Representing an element in F_q[t] as the sum of two irreducibles

TL;DR

This paper proves that for sufficiently large finite fields, any monic polynomial can be expressed as the sum of two irreducible monic polynomials of specific degrees, using a new criterion for simultaneous primality.

Contribution

It introduces a sufficient condition for the simultaneous primality of two polynomials, enabling the representation of polynomials as sums of two irreducibles in finite fields.

Findings

Such representations exist when q exceeds a bound depending on n.

The method applies to polynomials over finite fields of odd characteristic.

Provides a new criterion for simultaneous primality of polynomials.

Abstract

A monic polynomial in F_q[t] of degree n over a finite field F_q of odd characteristic can be written as the sum of two irreducible monic elements in F_q[t] of degrees n and n-1 if q is larger than a bound depending only on n. The main tool is a sufficient condition for simultaneous primality of two polynomials in one variable x with coefficients in F_q[t].