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Discriminant and integral basis of number fields defined by exponential Taylor polynomials
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 2 / May 2024
- Published online by Cambridge University Press:
- 08 April 2024, pp. 528-541
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Let
$K_n=\mathbb{Q}(\alpha_n)$ be a family of algebraic number fields where 
$\alpha_n\in \mathbb{C}$ is a root of the nth exponential Taylor polynomial 
$\frac{x^n}{n!}+ \frac{x^{n-1}}{(n-1)!}+ \cdots +\frac{x^2}{2!}+\frac{x}{1!}+1$, 
$n\in \mathbb{N}$. In this paper, we give a formula for the exact power of any prime p dividing the discriminant of Kn in terms of the p-adic expansion of n. An explicit p-integral basis of Kn is also given for each prime p. These p-integral bases quickly lead to the construction of an integral basis of Kn.
