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

primitivemeans that there is no nonunit simultaneously dividing all the coefficients. – Pete L. Clark Jan 25 '11 at 4:48