The following theorem is usually attributed to Eduard Study:
Let $f(x,y)$ and $g(x,y)$ be polynomials in two variables over a field, with $f$ irreducible. If $f\nmid g$ then the curves $C_f:f=0$ and $C_g:g=0$ have finitely many points of intersection. Consequently, If the field is algebraically closed and $C_f\subseteq C_g$ (hence $C_f\cap C_g$ has infinitely many points) then $f|g$.
However, I have not been able to track down any reference to this result outside of modern textbooks.
- What is the original reference for this result?
- What did Study actually prove?
- What was the context?
- Did this result directly influence later versions of the Nullstellensatz?
Edit: I see that Study wrote a book on on the Theory of Ternary Forms (1889). I suppose the result must be in there somewhere.