Ring of entire functions is integrally closed or not? > Is the ring $\mathscr{O}(\mathbf{C})$ of entire functions integrally closed (in its field of fractions, the meromorphic functions)? I know it's not factorial, but this doesn't exclude the possibility of it being integrally closed.

Yes, $\mathscr{O}(\mathbb{C})$ is a GCD domain, and as such it is integrally closed.

You can also see it directly, if $f \in \mathscr{M}(\mathbb{C})$ satisfies

$$\sum_{k=0}^n a_k(z) f(z)^k \equiv 0$$

with $a_k \in \mathscr{O}(\mathbb{C})$ and $a_n = 1$, if it had a pole of order $m$ in $z_0$, you would have

$$1 + \sum_{k=0}^{n-1} a_k(z)f(z)^{k-n} \equiv 0$$

in a punctured neighbourhood of $z_0$, but

$$\lim_{z\to z_0} \sum_{k=0}^{n-1} a_k(z)f(z)^{k-n} = 0$$

since the $a_k$ are bounded.