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

where is a fourth root of and is a squareroot of , is a Galois extension whose automorphism group is the dihedral group of order eight.

A concrete example is .

### Description of the automorphisms

The dihedral group is given by the presentation:

.

Here, acts by fixing and sending to , while fixes and sends to .

In fact, the dihedral group acts on the set precisely the way it acts on the vertices of a square. In the concrete case of , these elements, when plotted in 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 .
- The center corresponds to the subfield .
- The two-element subgroup generated by corresponds to the subfield . The two-element subgroup generated by corresponds to the subfield . The other two-element subgroups correspond to the subfields and .
- The four-element subgroup generated by and corresponds to .
- The four-element subgroup generated by and corresponds to .
- The cyclic four-element subgroup generated by corresponds to .
- The whole group corresponds to .