On an embedding problem (Q1866702)

Let \(n\) be an odd integer. Then the splitting fields \(K\) of \(f(x)= x^4- 2nx-1\) over \(\mathbb{Q}\) has Galois group \(S_4\). The author proves that the nonsplit embedding problem of \(K/\mathbb{Q}\) with kernel of order 2 has a solution if in the prime decomposition of \(16+ 27n^4\) the primes of odd multiplicity are of the form \(8m+ 1\), \(8m+ 3\) or \(8m+ 5\).