Let $L/K$ be any separable extension of complete discrete valued fields of degree $p$. This work, is a study of some “standard over-extensions” of $L/K$, with the description of their Galois groups. The second target, which is the aim of this work, concerns the Galois closure of $L/K$. The study of the normal case has been done in some former work.
"On standard extensions of local fields." Tbilisi Math. J. 8 (2) 49 - 67, December 2015. https://doi.org/10.1515/tmj-2015-0010