Determine the Galois group of the field extension $E/\mathbb{Q}$, where $E$ is the splitting field of the polynomial $x^4-2\in \mathbb{Q}[x] $.
Here it is clear that $\Bbb Q(\sqrt[4]{2})$ is a splitting field. But I couldn't proceed further. Could somebody please give me hint? Thanks.
Consider the polynomial $x^4-2$. It has $\sqrt[4]{2}$ as a root; so $\Bbb Q(\sqrt[4]{2})\subseteq E$. On the other hand, $x^4-2$ has two complex roots; since $\Bbb Q(\sqrt[4]{2})$ is a real field (i.e. a subfield of $\Bbb R$), you have the extension $\Bbb Q(\sqrt[4]{2},i)/\Bbb Q(\sqrt[4]{2})$, which has degree $2$, and $x^4-2$ splits completely in this field; so $\Bbb Q(\sqrt[4]{2})\subset E\subseteq\Bbb Q(\sqrt[4]{2},i)$. Since the extension $\Bbb Q(\sqrt[4]{2},i)/\Bbb Q(\sqrt[4]{2})$ has no intermediate fields, you have $E=\Bbb Q(\sqrt[4]{2},i)$, and $[E:\Bbb Q]=8$.
Now, remember that the Galois group of $E/\Bbb Q$ is a transitive subgroup of $S_4$. By group theory it is known that any subgroup of order $8$ of $S_4$ is isomorphic to the dihedral group $D_8$. Therefore, the Galois group of $E/\Bbb Q$ is $D_8$.