The Goldbach \(3\)-primes property for polynomial rings over certain infinite fields (Q5955915)

In [Bull. Am. Math. Soc., New Ser. 24, 363-369 (1991; Zbl 0724.11065)], \textit{G. W. Effinger} and \textit{D. R. Hayes} proved a kind of Goldbach 3-primes property for polynomial rings of one variable over finite fields. Their result may be stated as follows (EH-theorem). Theorem. (i) Let \(F\) be a finite field of odd order and let \(F[x]\) be the polynomial ring of one variable over \(F\). Then every monic polynomial of degree \(d\geq 2\) in \(F[x]\) can be expressed as a sum of three irreducible monic polynomials in \(F[x]\), one of degree \(d\) and the other two of lesser degree. (ii) Let \(F\) be a finite field of even order. Then every monic polynomial of degree \(d\geq 3\) in \(F[x]\) can be expressed analogously as a sum of three irreducible monic polynomials in \(F[x]\). This is an analogue of Goldbach's 3-primes conjecture in number theory. For polynomial rings over infinite fields, the 3-primes property need not to be valid. Here the authors prove by model-theoretic methods the following: There exist infinitely many infinite fields \(F\) such that the 3-primes property is valid for \(F[x]\). Their proofs do not depend on methods of analytic number theory or computers, but only on the statement of the EH-theorem and on two theorems in model theory.