On a class of ideals in an algebraic number field

AbstractLet k be an algebraic number field and let θ be the ring of integers of k. We define for each positive n and each prime ideal p of θ a nonnegative integer rn(p) as follows: if n = Σi = 0tkiqi is the q-adic expansion of n where q = Np, set rn(p) = (n − Σi = 0tki)/(q − 1). We then set Jn(k) = Πpprn(p) where the product is taken over all prime ideals of θ. Of course Jn(Q) = (n!). Properties of the ideals Jn(k) are investigated. For instance, we prove: Suppose f(x) ∈ θ[x] is primitive of degree m and dk(f) = the ideal in θ generated by the elements f(α), α ∈ θ, then dk(f) divides Jm(k). We also associate with each extension K of k of degree m an ideal J(K/k) which divides Jm(k). Properties of J(K/k) are established