D+F 14.6.5

Determine the Galois group of $x^4+4$.

$\textit{Solution}$ : By problem 13.2.3, we know that the splitting field for $x^4+4$ is only $\mathbb{Q}[i]$, which is a degree 2 extension. The 4 roots of $x^4+4$ are $\pm 1 \pm i$ so $x^4+4$ is separable since none of the roots are repeated. By prop. 5 p.562, this means that $$|Aut(\mathbb{Q}[i])/\mathbb{Q})|=|\mathbb{Q}[i]:\mathbb{Q}|=2$$ $\Longrightarrow \mathbb{Q}[i]$ is Galois over $\mathbb{Q}$ and the Galois group is $\mathbb{Z}_2$.