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?

Yes. The easy version is noticing that $6=-1$, thus $$6x^6+1=-x^6+1$$ By Fermat's little theorem, $x^6-1$ has the six roots $1,2,3,4,5,6$ in $\Bbb F_7$.
