Let $F$ be the splitting field of a separable polynomial over $K$, and let $E$ be a subfield between $K$ and $F$. Show that if $[E:K]=2$, then E is the splitting field of some polynomial over $K$.
- I've seen solutions that just go through some rote calculation of some irreducible $f(x)$, but I would rather go about it by correlating the index of some subgroup H of $G=Gal(F/K)$ to the degree of $[E:K]$. Since normal subgroups correspond to normal field extensions by FTGT then it shows the extension is normal. I am however having a hard time with expressing this the correct way.
Hint. By the fundamental theorem of Galois theory, the degree $[E:K]$ equals the index $[G:H]$. Group theory: what do you know about subgroups of index $2$?