If $F/L$ is normal and $L/K$ is purely inseparable, then $F/K$ is normal
Solution 1:
Hint. Take a set of polynomials $f_i\in L[x]$ such that $F$ is generated by the roots of these polynomials. Since $L/K$ is purely inseparable, how can you make the $f_i$ into polynomials in $K[x]$ without affecting their roots?
You might find the fact that if $\text{char}(K)=p>0$, then $x\mapsto x^p$ is an endomorphism of $K$ (called the Frobenius endomorphism) useful.