Colourings of cyclotomic integers with class number one

Abstract

This paper continues the study of colourings of the sets of cyclotomic integers ℳ n = ℤ[ξ n ] (ξ n = e 2πi/n , a primitive nth root of unity) with class number one. We present results for the colour symmetry group and colour preserving group for a given ideal colouring of ℳ n , with φ(n) = 8 and 10, thus completing the characterisation of the colour preserving group for the cases φ(n) ≤ 10, where φ is Euler's totient function.