Galois extensions for dihedral group:D8

This article gives specific information, namely, Galois extensions, about a particular group, namely: dihedral group:D8.
View Galois extensions for particular groups | View other specific information about 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:

$⟨a,x\mid a^4=x^2=1,xax=a^{-1}⟩$.

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 $a^{2}x$ 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 $K$.