Some Problems Of Number Theory On Function Field

This paper is divided into two parts.Part one:Some properties on Dirichlet's theorem in arithmetic progressions in function fields are studied.Let Fq[t]denote the polynomial ring over the finite field IFq of q elements.Let a,b ?Fg[t]?0?.Suppose that deg b? 1 and a is prime to b.Let S denote the subset of Fq[t]containing all monic irreducible elements which is congruent to a modulo b.For x? 1,let ?(x,a,b)denote the number of elements of degree less or equal to logqx in S.Limit properties of ?(x,a,b)are investigated as x tends to infinity.It is proven that the relative density of the set S is 1/?(b),where ? denotes the Euler function.Part two:Some properties on intersective polynomial in function fields are studied.Let Z denote the ring of rational integers.Let 0 be a field,and let p be its characteristic.Let f(x)=?jn==0ajxj?(?)[X],the polynomial ring over(?)·Suppose that f(x)=a?ir=1(x-?i)e over some algebraic closure of 0,where a E 0,all the ?i are distinct,and r,e1,e2,…,er are positive integers with r?2 and n=?j=1rej.The semi discriminant ?(?)of f is defined by A(f)=a2n-1 ?(?)(?)(?i-?j)eiej.It is proved that if n<p,then there exist a positive integer m with m|n| and a polynomial G E Z[x0,x1,…,xa?,which depend only on the vector(e1,e2,…,er),such that ?(f)=1/mG(a0,a1,?,an)and m|n|.This result is applied to investigate a question on intersective polynomials over the ring(?)[x],where(?) is a finite field.