Related Discussion Of Irreducible Polynomials

Let Fq be a finite field with q elements.The set of monic irreducible polynomial of degree n on Fq is marked as In(n≥2).In this paper,we give a formula for calculating the orbital number of PGL2(Fq)action on In and prove that the necessary and sufficient condition for the group action to be transitive is n=2 or n=3 or(n,q)=(4,2)or(n,q)=(5,2).In addition,we can assume that f(x)is an irreducible polynomial on Fq.We define f0(x)=f(x),fi(x)=fi-1(x)Q is a polynomial obtained from the rational transformation of the polynomial fi-1(x)determined by Q(x)=(x+x-1)/2.If fi(x)(i≥0)is an irreducible polynomial sequence,then f(x)is a Q-polynomial.In this paper,the number of first and second Q-polynomials on Fq is calculated.