What does it mean to say that A is an $\Bbb F_3$ endomorphism of F .
Where $F:=\Bbb F_3(\beta)$, and we define $A:F\rightarrow F$ where $A(x)=\beta x $ $\forall x\in F$ ?
I know that an endomorphism is a homomorphism from an object to itself. the part that confuses me is that its an $\Bbb F_3$ endomorphism. It would make more sense to me if it was an $\Bbb F_{27}$ endomorphism as F is isomorphic to this field.
I'm supposing that the fact that $x$ is mapped onto $ \beta x$ is what makes it an $\Bbb F_3$ endomorphism but I dont really understand how as I would have thought that it still maps 27 elements onto 27 elements ?
$\mathbb{F}_3$ endomorphism means it's an endomorphism which additionally fixes $\mathbb{F}_3$, and is $\mathbb{F}_3$ linear. In particular, for any $k \in \mathbb{F}_3$, $f(k x) = k f(x)$ and $f(k) = k$.