The Congruence Question Of High-order Recurrence Sequence

In this paper we study the congruence question Of rational recurrence sequence un+k=a1un+k-1+…+akun,i.e.a1,a2:…,ak are rational numbers. Let f(x)= xk-a1xk-1-…-ak be the characteristic polynomial. The splitting field of f(x) over Q is denoted by K.If f(x) has no multiple toot,and K is an abelian exteasion of Q,then we can determine the prime number such that un+p-1(?)un(mod p).The key point is to find a positive integer N and a finite set S:if a prime p satisfys p (?) S and p (?) 1(mod N),then the p is what we want.The main theories we used is the ramification theory and class field theory.