A Galois extension of degree p has Galois group Z/pZ, so you are asking about abelian extensions of your local field. Thus the answer to your question can be obtained explicitly via local class field theory -- you can get the needed results out of Serre's Local Fields or many other books.

A Galois extension of degree $p$ has Galois group $\mathbb{Z}/p\mathbb{Z}$, so you are asking about abelian extensions of your local field. Thus the answer to your question can be obtained explicitly via local class field theory -- you can get the needed results out of Serre's Local Fields or many other books.