Reciprocal relations between cyclotomic fields

Abstract

We describe a reciprocity relation between the prime ideal factorization, and related properties, of certain cyclotomic integers of the type ϕ n ( c − ζ m ) in the cyclotomic field of the m-th roots of unity and that of the symmetrical elements ϕ m ( c − ζ n ) in the cyclotomic field of the n-th roots. Here m and n are two positive integers, ϕ n is the n-th cyclotomic polynomial, ζ m a primitive m-th root of unity, and c a rational integer. In particular, one of these integers is a prime element in one cyclotomic field if and only if its symmetrical counterpart is prime in the other cyclotomic field. More properties are also established for the special class of pairs of cyclotomic integers ( 1 − ζ p ) q − 1 and ( 1 − ζ q ) p − 1 , where p and q are prime numbers.