A Reciprocity Relation Between Some Cyclotomic Integers

Let p, q be two prime numbers, ζ_p, ζ_q primitive p-th, q-th roots of unity in and λ _p = 1 — ζ _p, λ_q = 1 — ζ_q, the distinguished prime elements of the cyclotomic fields (ζ_p), (ζ_q) above p, q, respectively. The elements 1— λn _p (n a positive integer) are well known to form a topological basis for the group of principal units in the λ_p -adic completion of (ζ_p) ([1], p. 247). But little is known of their multiplicative properties as global objects, i.e., of their factorization in (ζ_p). It turns out that there is a reciprocity relation between the factorization of 1 — λ_q p, in (ζ_p) and that of 1 — λ_pq in (ζ_p) (Corollary 11, Proposition 15). In particular, the latter is prime if and only if the former is (Corollary 12). There are also simple expressions for the norm Np (1 — λ_np) of 1 — λ_np in (ζ_p)| for some small values of n (formulas (3.2), (3.4)—(3.8)). for distinct prime numbers p, q, the integers Np (1— λ_qq) generalize the Mersenne numbers 2 ^q — 1 = N_p (1 − λ_p 2) and share some of their properties (Proposition 14); numerical evidence seems to suggest that they are always squarefree. They satisfy the relation Np (1 — λ^q_p).