Towards a Schinzel–Wójcik theorem for number fields

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