\subsection*{Exercise 13.4.4}

Determine the splitting field and its degree over $\mathbb{Q}$ for $x^6 - 4$.

\subsubsection*{Solution}

We have $4^{1/6} = (2^2)^{1/6} = \pm 2^{1/3} \exp \left( \frac{i2\pi k}{3} \right) = \pm 2^{1/3} \left( \cos \frac{2\pi k}{3} + i \sin \frac{2\pi k}{3} \right) \enspace,$ where naturally $k \in { 0, 1, 2 }$. Thus $x^6 - 4$ splits completely in $\mathbb{Q}(\sqrt[3]{2}, \sqrt{-3})$. Since $x^3 - 2$ and $x^2 + 3$ are both irreducible in $\mathbb{Q}$ by Eisenstein’s criterion, the degree of this extension over the rationals is $2 \times 3 = 6$.