# Galois group of the field extension

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 $\Bbb Q(\sqrt{2})$ is a splitting field. But I couldn't proceed further. Could somebody please give me hint? Thanks.

• Hint: the extension $E/\Bbb Q(\sqrt{-1})$ is cyclic of degree $4$. – Ángel Valencia Mar 28 '16 at 23:39
• The splitting field is $\mathbf Q[\sqrt2,\mathrm i]$. It has degree $8$ over $\mathbf Q$, hence the Galois group has order $8$. – Bernard Mar 28 '16 at 23:41
• @ Ángel Valencia.. I see but what would be its Galois group? – UserAb Mar 28 '16 at 23:48
• Note that the Galois group of $E/\Bbb Q$ is a subgroup of $S_4$ of order 8. What group is it? – Ángel Valencia Mar 28 '16 at 23:58

Consider the polynomial $x^4-2$. It has $\sqrt{2}$ as a root; so $\Bbb Q(\sqrt{2})\subseteq E$. On the other hand, $x^4-2$ has two complex roots; since $\Bbb Q(\sqrt{2})$ is a real field (i.e. a subfield of $\Bbb R$), you have the extension $\Bbb Q(\sqrt{2},i)/\Bbb Q(\sqrt{2})$, which has degree $2$, and $x^4-2$ splits completely in this field; so $\Bbb Q(\sqrt{2})\subset E\subseteq\Bbb Q(\sqrt{2},i)$. Since the extension $\Bbb Q(\sqrt{2},i)/\Bbb Q(\sqrt{2})$ has no intermediate fields, you have $E=\Bbb Q(\sqrt{2},i)$, 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 $D_8$.