INDEX CHARACTERIZATION AND MONOGENITY OF SEPTIC NUMBER FIELDS DEFINED BY x7 + ax4 + b

In this paper, we calculate the index of any septic number field K generated by a root alpha of a monic irreducible trinomial F(x) = x(7) + ax(4) + b is an element of Z[x]. Our approach is based on Engstrom's results and the factorization of any rational prime in K. In such a way we give a complete answer of Problem 22 of Narkiewicz ([28]) for this family of number fields. As an application of our results, if i(K) not equal 1, then K is not monogenic. Also, we give generators of power integral bases in some cases where i(K) = 1. Our results are illustrated by some computational examples.