## Compute the splitting field of x⁴-2 over QQ

Compute the splitting field of $p(x) = x^4-2$ over $\mathbb{Q}$, as well as its degree.

Let $\sqrt[4]{2}$ denote the positive real fourth root of 2. Evidently, $p(x) = (x-\sqrt[4]{2})(x+\sqrt[4]{2})(x-i\sqrt[4]{2})(x+i\sqrt[4]{2})$. So the splitting field of $p(x)$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt[4]{2}, i\sqrt[4]{2}) = \mathbb{Q}(i,\sqrt[4]{2})$.

Note that $\mathbb{Q}(i)$ has degree 2 over $\mathbb{Q}$. Over the UFD $\mathbb{Z}[i]$, $p(x)$ is Eisenstein at $1+i$, hence irreducible, and so by Gauss’ Lemma, is irreducible over $\mathbb{Q}(i)$ (the field of fractions of $\mathbb{Z}[i]$). So $\mathbb{Q}(i,\sqrt[4]{2})$ has degree 4 over $\mathbb{Q}(i)$. We can visualize this scenario in the following diagram.

A field diagram

So $\mathbb{Q}(i,\sqrt[4]{2})$ has degree 8 over $\mathbb{Q}$.