An effective version of the primitive element theorem

Abstract

Let $\alpha$ and $\beta$ be two algebraic numbers, $F={{\mathbb {Q}}}(\alpha ,\beta )$ and $d=[F:{{\mathbb {Q}}}] \ge 2$. By the primitive element theorem, for all but finitely many rational numbers r we have $F={{\mathbb {Q}}}(\alpha +r\beta )$. A straightforward argument implies that the number of exceptional r, namely, those $r \in {{\mathbb {Q}}}$ for which ${{\mathbb {Q}}}(\alpha +r\beta )$ is a proper subfield of F, is at most $(d-1)^2$. We show that the number of exceptional r is at most d. On the other hand, we give an example showing the number of exceptional r can be greater than $\big (\frac{\log d}{\log \log d}\big )^2$ for infinitely many $d \in {\mathbb {N}}$.