In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the inseparable case. In between the two cases he writes "a straightforward dévissage argument reduces one to the case of an inseparable extension.
What is dévissage argument? I saw in wikipedia something with the same name. Is there a connection?
If it matters, the proposition statement is as follows: Let $K$ be a complete field with respect to a discrete valuation $\nu$ with valuation ring $A$, and let $L/K$ be a finite extension. Let $B$ be the integral closure of $A$. Then $B$ is a discrete valuation ring and is a free $A$-module of rank $[L:K]$; also, $L$ is complete in the topology defined by $B$.
(So in this case the reduction really is easy, as one can break the extension into a separable part and an inseparable part. It is clear that if the statement holds for $L/E$ and for $E/K$ then it holds for $L/K$)