General semiaffine group of degree one
Definition
The general semiaffine group of degree one over a field , denoted
or
, is the general semiaffine group of degree one over
. It can be defined explicitly as the group of all permutations of
that can be expressed in the form:
It can be viewed as an iterated semidirect product:
The left parenthesized expression shows that the group can be viewed as a semidirect product with base the general affine group of degree one and acting group the Galois group:
The right parenthesized expression shows that the group can be viewed as a semidirect product with base the additive group of the field and acting group the general semilinear group of degree one:
Suppose is the prime subfield of
and
is a Galois extension of
. This always happens if
is a finite field. Then,
equal the Galois group
.
Alternative definition as automorphisms of a polynomial ring
The group can also be defined as the group of all ring automorphisms of the polynomial ring
. The subgroup
of those automorphisms that fix the base field
can be identified with the general affine group of degree one
.