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 x ⁶ + x + a into two cubics over the algebraic closure F are in natural Galois bijection with the 10 roots of x ¹⁰ + ^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 x ⁶ + x + a is isomorphic to a difference set constructed by Dillon using the polynomial x ¹⁰ + x + a; (3) obtain a natural realization for the accidental isomorphism between the alternating group A ₆ and the special linear group PSL₂(9); and (4) characterize how x ⁶ + x + a factors when F = GF (3^m) with m odd. For example, x ⁶ + x + a is irreducible if and only if a can be written as δ^{− 36} + δ⁴ with δ ∈ F ^× and Tr(δ⁵) ≠ 0.