Octahedral extensions with a given cubic subfield

Let K/Q be a non-Galois cubic extension with vertical bar d(K)vertical bar I a power of a prime p. We prove a conjecture of Wong, namely that the number of S-4-extensions of Q containing K and having discriminant a power of p is of the form 2(n) - 1 for some nonnegative n is an element of Z, and that n is positive if K is totally real. (C) 2016 Elsevier Inc. All rights reserved.