ACTA ARITHMETICA 108.1 (2003) On power residues by A. Schinzel and M. Skalba (Warszawa) Let n be a positive integer, K a number field, α i ∈ K (1 ≤ i ≤ k), β ∈ K. A simple necessary and sufficient condition was given in [7] in order that, for almost all prime ideals p of K, solubility of the k congruences x n i ≡ α i (mod p) should imply solubility of the congruence x n ≡ β (mod p), where n i | n. The aim of this paper is to extend that result to the case where the congruence x n ≡ β (mod p) is replaced by the alternative of l congruences x n ≡ β j (mod p). The general result is quite complicated, but it simplifies if n or K satisfy some restrictions. Here are precise statements, in which ζ n denotes a primitive nth root of unity, |A| is the cardinality of a set A, K n = {x n : x ∈ K} and F is the family of all subsets of {1,...,l}. Theorem 1. Let n and n i be positive integers with n i | n (1 ≤ i ≤ k), K be a number field and α i ,β j ∈ K ∗ (1 ≤ i ≤ k,1 ≤ j ≤ l). Consider the implication (i) solubility in K of the k congruences x n i ≡ α i (mod p) implies solu- bility in K of at least one of the l congruences x n ≡ β j (mod p). Then (i) holds for almost all prime ideals p of K if and only if (ii) for every unitary divisor m> 1 of n and , if n ≡ 0 (mod 4), for every m =2m ∗ , where m ∗ is a unitary divisor of the odd part of n, there exists an involution σ m of F such that for all A ⊂{1,...,l}, |σ m (A)|≡|A| + 1 (mod 2), (1)  j ∈σ m (A) β j =  j ∈A β j k  i=1 α a i m/(m,n i ) i Γ m , (2) where a i ∈ Z, Γ ∈ K(ζ m ) ∗ . Corollary 1. Let w n (K) be the number of nth roots of unity contained in K and assume that (w n (K), lcm[K(ζ q ): K]) = 1, (3) 2000 Mathematics Subject Classification : 11R20, 11A15. [77]