Dimension of a monic plane hypersurface
= Dimension of a monic plane hypersurface
{title2=$\dim k[X,Y]/(f)=1$}
If $f\in k[Y][X]$ is monic of positive degree in $X$, monic division makes $k[Y][X]/(f)$ free of rank $\deg_Xf$ over $k[Y]$. The inclusion of $k[Y]$ is injective and the quotient is an <integral extension>, so its <Krull dimension> is one. This argument applies even when the polynomial factors or the <field> has positive characteristic.