Galois extensions for dihedral group:D8

Fourth roots of a non-square

Suppose $K$ is a field of characteristic not equal to two, such that $-1$ is not a square in $K$ . Suppose $b$ is an element of $K$ such that neither $b$ nor $-b$ is a square in $K$ . Then, the extension:

$K(\theta,i)$

where $\theta$ is a fourth root of $b$ and $i$ is a squareroot of $-1$ , is a Galois extension whose automorphism group is the dihedral group of order eight.

A concrete example is $\mathbb{Q}(2^{1/4},i)$ .

Description of the automorphisms

The dihedral group is given by the presentation:

$\langle a,x \mid a^4 = x^2 = 1, xax = a^{-1} \rangle$ .

Here, $a$ acts by fixing $i$ and sending $\theta$ to $i\theta$ , while $x$ fixes $\theta$ and sends $i$ to $-i$ .

In fact, the dihedral group acts on the set $\{ \theta, i\theta, -\theta, -i\theta \}$ precisely the way it acts on the vertices of a square. In the concrete case of $\mathbb{Q}(2^{1/4},i)$ , these elements, when plotted in $\mathbb{C}$ do form the vertices of a square, so the Galois automorphisms correspond to the usual rotations and reflections.

Galois correspondence for subgroups

For more on the subgroup structure, refer subgroup structure of dihedral group:D8.

The trivial subgroup corresponds to the whole field $K(\theta,i)$ .
The center corresponds to the subfield $K(\theta^2,i)$ .
The two-element subgroup generated by $x$ corresponds to the subfield $K(\theta)$ . The two-element subgroup generated by $a^2x$ corresponds to the subfield $K(i\theta)$ . The other two-element subgroups correspond to the subfields $K(\theta(1+i)/2)$ and $K(\theta(1-i)/2)$ .
The four-element subgroup generated by $a^2$ and $x$ corresponds to $K(\theta^2)$ .
The four-element subgroup generated by $a^2$ and $ax$ corresponds to $K(i\theta^2)$ .
The cyclic four-element subgroup generated by $a$ corresponds to $K(i)$ .
The whole group corresponds to $K$ .