Problem 6

Consider the polynomial $f(x)=x^{3}-2$. Let $E$ be the splitting field of $f(x)$ over $\mathbb{Q}$.

1. Find a field $F$ such that $F$ is a simple extension of $\mathbb{Q}$, its primitive element is a real number, and $\mathbb{Q}\subset F\subset E$.

2. Show that $F$ is not a splitting field of any polynomial in $\mathbb{Q}[x]$.