Show that $\mathbb{Z}'_\mathbb{C}$ is not a finitely generated $\mathbb{Z}$-module.

Question

Let $\mathbb{Z}\subset \mathbb{C}$ an extension and $\mathbb{Z}'_\mathbb{C}$ the integral closure of $\mathbb{Z}$ in $\mathbb{C}$.

Show that $\mathbb{Z}'_\mathbb{C}$ is not a finitely generated $\mathbb{Z}$-module.

Answer 1

$\mathbb{Z}'_\mathbb{C}$ in particular would have to contain all the roots of unity, and so contain $\Bbb Z[\zeta_n]$ so the number of generators is greater than $\phi(n)$ for all $n$, which of course is impossible as $\phi(n)\to\infty$ as $n\to\infty$.

Answer 2

More-or-less the same idea as Adam Hughes.

$\mathbb{Z}'_\mathbb{C}$ contains $\Bbb Z[2^{1/n}]$ for all $n\in\Bbb N$. This has rank $n$ (ultimately down to Eisenstein's criterion), so the rank of $\mathbb{Z}'_\mathbb{C}$ must be infinite.