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I am stuck with this question:

"Determine if $GF(2011^2) [x] /<x^4-6x+12>$ 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.

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    $\begingroup$ If the polynomial is reducible, then what are the possible forms for its factors? Can you show that those factors can’t exist? $\endgroup$ – Santana Afton Oct 16 '18 at 13:14
  • $\begingroup$ 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.... $\endgroup$ – peter peter Oct 16 '18 at 20:19
  • $\begingroup$ 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. $\endgroup$ – Santana Afton Oct 16 '18 at 20:37
  • $\begingroup$ 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? $\endgroup$ – peter peter Oct 16 '18 at 23:02

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