  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.
