Capital Region Algebra/Number Theory Seminar

Speaker:    Gove Effinger, Skidmore College
 
Time:    Wednesday, March 19, at 4:15 pm
 
Place:    Bailey Hall 207, Union College
 
Title:     Toward a Complete Twin Primes Theorem for Polynomials over Finite Fields

Abstract

t monic irreducible polynomials in one variable over the finite field F_{q} with q > 2 are called t-tuplets provided that they all are identical except in their constant terms. If t = 2, they are, naturally, called twins. Over F_{2}, two irreducibles are called twins provided that they are identical except in their linear and quadratic terms. We prove, based on a theorem of Lidl and Niederreiter on generating new irreducibles from old, that over every finite field F_{q} with q > 2 there are infinitely many pairs of twin irreducible polynomials, and we give precise counts of how many t-tuplets exist depending on q and on a prime p dividing q-1. Finally, we discuss the distinct and seemingly difficult case of twins over F_{2}.