TITLE: Tensor products of fields
QUESTION [5 upvotes]: Let $K/F$ be a field extension.  I am interested in the situation where there exists a field extension $L/F$ such that the ring $L \otimes_FK$ is not a field.
If there exists $z\in K \setminus F$ such that $z$ is algebraic over $F$, then I think I can show that $K \otimes_F K$ is not a field.  So my question is: If no such element exists in $K$, does it follow that for any field extension $L$ of $F$, $L \otimes_F K$ is a field?

REPLY [7 votes]: As a graduate student I remember being disappointed that it was hard to find much information concerning tensor products of fields.  Later, as with many things, I realized that it depends a good bit on knowing where to look: it turns out that the more standard topic (found in most "serious" treatments of field theory) of linear disjointness is closely related.  
You can find some material on when $L \otimes_F K$ is a field (and also when it is a domain) in $\S 12$ of my field theory notes.