A Mild Tchebotarev Theorem for GL$(n)$

It is well known that the Tchebotarev density theorem implies that an irreducible $\ell$-adic representation $\rho$ of the absolute Galois group of a number field $K$ is determined (up to isomorphism) by the characteristic polynomials of Frobenius elements at any set of primes of density 1. In this Note we make some progress on the automorphic side for GL$(n)$ by showing that, given a cyclic extension $K/k$ of number fields of prime degree $p$, a cuspidal automorphic representation $\pi$ of GL$(n,{\mathbb A}_K)$ is determined up to twist equivalence by the knowledge of its local components at the (density one) set $S_{K/k}$ of primes of $K$ of degree $1$ over $k$, and moreover that $\pi$ is determined even up to isomorphism if $p=2$. The proof uses the Luo-Rudnick-Sarnak bound for the Hecke roots of $\pi$, applied to certain Rankin-Selberg $L$-functions of positive type, in conjunction with some Kummer theory and descent along suitable $p$-power extensions arising as nested sequences of cyclic $p^2$-extensions.