Origin of the theorem on the existence of the smallest field of definition of an affine variety
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 $k[X_1,\dots,X_n]$.