ON COMMON INDEX DIVISORS ANDMONOGENITY OF SEPTIC NUMBER FIELDSDEFINED BY TRINOMIALS OF TYPE x7 + αx2 + b

We study the index i(K) of any septic number field K generated by a root of an irreducible trinomial of type F(x) = x(7 )+ alpha x(2) + b is an element of Z[x]. We show that the unique prime which can divide i (K) is 2. Moreover, we give necessary and sufficient conditions on a and b so that 2 is a common index divisor of K. Further, we show that i(K) = 2 whenever 2 divides i( K). In this way, we answer completely Problem 6 and Problem 22 of Narkiewicz [34] for these families of number fields. As an application of our results, if 2 divides i (K), then the ring OK of integers of K has no power integral basis. We illustrate our results by giving some numerical examples.