Show that the ideal generated by a polynomial and a prime is maximal

Question

I need to show that the ideal generated by $x^2 - x + 1$ and 17 in $\mathbb{Z}[x]$ is maximal

As far as I know, it would be sufficient to show that the polynomial is irreducible mod 17, what could be a good approach to prove this?

Answer 1

Your polynomial is of degree $2$, and so it is irreducible over $\mathbb{Z}_{17}$ if and only if it does not have any roots in $\mathbb{Z}_{17}$, so you only need to plug in $17$ numbers into the polynomial and check that you don't get any zeros.