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.
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