I've got this question:
"Write $2+4x+6x^2+x^3$ as a product of irreduible elements, considering it an element in each of the following rings: $\mathbb{Z}[x]$,$\mathbb{Q}[x]$ and $\mathbb{Z}_3[x]$."
My solution:
I used Eisenstein’s Criterion on the first two $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ where I can conclude it's irreducible, I mean, there exists no roots in the rings.
But however in $\mathbb{Z}_3[x]$ I've used modulo for the polynomial $2+4x+6x^2+x^3$ and I get: $2+x+x^3$. I see that $x=-1$ is a solution which will be $x=2$ in the ring $\mathbb{Z}_3[x]$. By applying polynomial division I get $\frac{2+x+x^3}{x+1}=x^2-x+2$ so are my answer correct? Sum up:
In the ring $\mathbb{Z}_3[x]$ the polynomial $2+4x+6x^2+x^3$ are: $(x^2-x+2)(x+1)$ because $(x^2-x+2)(x+1)=2+x+x^3$.
Hope it's clear, thank you.