Weil proved the following theorem in his book Foundations of Algebraic Geometry, p.19. The proof is somewhat involved. I wonder if the theorem is his original.

Theorem Let $K[X_1,\dots, X_n]$ be the polynomial ring over a field $K$. Let $I$ be an ideal of $K[X_1,\dots, X_n]$. There exists a smallest subfield $k$ of $K$ such that $I$ is generated by polynomials in $k[X_1,\dots,X_n]$.