JP Journal of Algebra, Number Theory and Applications
Volume 28, Issue 1, Pages 45 - 80
(February 2013)
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SUBORDER POLYNOMIALS MODULO PRIMES
J. Larry Lehman
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Abstract: 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. |
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