The content of a Gaussian polynomial is invertible

Abstract

Let R R be an integral domain and let f ( X ) f(X) be a nonzero polynomial in R [ X ] R[X] . The content of f f is the ideal c ( f ) \mathfrak c(f) generated by the coefficients of f f . The polynomial f ( X ) f(X) is called Gaussian if c ( f g ) = c ( f ) c ( g ) \mathfrak c(fg) = \mathfrak c(f)\mathfrak c(g) for all g ( X ) ∈ R [ X ] g(X) \in R[X] . It is well known that if c ( f ) \mathfrak c(f) is an invertible ideal, then f f is Gaussian. In this note we prove the converse.