Power integral basis for relative extensions of p^{n}-power Number Fields

Let $L = K(\alpha)$ be an extension of the number field K, where $\alpha$ satisfies the monic irreducible polynomial $f(x)=x^{p^n}-\beta$ of power-prime degree belonging to ${\mathfrak{o}}_{K}[x]$ and $\mathfrak{o}_K$ is the integral closure of $K$. The purpose of this paper is to study the monogenity of $L/K$ by using a new version of Dedekind's criterion. So, we give an integral basis of a family of number field of degree $2 p^n$ for some positive integer n. As an illustration, we get a slightly simpler computation of relative discriminant $D_{L/K }$.