Linear Recurring Sequences and Explicit Factors of x2nd—1 in Fq[x]

Let F_(q) be a finite field of odd characteristic containing q elements,and n be a positive integer.An important problem in finite field theory is to factorize x^(n)-1 into the product of irreducible factors over a finite field.Beyond the realm of theoretical needs,the availability of coefficients of irreducible factors over finite fields is also very important for applications.In this paper,we introduce second order linear recurring sequences in F_(q) and reformulate the explicit factorization of x^(2nd)-1 over in such a way that the coefficients of its irreducible factors can be determined from these sequences when d is an odd divisor of q+1.