Let \(p\) be a prime number and \(f (x) = x^{p^ s}- ax^m- b\) belonging to \(\mathbb Z[x]\) be an irreducible polynomial. Let \( K = \mathbb Q(\theta )\) be an algebraic number field with \(\theta\) a root of \( f (x)\). Let \(r_1\) stand for the highest power of \(p\) dividing \(b^{p^s}- b -ab^m.\) This paper gives some explicit conditions involving only \(a, b, m, r_1,s\) for which \(K\) is not monogenic. In particular, it is shown that when \(p\) is an odd prime and \(s \geq r_1 > p\), then \(K\) is not monogenic. Like in several papers giving non-monogenic fields, the proof is based on a theorem of product of Ore and Dedekind's Theorem on splitting of primes.