Explicit Kummer theory for the rational numbers

Antonella Perucca, Pietro Sgobba, Sebastiano Tronto*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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 languageEnglish
Pages (from-to)2213-2231
Number of pages19
JournalInternational Journal of Number Theory
Issue number10
Publication statusPublished - 1 Nov 2020
Externally publishedYes


  • Kummer theory
  • Number fields
  • cyclotomic fields
  • degree


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