数域的可分性参数与Kummer扩展的度

Antonella Perucca, Pietro Sgobba, S. Tronto
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引用次数: 0

摘要

摘要设K是一个数域,设r是一个素数。修复一些元素α1,…,αr (kx)生成秩为r的kx子群,设n1,…,nr, m是每个i的m大于或等于ni的正整数。我们表明存在可计算的参数公式(仅涉及有限情况区别)来表示Kummer扩展K(ζ ζ m, α1∑n1,…,αr∑nr \root {{\ell ^{n_1}}}} \of {{\ α _1}},\ ldots,\root {{\ell ^{{n_r}}}} \of {{\ α _r}})在K(ζ∑m)上的所有n1,…这是通过一种相对于先前工作的新方法实现的,即我们确定了起作用的可整除性参数的显式公式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Divisibility Parameters and the Degree of Kummer Extensions of Number Fields
Abstract Let K be a number field, and let ℓ be a prime number. Fix some elements α1,...,αr of K× which generate a subgroup of K× of rank r. Let n1,...,nr, m be positive integers with m ⩾ ni for every i. We show that there exist computable parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extension K(ζℓm, α1ℓn1,…,αrℓnr \root {{\ell ^{{n_1}}}} \of {{\alpha _1}} , \ldots ,\root {{\ell ^{{n_r}}}} \of {{\alpha _r}} ) over K(ζℓm) for all n1,..., nr, m. This is achieved with a new method with respect to a previous work, namely we determine explicit formulas for the divisibility parameters which come into play.
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