Suppose that $E$ is a splitting field over $\mathbb{Q}$ for $f(x)=x^3-2$. We want to show using Galois theory that $g(x)=x^2-5$ is irreducible in $E[x]$.
Here is what I have so far. Assume for contradiction that $g(x)$ is reducible in $E[x]$. $g(x)$ has roots $\sqrt 5$ and $-\sqrt 5$. Then we have a subfield $\mathbb{Q}(\beta)$ in $E$. We have that $\mathbb{Q} \subset \mathbb{Q}(\beta) \subset E$. I then hope to use The Fundamental Theorem of Galois Theory to reach a contradiction using the degrees of the extensions although I am not sure how I would go about this?