ON NONMONOGENIC ALGEBRAIC NUMBER FIELDS

Let q be a prime number and f (x) = xqs − axm − b be a monic irreducible polynomial of degree qs having integer coefficients. Let K = Q(θ) be an algebraic number field with θ 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 θ is a root of a monic polynomial f (x) = x2s − 32cpx2r − p ∈ Z[x] with s > 4, 2 ∤ c, s ̸= 5 + r, then Q(θ) is not monogenic.