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.