Abstract
In this article, we study the monogenity of a tower of number fields defined by the iterates of a stable polynomial. We give a necessary condition for the monogenity of the number fields defined by the iterates of a stable polynomial. When the stable polynomial is of certain type, we also give a sufficient condition for the monogenity of the fields defined by each of its iterate. As a consequence, we obtain an infinite 3-tower of monogenic number fields. Moreover, we construct an infinite family of stable polynomials such that each of its iterate is non-monogenic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmadi, O., Monsef-Shokri, K.: A note on the stability of trinomials over finite fields. Finite Fields Appl. 63, 101649, 13 pp. (2020)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24(3–4), 235–265 (1997)
Castillo, M.: A dynamical characterization for monogenity at every level of some infinite-towers. Canad. Math. Bull. 65(3), 806–814 (2022)
Cohen, H.: Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics, 193. Springer, New York (2000)
Conrad, K.: Totally ramified primes and Eisenstein polynomials. http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/totram.pdf
Jones, L.: Monogenic polynomials with non-squarefree discriminant. Proc. Amer. Math. Soc. 148(4), 1527–1533 (2020)
Khanduja, S.K.: A Textbook of Algebraic Number Theory. La Matematica per il 3+2, 135. Unitext. Springer, Singapore (2022)
Li, R.: On number fields towers defined by iteration of polynomials. Arch. Math. (Basel) 119(4), 371–379 (2022)
Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers. Third Edition. Springer Monographs in Mathematics. Springer, Berlin (2004)
Uchida, K.: When is Z[\(\alpha \)] the ring of the integers? Osaka Math. J. 14(1), 155–157 (1977)
Acknowledgements
We thank the anonymous referee for the helpful remarks and comments for a better exposition of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sharma, H., Sarma, R. & Laishram, S. Monogenity of iterates of polynomials. Arch. Math. 122, 295–305 (2024). https://doi.org/10.1007/s00013-023-01949-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-023-01949-9