If $a \in \mathbb{F}_q$ and $n \in \mathbb{N}$. Then $x^{q^n}-x+na$ is divisible by $x^q-x+ a$ over $\mathbb{F}_q$.

Question

If $\beta$ is root $x^q-x+a$ then $\beta^q-\beta \in \mathbb{F}_q$. I thought about the frobenius map but could not get anywhere. Would you please help me with the remaining?

Answer 1

If $\;\beta^q-\beta=-a\iff \beta^a=\beta-a\in\Bbb F_q\;$ , then

$$-a=(-a)^q=(\beta^q-\beta)^q=\beta^{q^2}-\beta^q=\beta^{q^2}-\beta+a\implies\beta^{q^2}-\beta+2a=0$$

and there you have the hint (first non-trivial example) for proving inductively that $\;\beta\;$ root of $\;\beta^q-\beta+a\implies\beta\;$ root of $\;\beta^{q^n}-\beta+na\;$