Exercise 5.7.4 (Rational curve passing through given rational points)

Proof. If we apply Problem 3 in the field $K=\mathbb{Q}$, we obtain that if $\left(\genfrac{}{}{0.0pt}{}{d+2}{2}\right)-1$ rational points are given in the plane, then there exists a polynomial $g(x,y)\in \mathbb{Q}[x,y]$ of degree at most $d$, with rational coefficients, not all $0$, such that the given points all lie on the curve ${𝒞}_g(\mathbb{Q})$. If $k\ne 0$ is a common multiple of the denominators of the coefficients of $g$, and $g(x,y)=\mathit{kf}(x,y)$, then ${𝒞}_f(\mathbb{Q})={𝒞}_g(\mathbb{Q})$, and $f$ has integral coefficients.

There exists a polynomial $f(x,y)$ of degree at most $d$, with integral coefficients, not all $0$, such that the given points all lie on the curve ${𝒞}_f(\mathbb{Q})$. □