Let \(K\) be the number field defined by a complex root \(\alpha\) of a monic irreducible polynomial \(x^{p^r} - m\), where \(m \ne 1\) a square free rational integer, \(p\) is a prime number and \(r\) is a positive integer. Let \(\nu_p(n)\) be the power of the prime \(p\) in the expansion of the integer \(n\). The authors prove that if \(\nu_p(m^p-m)=1\) then \({\mathbb Z}[\alpha]\) is the ring of integers of \(K\), and so \(K\) is monogenic and \(\alpha\) generates a power integral basis. In particular, for \(p=2\), this implies the above assertion for \(m \equiv 2,3 \pmod 4\) and any positive integer \(r\). The authors also show that if \(p\) is an odd prime that does not divide \(m\) and also \(\nu_p(m^{p-1}-1)>p\) and \(r \geq p\), then the corresponding field \(K\) is not monogenic.