ON FACTORIZATION OF SOME PERMUTATION POLYNOMIALS OVER FINITE FIELDS

Tekgül Kalaycı
Mathematics, PhD Dissertation, 2019

Thesis Jury

Prof. Dr. Alev Topuzoğlu (Thesis Advisor), Prof. Dr. Cem Güneri, Prof. Dr. Erkay Savaş,

Assoc. Prof. Dr. Wilfried Meidl, Prof. Dr. Ayşe Berkman

Date & Time: 7th, January 2019 –  13 : 00

Place: FENS L055

Keywords : Finite fields, permutation polynomials, factorization of polynomials, irreducible polynomials

Abstract

Factorization of polynomials over finite fields is a classical problem, going back to the 19th century. However, factorization of an important class, namely, of permutation polynomials was not studied previously. In this thesis, we present results on factorization of permutation polynomials of Fq, for q greater than or equal to 2.

     In order to tackle this problem, we consider permutation polynomials fn(x) in Fq[x], which are defined recursively as compositions of monomials of degree d with gcd(d, q – 1)=1, and linear polynomials, for n greater than or equal to 0. Extensions of Fq defined by using the recursive structure of fn(x), satisfy particular properties that enable us to employ techniques from Galois theory. In consequence, we obtain a variety of results on degrees and number of irreducible factors of the polynomials fn(x).