# determining if quotient ring of polynomials over a finite field is a field or not

I am stuck with this question:

"Determine if $$GF(2011^2) [x] /$$ is a field or not."

I know that since this polynomial ring is defined over a field, I only have to determine if $$x^4-6x+12$$ is irreducible or not. But I have no idea how to do this.

Professor told me to check what happens if I assume this polynomial to be reducible. But I'm still in a mist.

Thank you in advance for help.

• If the polynomial is reducible, then what are the possible forms for its factors? Can you show that those factors can’t exist? – Santana Afton Oct 16 '18 at 13:14
• I have tried both (linear)(cubic) and (quadratic)(quadratic) and compared coefficients with original polynomial but ended up with nothing. I might have gone wrong in the middle but I couldn't make a conclusion doing that.... – peter peter Oct 16 '18 at 20:19
• For the quadratics case, it looks like it comes down to finding some element $a$ such that $a^3 = -6$. Can you prove that this polynomial has no root in the given field? Note that I'm sure there's a far more slick way of proving this. – Santana Afton Oct 16 '18 at 20:37
• How did you get that? I just tried once again but still couldn't get to that far... and assuming I get that, if I bring that down to (mod 4) then $a^3=2(mod4)$ has no solution, I see. Is it right? Moreover how do I exclude linear factor case? – peter peter Oct 16 '18 at 23:02