Exercise 5.2.1

Proof. This is equivalent to show that $\mathbb{Q}(\sqrt[4]{2})$ is not a normal extension of $\mathbb{Q}$.

$x^4-2$ is an irreducible polynomial over $\mathbb{Q}$ by Schönemann-Eisenstein Criterion with $p=2$.

The roots of the minimal polynomial of $\sqrt[4]{2}$ over $\mathbb{Q}$ are $\sqrt[4]{2},i\sqrt[4]{2},-\sqrt[4]{2},-i\sqrt[4]{2}$.

As the root $i\sqrt[4]{2}$ is a non real complex, it is not in $\mathbb{Q}(\sqrt[4]{2})\subset ℝ$. So $\mathbb{Q}\subset \mathbb{Q}(\sqrt[4]{2})$ is not a normal extension, thus $\mathbb{Q}(\sqrt[4]{2})$ is not the splitting field of any polynomial in $\mathbb{Q}[x]$. □