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

### Solution

We have

Since

we find that $x^4 - 2$ splits completely in $\mathbb{Q}(2^{1/4}, i)$. To see that

we note that $\mathbb{Q}(2^{1/4}) \subset \mathbb{R}$ and, as $i \notin \mathbb{R}$, adjoining $i$ to $\mathbb{Q}(2^{1/4})$ gives a proper extension of degree at least 2. But $i$ is a root of the quadratic $x^2 + 1 \in \mathbb{Q}(2^{1/4})$, hence this extension is of degree at most 2, so it must be that its degree is precisely 2. Now $x^4 - 2$ is irreducible in $\mathbb{Q}$ by Eisenstein’s criterion, implying

By the multiplicative property of field extensions’ degrees, we then conclude