Discriminant is a basic invariant associated with an algebraic number field. Its notion was
first introduced by Dedekind in 1871. The problem of effective computation of discriminant as
well as an integral basis of an infinite family of algebraic number fields which are defined over
the field $\mathbb{Q}$ of rational numbers by certain types of irreducible polynomials has been tackled
by several mathematicians. In this lecture we shall review the progress made regarding this problem in the case of pure fields. We shall present a formula for the exact power of any prime $p$ dividing the discriminant of the field  $K_n=\mathbb{Q}(\alpha_n),$ where $\alpha_n\in\mathbb{C}$ is a root of the $n$th exponential Taylor polynomial
$\frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+\cdots+\frac{x^2}{2!}+\frac{x}{1!}+1,
$ in terms of the $p $-adic expansion of positive integer $n$. We also describe an explicit $p$-integral basis of $K_n$ for each prime $p$. These local bases lead naturally to the construction of an integral basis of $K_n$.