The ring of integers of a cyclotomic field $\mathbb{Q}(\zeta_n)$ is the unique maximal order of that field. Kummer's attempt at proving FLT fails for prime exponents which are irregular, i.e. divide the class number of the ring of integers.
Is there simple argument that prevents one to find, for an irregular prime $p$, a (non-maximal) order $\mathcal{O}$ of $\mathbb{Q}(\zeta_n)$ such that:
- $\mathcal{O}$ is a Dedekind domain
- $\mathcal{O}$ contains a subring isomorphic to $\mathbb{Z}$
- $p$ does not divide the class number of $\mathcal{O}$
In general, if $K$ is a number field and $\mathcal{O}$ a nonmaximal order, then $\mathcal{O}$ is not integrally closed so is not a Dedekind domain. See for example this paper.
A proof (taken from that paper) is as follows: Since $\mathcal{O}\subsetneq \mathcal{O}_K$, choose $r\in\mathcal{O}_K\backslash\mathcal{O}$. Then $r$ satisfies a monic polynomial $g(x)\in\mathbb{Z}[x]\subset\mathcal{O}[x]$. Hence $\mathcal{O}$ is not integrally closed.