Fields1

From Exampleproblems

Let $\alpha=\sqrt[3]{5}$ and $\zeta_3=\frac{1}{2}(-1+\sqrt{-3})$ and note that the roots of $f\,$ in $\mathbb{C}$ are $\alpha,\,\alpha\zeta_3,\,\alpha\zeta_3^2$ . Then the splitting field of $f(x)=x^3-5\,$ over $\mathbb{Q}$ is $E=\mathbb{Q}(\alpha,\alpha\zeta_3,\alpha\zeta_3^2)=\mathbb{Q}(\alpha,\alpha\zeta_3)=\mathbb{Q}(\alpha,\zeta_3)$ .

Note the minimal polynomial of $\alpha=\sqrt[3]{5}$ over $\mathbb{Q}$ is $m_{\alpha,\mathbb{Q}}(x)=x^3-5\,$ , so $[\mathbb{Q}(\alpha):\mathbb{Q}]=3$ . Further, the minimal polynomial of $\zeta_3\,$ over $\mathbb{Q}(\alpha)$ is $m_{\zeta_3,\mathbb{Q}(\alpha)}(x)=x^2+x+1$ , so $[\mathbb{Q}(\alpha)(\zeta_3):\mathbb{Q}(\alpha)]=2$ . Thus, $[E:\mathbb{Q}]=[\mathbb{Q}(\alpha,\zeta_3):\mathbb{Q}(\alpha)]\cdot[\mathbb{Q}(\alpha):\mathbb{Q}]=2\cdot3=6$ .