On Permutable Polynomials

Abstract

In this thesis, we studied some concept of the theory of permutable polynomials which is a part of the decomposition theory, that is, a case where the operation of composition function is commutative. We let ourselves to study this subject over algebraically closed fleld of characteristic zero. We gave a historical survey to the theory and joined the work of previous mathematicians to give a new algebraically proof and generalization of previous results. We studied some special classes of polynomials like the class of chebyshev polynomials of the flrst kind, the set of monic monomials and m- odd polynomials. We studied the concepts of gap-form and gap-degree of polynomials and some properties of them. We studied the number of polynomials which commute with a flxed polynomials. Finally, we gave a new proof to show that there is only two chains up to conjugation.