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### (how/can) i declair this isomorphism

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

$\psi$: $\mathbb F_{q^m}$ $\rightarrow$ $\mathbb{F}_{q}[U]$

$\alpha$ $\mapsto$ & $\psi(\alpha)=U$

### (how/can) i declair this isomorphism

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

$\psi$: $\mathbb F_{q^m}$ $\rightarrow$ $\mathbb{F}_{q}[U]$

$\alpha$ $\mapsto$ & $\psi(\alpha)=U$ 3 retagged FrédéricC 2604 ●3 ●29 ●54

### (how/can) i declair this isomorphism

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

$\psi$: $\mathbb F_{q^m}$ $\rightarrow$ $\mathbb{F}_{q}[U]$

$\alpha$ $\mapsto$ $\psi(\alpha)=U$