Prove that $(x^2 + x+2)$ ideal is a maximal ideal in $F_3[x]$ ring.

I know the theorem about factor rings and fields

B ideal from A ring is maximal iff A/B factor-ring is a field

How can I use the theorem here?

  • $\begingroup$ Any particular reason you want to use that theorem? $\endgroup$ – Ittay Weiss Dec 24 '17 at 19:27
  • $\begingroup$ I stacked there that, if $F_3[x]/(x^2+x+2)$ is a field, from theorem the ideal will be the maximal $\endgroup$ – Spike Bughdaryan Dec 24 '17 at 19:29
  • $\begingroup$ No particular reasons $\endgroup$ – Spike Bughdaryan Dec 24 '17 at 19:30
  • $\begingroup$ What do you know about irreducible polynomials? $\endgroup$ – Ittay Weiss Dec 24 '17 at 19:32
  • 2
    $\begingroup$ The polynomial has no root in $F_3$. So, firstly ask yourself why this implies irreducibility. ( irreducibility in finite filed an infinite field.) Secondly, you can look quotient ring with ideal generated by irreducible polynomial. $\endgroup$ – 1ENİGMA1 Dec 24 '17 at 19:40

If $F$ is a field then a non-zero polynomial $f(x) \in F[x]$ is irreducible if and only if the ideal $(f)$ is prime if and only if the ideal $(f)$ is maximal. All you need to do is show that $x^2 + x + 2$ is irreducible. This is easy because all the possible factors are linear.


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