Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 24F / c / Solution

Suppose an affine line lies in $W$. Parametrize it asThe defining equation would give the polynomial identityThe only units of $k[t]$ are nonzero constants. Since a product is a unit only when every factor is a unit, all three linear polynomials must be constant, so $b_1=b_2=b_3=0$. The alleged line is a point, a contradiction. Therefore