Abstract: | Let K and K′ be number fields with L = K · K′ and F = KφK′. Suppose that and are normal extensions of degree n. Let be a prime ideal in L and suppose that is totally ramified in and in . Let π be a prime element for K = φ K, and let f(x) ∈ Fx] be the minimum polynomial for π over F. Suppose that K · L = (≠)e. Then, , where and m is the largest integer such that (K′)nm/e φ f(K′) ≠ {φ}.If we assume in addition to the above hypotheses that K : F] = K′: F] = pn, a prime power, and that divides p and is totally ramified in , then , with t = t( : L/F). |