The content of a Gaussian polynomial is invertible

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