On multiplicatively independent bases in cyclotomic number fields

Recently the authors [ 12 ] showed that the algebraic integers of the form - m + ζ k are bases of a canonical number system of Z [ ζ k ] provided m ≧ ϕ ( k ) + 1 , where ζ k denotes a k -th primitive root of unity and ϕ is Euler’s totient function. In this paper we are interested in the questions whether two bases - m + ζ k and - n + ζ k are multiplicatively independent. We show the multiplicative independence in case that 0