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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

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