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Discriminant and integral basis of number fields defined by exponential Taylor polynomials
Published online by Cambridge University Press: 08 April 2024
Abstract
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.
MSC classification
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- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society
Footnotes
Dedicated to the memory of Peter Roquette.
References
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