If p is a prime number congruent to 1 modulo 3, then we explicitly describe an element of the cyclotomic field  whose third root generates the cubic subextension of Similarly, if p is a prime number congruent to 1 modulo 4, then we explicitly describe an element of the cyclotomic field  whose fourth root generates the quartic cyclic subextension of  For further number fields, we express generators of Kummer extensions inside cyclotomic fields in terms of Gauss sums.

Kummer theory, Kummer extension, number field, cyclotomic field, quadratic field, degree.

Received: September 3, 2021; Revised: November 4, 2021; Accepted: November 18, 2021; Published: December 31, 2021

How to cite this article: Fritz Hörmann, Antonella Perucca, Pietro Sgobba and Sebastiano Tronto, Explicit Kummer generators for cyclotomic extensions, JP Journal of Algebra, Number Theory and Applications 53(1) (2022), 69-84. DOI: 10.17654/0972555522004

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

[1] I. B. Fesenko and S. V. Vostokov, Local fields and their extensions, Volume 121 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 2nd ed., 2002. https://doi.org/10.1090/mmono/121.
[2] A. Fröhlich, Stickelberger without Gauss sums, Algebraic Number Fields: L-functions and Galois Properties (Proc. Sympos., Univ. Durham, Durham, 1975), 1977, pp. 589-607.
[3] M. Hindry and J. H. Silverman, Diophantine geometry - an introduction, Graduate Texts in Mathematics, 201, Springer-Verlag, New York, 2000.
[4] K. Ireland and M. Rosen, A classical introduction to modern number theory, Volume 84 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2nd ed., 1990. https://doi.org/10.1007/978-1-4757-2103-4.
[5] J. M. Masley, Class numbers of real cyclic number fields with small conductor, Compos. Math. 37(3) (1978), 297-319.
[6] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), 2021. https://www.sagemath.org.
[7] L. C. Washington, Introduction to cyclotomic fields, Volume 83 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982. https://doi.org/10.1007/978-1-4684-0133-2.