A Goldbach 3-primes theorem for polynomials of low degree over finite fields of characteristic 2

Abstract

A monic polynomial M in one variable x over the finite field k q of q elements is called even if q = 2 and if x or x + 1 divides M, otherwise M is called odd. We prove: Theorem. Every odd monic polynomial M of degree 2, 3, 4, 5, and 6 over every finite field k q of characteristic 2 can be written a sum of 3 irreducible monic polynomials except for the case of M = x 2 + α ( α ϵ k q ). Together with previous results, this theorem reduces the following conjecture to a finite, tractible calculation: The Polynomial 3-Primes Conjecture. Every odd monic polynomial M of degree ≥ 2 over every finite field k q can be written as a sum of 3 irreducible monic polynomials except for the case of M = x 2 + α, α ϵ k q , q even.