Towards a Schinzel–Wójcik theorem for number fields

Pollack, Paul

doi:10.1007/s40879-025-00819-8

Abstract 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 \in Q^{\times} \ {\pm 1}

, there are infinitely many primespwhere$$v_p(\alpha )=v_p(\beta )=0$$

v_{p} (α) = v_{p} (β) = 0

and where$$\alpha $$

α

and$$\beta $$

β

generate the same multiplicative group modp. 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 \in {\bar{Q}}^{\times}

, and suppose$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\alpha )|\ne 1$$

| N_{Q (α, β) / Q} (α) | \neq 1

and$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\beta )|\ne 1$$

| N_{Q (α, β) / Q} (β) | \neq 1

. Then for some positive integer$$C = C(\alpha ,\beta )$$

C = C (α, β)

, there are infinitely many prime idealsPof Equation missing<#comment/>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 ⟩]

dividingC. A key component of the proof is a theorem of Corvaja and Zannier bounding the greatest common divisor of shiftedS-units.

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