Stichtenoth, Henning and Topuzoğlu, Alev (2012) Factorization of a class of polynomials over finite fields. Finite Fields and Their Applications, 18 (1). pp. 108122. ISSN 10715797
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Official URL: http://dx.doi.org/10.1016/j.ffa.2011.07.005
Abstract
We study the factorization of polynomials of the form F(r)(X) = bx(qr+1)  ax(qr) + dx  c over the finite field F(q). We show that these polynomials are closely related to a natural action of the projective linear group PGL(2, q) on nonlinear irreducible polynomials over F(q). Namely, irreducible factors of F(r)(X) are exactly those polynomials that are invariant under the action of some nontrivial element [A] is an element of PGL(2, q). This connection enables us to enumerate irreducibles which are invariant under [A]. Since the class of polynomials F(r)(x) includes some interesting polynomials like x(qr)  x or x(qr+1)  1, our work generalizes wellknown asymptotic results about the number of irreducible polynomials and the number of selfreciprocal irreducible polynomials over F(q). At the same time, we generalize recent results about certain invariant polynomials over the binary field F(2).
Item Type:  Article 

Uncontrolled Keywords:  Polynomial factorization; Selfreciprocal polynomial; Group action on irreducible polynomials; Invariant polynomial 
Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences 
Depositing User:  Henning Stichtenoth 
Date Deposited:  29 Mar 2012 12:07 
Last Modified:  31 Jul 2019 10:27 
URI:  https://research.sabanciuniv.edu/id/eprint/18941 
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Factorization of a class of polynomials over finite fields. (deposited 23 Nov 2011 10:25)
 Factorization of a class of polynomials over finite fields. (deposited 29 Mar 2012 12:07) [Currently Displayed]