Abstract
Let K be a number field, and let G be a finitely generated subgroup of \(K^\times \). We are interested in computing the degree of the cyclotomic-Kummer extension \(K(\root n \of {G})\) over K, where \(\root n \of {G}\) consists of all n-th roots of the elements of G. We develop the theory of entanglements introduced by Lenstra, and we apply it to compute the above degrees.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Buchmann, J.: Complexity of algorithms in algebraic number theory. In: Proceedings of the First Conference of the Canadian Number Theory Association, pp. 37–53. De Gruyter (1990)
Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Berlin (2013)
Debry, C., Perucca, A.: Reductions of algebraic integers. J. Number Theory 167(1), 259–283 (2016)
Hindry, M., Silverman, J.H.: Diophantine Geometry—An introduction. Graduate Texts in Mathematics, vol. 201. Springer-Verlag, New York (2000)
Fried, M.D., Jarden, M.: Field arithmetic, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, A Series of Modern Surveys in Mathematics, vol. 11. Springer, Berlin (2008)
Lenstra, H.W.: Factoring polynomials over algebraic number fields. In: Computer Algebra, pp. 245-254 (1983)
Lenstra, H. W.: Entangled radicals. In: Colloquium Lectures, AMS 112th Annual Meeting, San Antonio, January 12–15 (2006). https://www.math.leidenuniv.nl/~hwl/papers/rad.pdf
Palenstijn, W.J.: Radicals in arithmetic. PhD thesis, University of Leiden (2014). https://openaccess.leidenuniv.nl/handle/1887/25833
Perucca, A., Sgobba, P., Tronto, S.: Explicit Kummer theory for the rationals. Int. J. Number Theory 16(10), 2213–2231 (2020)
Perucca, A., Sgobba, P., Tronto, S.: The degree of Kummer extensions of number fields. Int. J. Number Theory 17(5), 1091–1110 (2021)
Schinzel, A.: Abelian binomials, power residues and exponential congruences, Acta Arith. 32(3), 245-274 (1977). Addendum, ibid. 36(1), 101–104 (1980). See also Andrzej Schinzel Selecta Vol.II, European Mathematical Society, Zürich, 2007, 939–970
Acknowledgements
We would like to thank Peter Stevenhagen for introducing us to the work of Lenstra on radical entanglements, and the referee for many useful comments.
Funding
No funding was received to assist with the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors have no conflicts of interest to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Perucca, A., Sgobba, P. & Tronto, S. Kummer theory for number fields via entanglement groups. manuscripta math. 169, 251–270 (2022). https://doi.org/10.1007/s00229-021-01328-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-021-01328-0