing Subfields

The purpose of this article is to determine all subfields Q(β) of fixed degree of a given algebraic number field Q(α). It is convenient to describe each subfield by a pair (h,g) of polynomials in Q[t] resp. Z[t] such thatgis the minimal polynomial of β = h(α). The computations are done in unramifiedp-adic extensions and use information concerning subgroups of the Galois group of the normal closure of Q(α) obtained from the van der Waerden criterion.