Corresponding residue systems in normal extensions

AbstractLet K and K′ be number fields with L = K · K′ and F = K φ K′. Suppose that KF and K′F are normal extensions of degree n. Let B be a prime ideal in L and suppose that B is totally ramified in KF and in K′F. Let π be a prime element for BK = B φ K, and let f(x) ∈ F[x] be the minimum polynomial for π over F. Suppose that BK · DL = (B≠)e. Then, M(B# : K, K′) = min{m, e(t + 1)}, where t = min{t(KF), t(K′F)} and m is the largest integer such that (BK′)nm/e φ f(DK′) ≠ {φ}.If we assume in addition to the above hypotheses that [K : F] = [K′: F] = pn, a prime power, and that B divides p and is totally ramified in LF, then M(B# : K, K′) ⩾ pn−1[(p − 1)(t + p], with t = t(B : L/F)