Factorization of a class of polynomials over finite fields

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Stichtenoth, Henning and Topuzoğlu, Alev (2012) Factorization of a class of polynomials over finite fields. Finite Fields and Their Applications, 18 (1). pp. 108-122. ISSN 1071-5797

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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 non-linear irreducible polynomials over F(q). Namely, irreducible factors of F(r)(X) are exactly those polynomials that are invariant under the action of some non-trivial 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 well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal 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; Self-reciprocal 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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