AN EFFECTIVE VERSION OF THE PRIMITIVE ELEMENT THEOREM

Abstract Let K be an infinite field. If $\alpha $ and $\beta $ are algebraic and separable elements over K , then by the primitive element theorem, it is well known that $\alpha +u\beta $ is a primitive element for $K(\alpha , \beta )$ for all but finitely many elements $u\in K$ . If we let $$ \begin{align*}\xi_K(\alpha, \beta) = \{u\in K : K(\alpha, \beta) \ne K(\alpha+u\beta)\}\end{align*} $$ be the exceptional set, then by the primitive element theorem, $|\xi _K(\alpha , \beta )| < \infty $ . Dubickas [‘An effective version of the primitive element theorem’, Indian J. Pure Appl. Math. 53 (3) (2022), 720–726] estimated the size of this set when $K = \mathbb {Q}$ . We take K to be a finite extension over $\mathbb {Q}$ or $\mathbb {Q}_p$ , the field of p -adic numbers for some prime p , and estimate the size of the exceptional set.