On x 6 + x + a in Characteristic Three

Abstract

Let F be a field of characteristic 3 and 0 ≠ a ∈ F. We show that the 10 ways to factor x6 + x + a into two cubics over the algebraic closure F are in natural Galois bijection with the 10 roots of x10 + ax + 1. We use this to (1) prove the two polynomials have the same splitting field; (2) prove that a difference set constructed by Arasu and Player using the polynomial x6 + x + a is isomorphic to a difference set constructed by Dillon using the polynomial x10 + x + a; (3) obtain a natural realization for the accidental isomorphism between the alternating group A6 and the special linear group PSL2(9); and (4) characterize how x6 + x + a factors when F e GF (3m) with m odd. For example, x6 + x + a is irreducible if and only if a can be written as δ− 36 + δ4 with δ ∈ F× and Tr(δ5) ≠ 0.