The following theorem is usually attributed to [Eduard Study](https://en.wikipedia.org/wiki/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.


Questions:


* 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?


Thanks.


Edit: I see that Study wrote a book on on the [Theory of Ternary Forms](https://archive.org/details/methodenzurtheo00studgoog/page/n6/mode/2up) (1889). I suppose the result must be in there somewhere.



History of Study's Lemma?
I found the lemma on page 63 of Study's [Einleitung in die Theorie der Invarianten linearer Transformationen auf Grund der Vektorenrechnung](https://archive.org/details/einleitungindiet01studuoft/page/62/mode/2up) (1923).


![](https://ilorentz.org/beenakker/MO/EduardStudy_1.png)
The source cited for the proof is page 202 of Study's [Methoden zur Theorie der ternären Formen](https://archive.org/details/methodenzurtheo00studgoog/page/n220/mode/2up?q=teilbar) (1889), where it appears as an [endnote](https://ilorentz.org/beenakker/MO/Eduard_Study_2.png) and refers to a "less elementary" proof by



> 
> O. Hölder, Z*um Invariantenbegriff*, Math. Naturwiss. Mitteilungen
> **1**, 59–65 (1884).
> 
> 
> 


Volume 1 of the [journal](https://www.worldcat.org/search?q=au%3AMathematisch-Naturwissenschaftlicher+Verein+in+W%C3%BCrtemberg%2C+Stuttgart.&qt=hot_author) is not online, but there is a review at [zbMath](https://zbmath.org/?q=an%3A16.0092.02) [thanks, @spin] that explicitly mentions the lemma:



> 
> Sind die ganzen rationalen Functionen $F(x\_1,x\_2,\ldots,x\_m)$,
> $G(x\_1,x\_2,\ldots,x\_m)$ so beschaffen, dass für alle (reellen oder
> complexen) Wertsysteme $x\_1,x\_2,x\_m$, wofür $G=0$ ist, auch die
> Gleichung $F=0$ gilt, und ist G irreduzibel, so ist $G$ ein Teiler
> von $F$.  
> 
>  Let the entire rational functions $F(x\_1,x\_2,\ldots,x\_m)$ and
> $G(x\_1,x\_2,\ldots,x\_m)$ be such, that for any (real or complex) values $x\_1,x\_2,\ldots x\_m$ where $G=0$ also $F=0$, and let $G$ be irreducible, then $G$ divides $F$.
> 
> 
> 


It's too late (and not productive) to change common practice, but priority wise I would conclude that Study's lemma is Hölder's lemma.


