Discriminant and integral basis of number fields defined by exponential Taylor polynomials (Q6543258)

Let \(n\) be a positive integer and \(K_n=\mathbb{Q}(\alpha_n)\), where \(\alpha_n\) is a root of the Taylor polynomial \(\frac{x^n}{n!} + \frac{x^{n-1}}{(n-1)!} + \cdots + \frac{x^2}{2!} + \frac{x}{1!} + 1\). It is proved that a prime number \(p\) divides the discriminant of \(K_n\) if and only if \(p\leq n\). Also the exact power of every prime number dividing this discriminant is determined, as well as the exact power of every prime dividing the index of \(\mathbb{Z}[\alpha_n]\) in the ring of integers of \(K_n\). This implies that the ring of integers of \(K_n\) coincides with \(\mathbb{Z}[\alpha_n]\) if and only if \(n=2\) or 3. Moreover, for every prime number \(p\) the \(p\)-integral basis of \(K_n\) is explicitly constructed. This allows the construction of an integral basis of \(K_n\). To illustrate this, two examples are given where explicit integral basis is constructed for \(K_6\) and \(K_7\).