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
.