General affine group of degree one

For a field
For a field $$K$$, the general affine group of degree one over $$K$$ is defined as the general affine group over $$K$$ of degree one. Equivalently, it is the external semidirect product of the additive group of $$K$$ by the multiplicative group of $$K$$, where the latter acts naturally on the former by field multiplication. Explicitly, it is denoted $$GA(1,K)$$ or $$AGL(1,K)$$, and can be written as:

$$GA(1,K) = K \rtimes K^\ast$$

Alternative definition as automorphisms of a polynomial ring
For a field $$K$$, the general affine group of degree one $$GA(1,K)$$ can be defined as the group $$\operatorname{Aut}_K(K[x])$$.

Note that this definition does not extend to general affine groups of higher degree. For $$n > 1$$, $$GA(n,K)$$ naturally sits as a subgroup inside $$\operatorname{Aut}_K(K[x_1,x_2,\dots,x_n])$$ but is not the whole automorphism group.

For a finite number
Let $$p$$ be a prime number and $$q = p^r$$ be a power of $$p$$. The general affine group or collineation group $$GA(1,q)$$ is defined as follows. Let $$\mathbb{F}_q$$ denote the field with $$q$$ elements. Then $$GA(1,q)$$ is the semidirect product of the additive group of $$\mathbb{F}_q$$ with its multiplicative group.

Equivalently it is the general affine group of degree $$1$$ over the field of $$q$$ elements.

Arithmetic functions
Below, $$q$$ is the size of the field and $$p$$ is the underlying prime (the characteristic of the field). We have $$q = p^r$$ where $$r$$ is a positive integer.