© Springer Nature Switzerland AG 2019.In this work the unsolvability of certain equations is studied in the case of cyclotomic number fields whose ring of integers is not a principal ideal domain. Winterhof et al. considered the equations for (Formula presented). We first extend this result to (Formula presented) by using a new method from algebraic number theory. Then we present its applications to ᵧ -Butson-Hadamard matrices, ᵧ -Conference matrices and type ᵧ nearly perfect sequences for (Formula presented).