Explicit Kummer theory for the rational numbers

Antonella Perucca; Pietro Sgobba; Sebastiano Tronto

doi:10.1142/S1793042120501146

Explicit Kummer theory for the rational numbers

Antonella Perucca, Pietro Sgobba, Sebastiano Tronto^*

^*Corresponding author for this work

University of Luxembourg

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

Let G be a finitely generated multiplicative subgroup of as× having rank r. The ratio between nr and the Kummer degree [as(ζm,Gn): as(ζm)], where n divides m, is bounded independently of n and m. We prove that there exist integers m0,n0 such that the above ratio depends only on G, gcd(m,m0), and gcd(n,n0). Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).

Original language	English
Pages (from-to)	2213-2231
Number of pages	19
Journal	International Journal of Number Theory
Volume	16
Issue number	10
DOIs	https://doi.org/10.1142/S1793042120501146
Publication status	Published - 1 Nov 2020
Externally published	Yes

Keywords

Kummer theory
Number fields
cyclotomic fields
degree

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10.1142/S1793042120501146

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title = "Explicit Kummer theory for the rational numbers",

abstract = "Let G be a finitely generated multiplicative subgroup of as× having rank r. The ratio between nr and the Kummer degree [as(ζm,Gn): as(ζm)], where n divides m, is bounded independently of n and m. We prove that there exist integers m0,n0 such that the above ratio depends only on G, gcd(m,m0), and gcd(n,n0). Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).",

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AU - Sgobba, Pietro

AU - Tronto, Sebastiano

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N2 - Let G be a finitely generated multiplicative subgroup of as× having rank r. The ratio between nr and the Kummer degree [as(ζm,Gn): as(ζm)], where n divides m, is bounded independently of n and m. We prove that there exist integers m0,n0 such that the above ratio depends only on G, gcd(m,m0), and gcd(n,n0). Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).

AB - Let G be a finitely generated multiplicative subgroup of as× having rank r. The ratio between nr and the Kummer degree [as(ζm,Gn): as(ζm)], where n divides m, is bounded independently of n and m. We prove that there exist integers m0,n0 such that the above ratio depends only on G, gcd(m,m0), and gcd(n,n0). Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).

KW - Kummer theory

KW - Number fields

KW - cyclotomic fields

KW - degree

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JO - International Journal of Number Theory

JF - International Journal of Number Theory

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ER -

Explicit Kummer theory for the rational numbers

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Keywords

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