# Arithmetic Properties of Generalized Rikuna Polynomials

Z. Chonoles[1]; J. Cullinan[2]; H. Hausman[3]; A.M. Pacelli[3]; S. Pegado[3]; F. Wei[4]

• [1] Department of Mathematics, The University of Chicago, 5734 S. University Avenue Chicago, IL 60637, USA
• [2] Department of Mathematics, Bard College, Annandale-On-Hudson, NY 12504, USA
• [3] Department of Mathematics, Williams College, Williamstown, MA 01267, USA
• [4] Department of Mathematics, Harvard University, One Oxford Street, Cambridge MA 02138, USA
• Issue: 1, page 19-33
• ISSN: 1958-7236

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## Abstract

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Fix an integer $\ell \ge 3$. Rikuna introduced a polynomial $r\left(x,t\right)$ defined over a function field $K\left(t\right)$ whose Galois group is cyclic of order $\ell$, where $K$ satisfies some mild hypotheses. In this paper we define the family of generalized Rikuna polynomials${\left\{{r}_{n}\left(x,t\right)\right\}}_{n\ge 1}$ of degree ${\ell }^{n}$. The ${r}_{n}\left(x,t\right)$ are constructed iteratively from the $r\left(x,t\right)$. We compute the Galois groups of the ${r}_{n}\left(x,t\right)$ for odd $\ell$ over an arbitrary base field and give applications to arithmetic dynamical systems.

## How to cite

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Chonoles, Z., et al. "Arithmetic Properties of Generalized Rikuna Polynomials." Publications mathématiques de Besançon (2014): 19-33. <http://eudml.org/doc/275709>.

@article{Chonoles2014,
abstract = {Fix an integer $\ell \ge 3$. Rikuna introduced a polynomial $r(x,t)$ defined over a function field $K(t)$ whose Galois group is cyclic of order $\ell$, where $K$ satisfies some mild hypotheses. In this paper we define the family of generalized Rikuna polynomials$\lbrace r_n(x,t) \rbrace _\{n \ge 1\}$ of degree $\ell ^n$. The $r_n(x,t)$ are constructed iteratively from the $r(x,t)$. We compute the Galois groups of the $r_n(x,t)$ for odd $\ell$ over an arbitrary base field and give applications to arithmetic dynamical systems.},
affiliation = {Department of Mathematics, The University of Chicago, 5734 S. University Avenue Chicago, IL 60637, USA; Department of Mathematics, Bard College, Annandale-On-Hudson, NY 12504, USA; Department of Mathematics, Williams College, Williamstown, MA 01267, USA; Department of Mathematics, Williams College, Williamstown, MA 01267, USA; Department of Mathematics, Williams College, Williamstown, MA 01267, USA; Department of Mathematics, Harvard University, One Oxford Street, Cambridge MA 02138, USA},
author = {Chonoles, Z., Cullinan, J., Hausman, H., Pacelli, A.M., Pegado, S., Wei, F.},
journal = {Publications mathématiques de Besançon},
keywords = {postcritically finite; Galois group; cyclotomic field},
language = {eng},
number = {1},
pages = {19-33},
publisher = {Presses universitaires de Franche-Comté},
title = {Arithmetic Properties of Generalized Rikuna Polynomials},
url = {http://eudml.org/doc/275709},
year = {2014},
}

TY - JOUR
AU - Chonoles, Z.
AU - Cullinan, J.
AU - Hausman, H.
AU - Pacelli, A.M.
AU - Wei, F.
TI - Arithmetic Properties of Generalized Rikuna Polynomials
JO - Publications mathématiques de Besançon
PY - 2014
PB - Presses universitaires de Franche-Comté
IS - 1
SP - 19
EP - 33
AB - Fix an integer $\ell \ge 3$. Rikuna introduced a polynomial $r(x,t)$ defined over a function field $K(t)$ whose Galois group is cyclic of order $\ell$, where $K$ satisfies some mild hypotheses. In this paper we define the family of generalized Rikuna polynomials$\lbrace r_n(x,t) \rbrace _{n \ge 1}$ of degree $\ell ^n$. The $r_n(x,t)$ are constructed iteratively from the $r(x,t)$. We compute the Galois groups of the $r_n(x,t)$ for odd $\ell$ over an arbitrary base field and give applications to arithmetic dynamical systems.
LA - eng
KW - postcritically finite; Galois group; cyclotomic field
UR - http://eudml.org/doc/275709
ER -

## References

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