General semiaffine group of degree one

Definition

The general semiaffine group of degree one over a field $K$, denoted $\Gamma A(1,K)$ or $A\Gamma L(1,K)$, is the general semiaffine group of degree one over $K$. It can be defined explicitly as the group of all permutations of $K$ that can be expressed in the form:

$x \mapsto a \sigma(x) + b, a \in K^\ast, b \in K, \sigma \in \operatorname{Aut}(K)$

It can be viewed as an iterated semidirect product:

$(K \rtimes K^\ast) \rtimes \operatorname{Aut}(K) = K \rtimes (K^\ast \rtimes \operatorname{Aut}(K))$

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:

$\Gamma A(1,K) = GA(1,K) \rtimes \operatorname{Aut}(K)$

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:

$\Gamma A(1,K) = K \rtimes \Gamma L(1,K)$

Suppose $k$ is the prime subfield of $K$ and $K$ is a Galois extension of $k$. This always happens if $K$ is a finite field. Then, $\operatorname{Aut}(K)$ equal the Galois group $\operatorname{Gal}(K/k)$.

Alternative definition as automorphisms of a polynomial ring

The group $\Gamma A(1,K)$ can also be defined as the group of all ring automorphisms of the polynomial ring $K[x]$. The subgroup $\operatorname{Aut}_K(K[x])$ of those automorphisms that fix the base field $K$ can be identified with the general affine group of degree one $GA(1,K)$.