<s>[INST] factorization of polynomials in $ \mathbb F_7$ I am not sure in finding a factorization of $f(X) = 6 x^6 +1 \in \mathbb F_7$.
I got $f(X) =6\cdot (x-1)\cdot(x+1)\cdot(x-2)\cdot(x+2)\cdot(x-3)\cdot(x+3) $
Is that correct?
 [/INST] In $\Bbb{F}_7$ one has on one hand
$$\begin{align}
6x^6+1&=6(x^6-1)\\
&=6(x^2-1)(x^4+x^2+1)\\
&=6(x-1)(x+1)(x^4+x^2+1)
\end{align}$$
On the other hand
$$(x-2)(x+2)(x-3)(x+3)=(x^2-4)(x^2-2)=x^4-6x+8=x^4+x^2+1$$
So the factorisation is correct.
</s>