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MÖBIUS–FROBENIUS MAPS ON IRREDUCIBLE POLYNOMIALS

Part of: Finite fields and commutative rings (number-theoretic aspects) General field theory

Published online by Cambridge University Press:  14 December 2020

F. E. BROCHERO MARTÍNEZ Open the ORCID record for F. E. BROCHERO MARTÍNEZ [Opens in a new window] ,
DANIELA OLIVEIRA Open the ORCID record for DANIELA OLIVEIRA [Opens in a new window]  and
LUCAS REIS Open the ORCID record for LUCAS REIS [Opens in a new window]
F. E. BROCHERO MARTÍNEZ
Affiliation:
Departamento de Matemática, Universidade Federal de Minas Gerais, UFMG, Belo HorizonteMG, 31270-901, Brazil e-mail: fbrocher@mat.ufmg.br
DANIELA OLIVEIRA
Affiliation:
Departamento de Matemática, Universidade Federal de Minas Gerais, UFMG, Belo HorizonteMG, 31270-901, Brazil e-mail: danielaalvesoliveira@gmail.com
LUCAS REIS*
Affiliation:
Departamento de Matemática, Universidade Federal de Minas Gerais, UFMG, Belo Horizonte, MG, 31270-901, Brazil
*
e-mail: lucasreismat@mat.ufmg.br
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Abstract

Let n be a positive integer and let $\mathbb{F} _{q^n}$ be the finite field with $q^n$ elements, where q is a prime power. We introduce a natural action of the projective semilinear group ${\mathrm{P}\Gamma\mathrm{L}} (2, q^n)={\mathrm{PGL}} (2, q^n)\rtimes {\mathrm{Gal}} ({\mathbb F_{q^n}} /\mathbb{F} _q)$ on the set of monic irreducible polynomials over the finite field $\mathbb{F} _{q^n}$ . Our main results provide information on the characterisation and number of fixed points.

Keywords

enumeration formulafixed pointgroup actionirreducible polynomials over finite fields

MSC classification

Primary: 11T06: Polynomials
Secondary: 12E20: Finite fields (field-theoretic aspects)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 66 - 77
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author was partially supported by FAPEMIG APQ-02973-17, Brazil. The third author was supported by FAPESP 2018/03038-2, Brazil.

References

Boripan, A., Jitman, S. and Udomkavanich, P., ‘Self-conjugate-reciprocal irreducible monic polynomials over finite fields’, Proc. 20th Annu. Meet. Mathematics 2015 (Silpakorn University, Nakhon Pathom, Thailand, 2015), 34–43.Google Scholar
Boripan, A., Jitman, S. and Udomkavanich, P., ‘Self-conjugate-reciprocal irreducible monic factors of ${x}^n-1$ over finite fields and their applications’, Finite Fields Appl. 55 (2019), 78–96.CrossRefGoogle Scholar
Garefalakis, T., ‘On the action of $\mathrm{GL}(2,q)$ on irreducible polynomials over ${F}_q$ ’, J. Pure Appl. Algebra 215 (2011), 1835–1843.CrossRefGoogle Scholar
Mattarei, S. and Pizzato, M., ‘Generalizations of self-reciprocal polynomials’ Finite Fields Appl. 48 (2017), 271–288.CrossRefGoogle Scholar
Meyn, H. and Götz, W., ‘Self-reciprocal polynomials over finite fields’, Publ. Inst. Rech. Math. Av. 413(21) (1990), 82–90.Google Scholar
Reis, L., ‘The action of ${\mathrm{GL}}_2({F}_q)$ on irreducible polynomials over ${F}_q$ , revisited’, J. Pure Appl. Algebra 222 (2018), 1087–1094.CrossRefGoogle Scholar
Stichtenoth, H. and Topuzoğlu, A., ‘Factorization of a class of polynomials over finite fields’, Finite Fields Appl. 18 (2012), 108–122.CrossRefGoogle Scholar