While studying Patrick Morandi's book "Field and Galois Theory", on page49, I came across the following question:
Let $F\subseteq L\subseteq K$ be fields such that $K/L$ is normal and $L/F$ is purely inseparable. Show that $K/F$ is normal.
What I know about normal extension is an equivalent conditon, which says that the following conditions are equivlent
- $K/F$ is normal extension.
- $M$ is an algebraic closure of $K$ and if $\tau :K\rightarrow M$ is an $F-$homo, then $\tau(K)=K$.
- If $F\subseteq L\subseteq K\subseteq N$ are fields and if $\sigma:L\rightarrow N$ is an $F-$ home, then $\sigma (L)\subseteq K$, and there is a $\tau \in Gal(K/F)$ with $\tau \mid_L=\sigma$.
- For any irreducible $f(x)\in F[x]$, if $f$ has a root in $K$, then $f$ splits over $K$.
But I just don't know how to use the above results. I am not familiar with these, so I hope someone could provide an answer with details. Thank you very much!