Hello all!


I recently had a question concerning algebraic dependence that has thus far gone unanswered from my professors and texts, that I hope I can phrase properly here. When answering, please reference any papers or texts that you may happen to be citing so that I can look them up later! The statement I would like to be true is the following:


Let {$\alpha$,$\beta$} be a set of real numbers that are algebraically dependent over $\mathbb{Q}$. Then $\exists! f \in \mathbb{Z}[x,y]$ of lowest degree such that $f$ is primitive (in the sense of ring theory) and $f(\alpha,\beta) = 0$.


Does anyone know if this statement, or a close cousin perhaps, is true? If it is false, can we add/remove some hypotheses to make it true?


Thank you all in advance for your time and help!


-Richard



Minimal Polynomials for Algebraic Dependence? 
It is true if both your numbers are transcendental. It follows from the fact that height one prime ideals of $\mathbb{Q}[x,y]$ are principal. This can be found in standard algebra textbooks. Otherwise, I believe it's false. There are already counterexamples in the comments. 


