ON NONMONOGENIC ALGEBRAIC NUMBER FIELDS

Abstract

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.