2
$\begingroup$

Given a field $F$ of 5 elements, show that $F[x]/\langle x^{3}+x+1 \rangle$ is not a field?

I went as follows: We can consider $F$ to be $\mathbb Z_{5}$, so $f(x)=x^{3}+x+1 $ is irreducible in $F$, then $\langle f(x) \rangle$ is maximal, so $F[x]/\langle f(x)\rangle$ is a field!! Where is my mistake??

  • 2
    Probably it should be $\ x^3 - x + 1\ =\ (x+2)\ (x^2 -2\ x - 2) \in \mathbb Z/5\:[x]$2011-07-05
  • 0
    Or, alternatively, the "not" could be a braino.2011-07-05

1 Answers 1