Showing that an ideal is maximal

Question

Let $k$ be an algebraically closed field and $f$ be the polynomial $x_1x_2+x_2x_3+x_3x_1$ in $k[x_1, x_2, x_3]$. Here $f$ is irreducible.

Then this polynomial ring is not a $PID$, it is only an $UFD$. But I have to show that the ideal generated by $f$ is maximal in $k[x_1, x_2, x_3]$. Could anyone help me with this?

Answer 1

The ideal generated by $f$ is not maximal. It is properly contained in the maximal ideal $(x_1,x_2,x_3)$.

Answer 2

Maybe the problem is showing that the ideal is prime. Since $k[x_1, x_2, x_3]$ is an UFD it's equivalent to $f$ is irreducible. Now $f$ is homogenous of degree $2$, a quadratic form of rank $3$ ( I stay away from $\text{char}\, 2$ but should work there too). Such a form cannot be written as a product of two linear forms since that would imply $\text{rank} \le 2$.