Square-free values of cubic polynomials in algebraic number fields

Abstract

Let K be an algebraic number field of finite degree over the rationals. The two themes of this paper are the problems about the existence of infinitely many power-free algebraic integers in polynomial sequences generated by irreducible polynomials of K with integer arguments and with prime arguments, respectively. In particular, it follows from the results that an irreducible cubic polynomial of K represents square-free numbers infinitely often, provided that an obvious necessary condition is fulfilled.