I am reading the notes of a course on local fields, and I came accross the following statement that may be implicitly given at some point.
Consider $L/K$ a finite Galois extension of local fields. Denote by $k_L$ and $k_K$ the respective associated residue fields. Assume that $k_K$ has prime characteristic $p>0$ and that the extension $k_L/k_K$ is separable. Then $k_L/k_K$ is actually Galois.
I really can't find out the reason what this extension of residue fields must be Galois in this case. Could someone give me a hint about it ?
Thank you in advance