Find the product of all quadratic irreducible polynomials of $\mathbf{Z}_{3}[x]$?

Question

I'm requested to find the product of all quadratic irreducible polynomials of $\mathbf{Z}_{3}[x]$ . How can I find them ? brute force ? check that every polynomial has no roots ?

Or , if I take for example $f(x)=ax^2 + bx +c $ , find restrictions for a,b & c ?

I'd appreciate your help

Regards

Answer 1

I think you are asking for all the polynomials of degree 2 irreducible over ${\bf Z}_3$. Let $f(x)=ax^2+bx+c$. You may assume $a=1$ (why?). You may assume $c\ne0$ (why?). That leaves you only six polynomials to check (why?).

Alternatively, you could argue that if it's not irreducible it must be a product of two linear factors. So once you get rid of $x^2,x(x-1),\dots,(x-2)^2$, the remaining (monic) polynomials must be irreducible.