Prove that $x^2+x+2$ is maximal ideal in $F_3[x]$ ring.
I think of using the theorem that $B$ is maximal ideal in $A$ if $A/B$ is field. Particularly in this case we need to prove that $F_3[x]/(x^2+x+2)$ is field. Could you please help?
2026-04-06 00:17:27.1775434647
Prove that $x^2+x+2$ is maximal ideal in $F_3[x]$ ring.
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Just verify that the polynomial does not have a root. If it was not maximal, it would have had a divisor of degree $1$, hence a root.