Schinzel and Wójcik have shown that for every \(\alpha ,\beta \in \mathbb {Q}^{\times }\hspace{0.55542pt}{\setminus }\hspace{1.111pt}\{\pm 1\}\) , there are infinitely many primes p where \(v_p(\alpha )=v_p(\beta )=0\) and where \(\alpha \) and \(\beta \) generate the same multiplicative group mod p. We prove a weaker result in the same direction for algebraic numbers \(\alpha , \beta \) . Let \(\alpha , \beta \in \overline{\mathbb {Q}} ^{\times }\) , and suppose \(|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\alpha )|\ne 1\) and \(|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\beta )|\ne 1\) . Then for some positive integer \(C = C(\alpha ,\beta )\) , there are infinitely many prime ideals P of where \(v_P(\alpha )=v_P(\beta )=0\) and where the group \(\langle \beta \bmod {P}\rangle \) is a subgroup of \(\langle \alpha \bmod {P}\rangle \) with \([\langle \alpha \bmod {P}\rangle \,{:}\, \langle \beta \bmod {P}\rangle ]\) dividing C. A key component of the proof is a theorem of Corvaja and Zannier bounding the greatest common divisor of shifted S-units.
