Let $K$ be a local field (of characteristic 0) with (finite) residue field of characteristic $l$ and let $p$ be a prime.
Considering the cases, whether the $p$-th roots of unity are in $K$ and whether $l$ equals $p$ (and maybe whether $p=2$) or not, my question is:
How many Galois extensions of $K$ of degree $p$ exist?
Best Answer
If $K$ contains the $p$-th roots of unity, then Kummer theory tells us that the degree $p$ Galois extensions of $K$ are in bijective correspondence with the subgroups of $K^{\times}/(K^{\times})^p$ of order $p$. The structure of $K^{\times}$ is well-known; see http://en.wikipedia.org/wiki/Local_fields or any decent book on local fields. So you can work out the answer in this case.
If $K$ doesn't contain the $p$-th roots of unity, then it becomes harder. Here are some special cases. For every $d \in \mathbb{N}$, $K$ has a unique unramified extension of degree $d$, which is necessarily cyclic - see Corollary 4.4 of these nice notes: http://websites.math.leidenuniv.nl/algebra/localfields.pdf So in particular there is a unique unramified degree p Galois extension of $K$.
If $l \neq p$, then any ramified extension of $K$ must be totally and tamely ramified. But then by 5.3 and 5.4 of the above notes, $\mathbb{Z}/p\mathbb{Z}$ must embed into the unit group of the residue field of $K$. Then by Hensel's Lemma, $K$ must contain $p$-th roots of unity and so we are reduced to the Kummer case above.
So we are left with the case $l=p$ and $K$ not containing $p$-th roots of unity. I'll think about this some more, but you should be able to use class field theory as mentioned above.