Determine the splitting field and its degree over $\mathbb{Q}$ for $x^4 + 2$.

### Solution

We have

Therefore all we need to adjoin to $\mathbb{Q}$ is $2^{1/4}$ and $i$. By Eisenstein’s criterion, $x^4 + 2$ is irreducible in $\mathbb{Q}$, hence by the same reasoning in Exercise 13.4.1, we find that $x^4 + 2$ splits completely in $\mathbb{Q}(2^{1/4}, i)$, which is an extension over the rationals of degree 8.