Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

Here is a belated answer (I came across this question only today) which doesn't require anything more than Kummer theory --- or Artin-Schreier theory, if you want to allow $K$ to be a finite extension of ${\mathbf F}_l((t))$. I will confine myself to the more interesting case of degree-$l$ extensions (where $l$ is the residual characteristic).

If $K$ contains a primitive $l$-th root of $1$, then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and the set of ${\mathbf F}_l$-lines in $K^\times/K^{\times l}$ (Kummer theory). The structure of this filtered ${\mathbf F}_l$-space is completely known; see for example Section V of arXiv:0711.3878 (where your $l$ is called $p$).

If $K$ does not contain a primitive $l$-th root $\zeta$ of $1$, then put $K'=K(\zeta)$, $\Delta={\rm Gal}(K'|K)$ and $\omega:\Delta\to{\mathbf F}_l^\times$ the cyclotomic character giving the action of $G$ on the $l$-th roots of $1$. Then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and ${\mathbf F}_l$-lines in the $\omega$-eigenspace for the action of $\Delta$ on $K^{\prime\times}/K^{\prime\times l}$. The structure of this filtered ${\mathbf F}_l[\Delta]$-module can be completely determined; see for example arXiv:0912.2829.

If you are interested more generally in all degree-$l$ (separable) extensions of $K$ (and not just the cyclic ones), then something similar can be done. Put $L=K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. There is a natural bijection between the set of (isomorphism classes of) degree-$l$ (separable) extensions of $K$ and the set of $G$-stable lines in the ${\mathbf F}_l$-space $L^\times/L^{\times l}$. Again, the structure of this filtered ${\mathbf F}_l[G]$-module is completely known: see for example arXiv:1005.2016.

Finally, if you allow $K$ to be a finite extension of ${\mathbf F}_l((t))$, there are similar results using Artin-Schreier theory instead of Kummer theory. See for example arXiv:0909.2541 for degree-$l$ cyclic extensions (which correspond to ${\mathbf F}_l$-lines in $K^+/\wp(K^+)$, where $\wp(x)=x^l-x$) and arXiv:1005.2016 for degree-$l$ separable extensions, which correspond to $G$-stable ${\mathbf F}_l$-lines in $L^+/\wp(L^+)$, where $L$ is still $K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. The filtered ${\mathbf F}_l$-space (resp. ${\mathbf F}_l[G]$-module) $K^+/\wp(K^+)$ (resp. $L^+/\wp(L^+)$) has been completely determined therein. These results allow you in particular to count the number of extensions with bounded ramification, of which there are only finitely many.