Hi,
let $U$ be a square matrix of order $m$ over $\mathbb F_{q}$, more precisely $U$ is the companion matrix of a monic irreducible polynomial over $\mathbb F_{q}$ that define $\mathbb F_{q^m}$ .
let $\alpha$ be a primitive element of $\mathbb F_{q^m}$
I wish to declare this morphism to compute some examples with SAGEMATH
$\psi$: $\mathbb F_{q^m}$ $\rightarrow$ $\mathbb{F}_{q}[U]$
$\alpha$ $\mapsto$ & $\psi(\alpha)=U$
thanks in advance;