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Let $F$ be a finite field of characteristic $2$. Let $m \geq 2$ be a positive integer. Seems unknown if the trinomial $$T(t,x) = x^{2m+1}+x^2+s(t) \in F[t][x]$$ (more explicitly, the constant coefficient $s(t)$ is a polynomial in $t$, i.e., $s(t) \in F[t]$)

has monic factors $D(t,x)$ in $F[t][x]$ with degree $3$ relative to $x$.

I.e., $$D(t,x) = x^3+ a_2(t)x^2+a_1(t)x + a_0(t)$$ with $a_2(t), a_1(t),a_0(t) \in F[t]$ and $T(t,x) = D(t,x)K(t,x)$ for some $K(t,x) \in F[t][x].$

We will then say that $T(t,x)$ have $D(t,x)$ as a factor or that $D(t,x)$ divides $T(t,x)$.

Question: Assume that $D(t,x)$ as above divides the trinomial $T(t,x).$

Do we have $$\deg(a_1(t)) = 2 \deg(a_2(t)),\;\;\deg(a_0(t))=3\deg(a_2(t)).$$ ???

I am aware of the work of Schinzel on trinomials. The trinomials $T(t,x)$ do not seem to be worked out in these papers.

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Reducible trinomials $x^{odd}+x^2+s(t)$ in characteristic $2$

Let $F$ be a finite field of characteristic $2$. Let $m \geq 2$ be a positive integer. Seems unknown if the trinomial $$T(t,x) = x^{2m+1}+x^2+s(t) \in F[t][x]$$ has monic factors $D(t,x)$ in $F[t][x]$ with degree $3$ relative to $x$.

I.e., $$D(t,x) = x^3+ a_2(t)x^2+a_1(t)x + a_0(t)$$ with $a_2(t), a_1(t),a_0(t) \in F[t]$ and $T(t,x) = D(t,x)K(t,x)$ for some $K(t,x) \in F[t][x].$

We will then say that $T(t,x)$ have $D(t,x)$ as a factor or that $D(t,x)$ divides $T(t,x)$.

Question: Assume that $D(t,x)$ as above divides the trinomial $T(t,x).$

Do we have $$\deg(a_1(t)) = 2 \deg(a_2(t)),\;\;\deg(a_0(t))=3\deg(a_2(t)).$$ ???

I am aware of the work of Schinzel on trinomials. The trinomials $T(t,x)$ do not seem to be worked out in these papers.