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Let $K$ be an algebraic number field of degree $n$. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\Bbb Z$-module of rank $n$.

In this case, we have $R^{\times}$ $\leq$ $O_K^{\times}$ as a subgroup, and the latter is a finite generated abelian group of rank $r_1+r_2-1$ by Dirichlet Unit Theorem. So $R^{\times}$ is a f.g Abelian group as well, but how could we compute the rank of R ? Is there some generalization of Dirichlet Unit Theorem?

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The rank of $R^\times$ is the same as that of $O_K^\times$.$\newcommand{\ep}{\varepsilon}$

There is a natural number $N$ such that $NO_K\subset R$. If $\ep$ is a unit of $O_K$ then $\ep^k\equiv1\pmod N$ for some $k>0$. Then $\ep^k\in R^\times$. So if $\ep_1,\ldots,\ep_r$ form a system of fundamental units in $O_K$ then there are $k_1,\ldots,k_r\ge1$ with $\ep_1^{k_1},\ldots,\ep_r^{k_r}$ independent units of $R$. Therefore $R^\times$ also has rank $r$.

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