TITLE: If $L/k$ is Galois and $k\subseteq K$ is any field extension, is $LK$ Galois over $L$
QUESTION [5 upvotes]: There is a theorem in Lang which says that if $L/k$ is Galois and $k\subseteq K$ is any field extension, if $L,K$ are subfields of a larger field, then $LK$ is Galois over $K$. I was wondering if $LK$ is also Galois over $L$?
I think this would be true if $K\subseteq L$ since then we can apply the same theorem to the extension $K\subseteq LK$ and use that $(LK)L=LK$. Is that right?

REPLY [3 votes]: No, in general $LK$ is not Galois over $L$.
Indeed, take $L=k$. Then $LK=K$ and the question becomes : 
$$\text {Is an arbitrary field extension } \;K/k  \; \text {Galois?} $$
Need I answer?