TY - JOUR

T1 - Explicit Kummer theory for the rational numbers

AU - Perucca, Antonella

AU - Sgobba, Pietro

AU - Tronto, Sebastiano

N1 - Publisher Copyright:
© 2020 World Scientific Publishing Company.

PY - 2020/11/1

Y1 - 2020/11/1

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

UR - http://www.scopus.com/inward/record.url?scp=85094838433&partnerID=8YFLogxK

U2 - 10.1142/S1793042120501146

DO - 10.1142/S1793042120501146

M3 - Article

AN - SCOPUS:85094838433

SN - 1793-0421

VL - 16

SP - 2213

EP - 2231

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 10

ER -