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Arithmetic Properties of Generalized Rikuna Polynomials
Z. Chonoles1; J. Cullinan2; H. Hausman3; A.M. Pacelli3; S. Pegado3; F. Wei4
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
Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 19-33.

Fix an integer ℓ≥3. Rikuna introduced a polynomial r(x,t) defined over a function field K(t) whose Galois group is cyclic of order ℓ, where K satisfies some mild hypotheses. In this paper we define the family of generalized Rikuna polynomials {r n (x,t)} n≥1 of degree ℓ 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 ℓ over an arbitrary base field and give applications to arithmetic dynamical systems.

Soit ℓ≥3 un nombre entier fixé. Rikuna a défini un polynôme r(x,t) sur un corps de fonctions K(t) dont le groupe de Galois est cyclique d’ordre ℓ, où K satisfait à certaines hypothèses pas très restrictives. Dans cet article, nous définissons la famille des polynômes de Rikuna généralisés {r n (x,t)} n≥1 de degré ℓ n . Les r n (x,t) sont construits de manière itérative à partir de r(x,t). Nous calculons les groupes de Galois des r n (x,t) pour ℓ impair sur un corps de base arbitraire et donnons des applications aux systèmes dynamiques arithmétiques.

Received:
Published online:
DOI: 10.5802/pmb.2
Classification: 11R32, 11S20
Keywords: postcritically finite, Galois group, cyclotomic field

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 
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     title = {Arithmetic {Properties} of {Generalized} {Rikuna} {Polynomials}},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
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Z. Chonoles; J. Cullinan; H. Hausman; A.M. Pacelli; S. Pegado; F. Wei. Arithmetic Properties of Generalized Rikuna Polynomials. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 19-33. doi : 10.5802/pmb.2. https://pmb.centre-mersenne.org/articles/10.5802/pmb.2/

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