Generalizations of Carmichaels in a field extension

The AKS conjecture states that (ζ − 1) ≡ ζ − 1 (mod n) for ζ an r-th root of unity with r prime, only if n is a prime and r n − 1. If n is a prime, n-th power map would be a Frobenius automorphism of Fn(ζ), and thus the congruence will hold true. In general, n-th power map is not a field automorphism, but under suitable conditions, it can be a group automorphism of the multiplicative group of certain residue fields and their direct products. This leads to consider certain generalizations of the notion of Carmichael numbers to number rings. 1 Congruence in number rings Definition 1.1. A composite n is a Carmichael number if it satisfies