Let $R=\mathbb{Z}[\sqrt {-11}]$ and $F = Quot(R) = \mathbb{Q}$$[\sqrt {-11}] = \{m+n\sqrt{-11}:m,n \in \mathbb{Q}\}$.
I have to show for Gauss Lemma that the domain of $R$ is not a unique factorization domain.
My attempt:
I used the polynomial $x^2-x+3$ and showed is irreducible in $R[x]$ but reducible in $F[x]$, Can I used for show the Gauss lemma?