On an embedding problem

The following theorem is proved. Let $n$ be an odd integer; if all primes which enter in the canonical decomposition of the integer $16+27n^4$ with odd multiplicities have the form $8m+1$, $8m+3$, or $8m+5$, then the decomposition field of the polynomial $f(x)=x^4-2nx-1$ is embeddable into a nonsplit Galois extension of degree 48.