Can a (higher) local field have uncountably many finite (seperable) extensions?



Finite extension of local fields
Let $k$ be any field and $K=k((x,y))$ the fraction field of the ring $k[[x,y]]$ of formal Laurent series in two variables. Then Harbater and Stevenson proved that the absolute Galois group of $K=k((x,y))$ is quasi-free of rank equal the cardinality of $m={\rm card}(K)>\aleph\_0$. In particular, for any nontrivial finite group $G$, there exist $m$ distinct Galois extensions of $K$ of Galois group $G$. In fact, these extensions can be chosen to be linearly disjoint (since the absolute Galois group is even "semi-free", as was shown by Harbater-Haran and myself). 


In particular if one takes $k$ to be a finite field, then $K$ is local field of dimension $2$. 


