Suppose that $F \subset K \subset E$ are fields and $E$ is the root field of some polynomial in $F[x]$. Show, by means of an example, that $K$ need not be the root field of some polynomial in $F[x]$.
I have tried many examples and I cannot come up with anything. I know some basic Galois theory. Any help would be appreciated! Thanks!
Let $F=\mathbb{Q}$, let $K=\mathbb{Q}(\sqrt[4]{2})$, and let $E$ be the root field over $\mathbb{Q}$ of the polynomial $x^4-2$.
Then $K$ is not a root field over $\mathbb{Q}$, since the smallest root field over $\mathbb{Q}$ containing $K$ is $E$, and $K$ is a proper subfield of $E$ (all elements of $K$ are real, and only two of the roots of $x^4-2$ are real).