On Narkiewicz problem 22 for number fields associated with certain quadrinomials

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>K</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="double-struck">Q</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> K=Q(θ ) be a number field corresponding to the minimal polynomial f ( x ) = x n + ax n −1 + bx + c over the field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">Q</m:mi> </m:math> Q , where θ is an algebraic integer. In this paper, we provide sufficient conditions for i ( K ) = 1 and i ( K ) = 2, which give a partial answer to the Narkiewicz Problem 22 for these number fields, where i ( K ) is the index of K . Finally, we provide examples of infinite families of number fields for which i ( K ) = 1 and i ( K ) = 2.