# 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 .