Let \(\pi\) be a cuspidal automorphic representation of \(GL(n,F_ A)\) where F is a global field. Suppose that E is a Galois extension of F and let g be a conjugacy class in Gal(E/F). The author considers the set C(g) of places for which the Frobenius conjugacy class is defined and equal to g. He seeks conditions under which he can identify the set of cuspidal automorphic representations \(\pi\) ’ for which \(\pi '_ v\cong \pi_ v\) for \(v\in C(g)'\), and, more especially, for a sufficiently large finite subset of C(g). He succeeds only under additional assumptions; for general n he can deal with the case E/F abelian (and needs also a weak form of the Ramanujan conjecture if \(n\geq 4)\). When \(n=2\) he can prove a corresponding result when E/F is soluble and g satisfies a further condition. The proofs are based on a Chebotarev density theorem due to Moreno and those parts of the theory of base change for GL(n) which have been established up to the present time.