Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 24 / 3 / a

For the affine polynomial $f(x,y)=y^2+y-x^3+x$, a singular point of an algebraic variety must satisfy $f=f_x=f_y=0$. Modulo $37$, the derivative equations are $2y+1=0$ and $3x^2=1$. Thus $y=18$ and $x^2=25$, giving candidates $x=5$ and $x=32$. At $(5,18)$, the two sides of the curve equation are both $9$ modulo $37$. At $(32,18)$, the right side is $28$, so that candidate is not on the curve. The point at infinity is nonsingular: for the homogeneous equation $Y^{2}Z+Y{Z}^2-X^3+X{Z}^2=0$, the derivative with respect to $Z$ is one at $(0:1:0)$. Therefore the unique singular point of an algebraic variety is