Galois extensions for dihedral group:D8

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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:

\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.