Consider the following statement:

Assume $E$ and $F$ are unramified (over some fixed prime) finite separable extensions of a field $K$. Then $EF$ is also unramified.

I always thought that it is called Abhyankar's lemma (and indeed it is a special case of it, see https://en.wikipedia.org/wiki/Abhyankar%27s_lemma).

My question is whether this statement is indeed named after Abhyankar. If so, a reference would be appreciated, if not, how is it called?

allmaximal ideals of the integral closure of $V$ in $EF$ (and the contractions of such to $E$ and $F$)? Whatever it is, this shouldn't be named after Abhyankar, since his lemma is all about how to go beyond this much more classical case. $\endgroup$ – nfdc23 Jan 30 '18 at 18:14Local Fields, or Thm. 2 + Cor. 2 in section 7 of Ch. I in Cassels-Frohlich, or ...) to conclude the one for $EF/K$ contains $E/K$ and $F/K$, hence is $EF$. $\endgroup$ – nfdc23 Jan 30 '18 at 22:08