An analogue of Bridges and Mena's theorem for local fields and a local-global principle

Let G be an abelian group of finite order n,K a field and R⊆K a ring. Let D=∑g∈Gagg∈R[G] such that χ(D)∈R for every character χ:G→K(ξn) (where χ(D)=∑g∈Gagχ(g) and ξn is a primitive nth root of unity). What does D look like? The case where K=Q and R=Z was settled by Bridges and Mena. Here we obtain a complete characterization for the case where K is a finite extension of the field Qp and R is its valuation ring under the condition that p does not divide n.As an application we obtain the following local-global principle for Z/q1q2Z (where q1 and q2 are distinct primes): If D∈Z[Z/q1q2Z], then χ(D)∈Z for every character χ:Z/q1q2Z→C× if and only if ψ(D)∈Zp for every prime p and every character ψ:Z/q1q2Z→Qp(ξn).