$\mathbb{Q}[t]$ is integrally closed in $Quot(\mathbb{Q}[t])$ I'm having trouble trying to show that $\mathbb{Q}[t]$ is integrally closed in $Quot(\mathbb{Q}[t])$. Where $Quot(\mathbb{Q}[t])$ is the field of fractions of $\mathbb{Q}[t]$. So I'm trying to show that $\mathbb{Q}[t]=$ integral closure of $\mathbb{Q}[t]$ in $Quot(\mathbb{Q}[t])$.

A unique factorization domain is always integrally closed in its field of fractions. The proof is exactly the same as for $\mathbb Z$.