The action of \rmGL₂(\mathbbF_q) on irreducible polynomials over \F_q

2016/08/12 by Lucas Reis, Reis, Lucas
Computer Science · Mathematics · #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1608.03915

openalex publication_date 2016/08/12 · openalex created_date 2016/09/16 · openalex updated_date 2026/07/28

Abstract

Let \F_q be the finite field with q elements, p=\Char \F_q. The group \GL₂(\F_q) acts naturally in the set of irreducible polynomials over \F_q of degree at least 2. In this paper we are interested in the characterization and number of the irreducible polynomials that are fixed by the elements of a subgroup H of \GL₂(\F_q). We make a complete characterization of the fixed polynomials in the case when H has only elements of the form (\beginmatrix 1&b 0&1 \endmatrix), corresponding to translations x↦ x+b and, as a consequence, the case when H is a p-subgroup of \GL₂(\F_q). This paper also contains alternative solutions for the cases when H is generated by an element of the form (\beginmatrix a&0 0&1 \endmatrix), obtained by Garefalakis (2010) and H=\rmPGL₂(\F_q), obtained by Stichtenoth and Topuzoglu (2011).

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