ON NONMONOGENIC ALGEBRAIC NUMBER FIELDS

Let q be a prime number and f ( x) = x(qs) - ax(m) - b be a monic irreducible polynomial of degree qs having integer coefficients. Let K = Q(theta) be an algebraic number field with theta a root of f (x). We give some explicit conditions involving only a, b, m, q, s for which K is not monogenic. As an application, we show that if p is a prime number of the form 32k + 1, k. Z and theta is a root of a monic polynomial f (x) = x(2s) - 32cpx(2r) - p is an element of Z[x] with s > 4, 2. c, s not equal 5+ r, then Q(theta) is not monogenic.