COMPUTING SUBFIELDS IN ALGEBRAIC NUMBER-FIELDS

Let K : = Q-(alpha) be an algebraic number field which is given by specifying the minimal polynomial f(X) for alpha over Q. We describe a procedure for finding the subfields L of K by constructing pairs (w(X), g(X)) of polynomials over Q such that L = Q(w(alpha)) and g(X) is the minimal polynomial for w(alpha). The construction uses local information obtained from the Frobenius-Chebotarev theorem about the Galois group Gal(f), and computations over p-adic extensions of Q.