If  is a polynomial with integer coefficients and p is prime, we define the suborder of  modulo p to be the smallest positive integer m for which  divides some polynomial  in  For quadratic polynomials  we associate an element a of  with and show that the suborder of  modulo p is determined by a, thus defining a suborder function on  Using properties of a recurrence relation that we can define from a quadratic polynomial, and calculations in an arbitrary quadratic extension of  we show that there is a family of polynomials,  for positive integers n, which we call suborder polynomials, whose roots modulo a prime p determine the values of the suborder function on  These suborder polynomials are easily defined recursively, or in terms of binomial coefficients, and we establish results on the number of solutions of  in  Legendre symbol properties of those solutions, connections between solutions and terms in certain recursive sequences, links to factorization of cyclotomic polynomials modulo primes, and relations between solutions for different values of n.

polynomials over finite fields, recurrence relations, cyclotomic polynomials.