In this paper, we prove the decidability of the theory of ℚ
in the language (+, −,⋅, 0, 1, P
(n ∈ ℕ)) expanded by a predicate for the multiplicative subgroup n
ℤ (where n is a fixed integer). There are two cases: if
$v_p \left( n \right) > 0$
then the group determines a cross-section and we get an axiomatization of the theory and a result of quantifier elimination. If
$v_p \left( n \right) = 0$
, then we use the Mann property of the group to get an axiomatization of the theory.