Composite of two purely inseparable extensions is purely inseparable. Let $L,F$ be extensions of the field $K$ and are contained in a common field. Prove that, if $L$ and $F$ are purely inseparable extensions over $K$ then $LF$ is also a purely inseparable extension over $K$. Is the converse true? How do I prove it? Thanks in advanced.

Assume the characteristic of your fields is $p>0$. You want to prove that, for a field extension $L/K$, the set $L^\prime$ of elements of $L$ that are purely inseparable over $K$ is a subfield of $L$. You do this with the following characterization of pure inseparability: an element $x\in L$ is purely inseparable over $K$ if and only if, for some integer $n\geq 0$, $x^{p^n}\in K$. A proof of this fact can be found here: Purely inseparable extension.

With this, your result can be proved as follows. Consider the _subfield_ $E$ of $LF$ consisting of elements purely inseparable over $K$. Because $L$ and $F$ are both purely inseparable over $K$, $L,F\subseteq E$, so $LF\subseteq E$, and thus $LF=E$ is purely inseparable over $K$.

The converse is also true. More generally, if $K\subseteq L\subseteq E$ and $E/K$ is purely inseparable, then $L/K$ is purely inseparable. This is immediate from the definition of pure inseparability.