Galois extensions for dihedral group:D8

This article discuss various examples of Galois extensions of fields for which the Galois group is 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.


 * 1) The trivial subgroup corresponds to the whole field $$K(\theta,i)$$.
 * 2) The center corresponds to the subfield $$K(\theta^2,i)$$.
 * 3) 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 $$K(\theta(1-i)/2)$$.
 * 4) The four-element subgroup generated by $$a^2$$ and $$x$$ corresponds to $$K(\theta^2)$$.
 * 5) The four-element subgroup generated by $$a^2$$ and $$ax$$ corresponds to $$K(i\theta^2)$$.
 * 6) The cyclic four-element subgroup generated by $$a$$ corresponds to $$K(i)$$.
 * 7) The whole group corresponds to $$K$$.