Extensions of degree p4 of a p-adic field

Let K be a p-adic field. Restricting to the case of no intermediate extensions, we obtain formulae counting the number of (totally and wildly) ramified extensions of degree p(4) of K up to K-isomorphism and in particular, we count the number of isomorphism classes of extensions for which the Galois closure has a prescribed Galois group. The principal tool used is a result, proved in Del Corso et al. (On wild extensions of a p-adic field, arXiv:1601.05939v1), which states that there is a one-to-one correspondence between the isomorphism classes of extensions of degree p(k) of K having no intermediate extensions and the irreducible H-sub-modules of dimension k of F*/F*(p), where F is the composite of certain fixed normal extensions of K and H is its Galois group over K.