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I have a question in Algebraic Number Theory. I somehow do not know how to proceed.

Let $O\subseteq \mathbb{Z}_K$ be subring where $K$ is a number field.

TFAE: $\quad$ i) $O$ is an order of $K$ $\quad$ and $\quad$ ii) Frac$(O)=K$

I should somehow use that $O$ is a free $\mathbb{Z}$-module of rank equal to $[K: \mathbb{Q}]$, I hope somebody knows how to prove this result.

Thank you in advance for your answer!

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up vote 3 down vote accepted


  1. If a set of elements of a vector space over $\mathbb{Q}$ are linearly independent over $\mathbb{Z}$, then they are linearly independent over $\mathbb{Q}$.
  2. To every element of an order there is another element of that order such that their product is in $\mathbb{Z}$.
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Thank you for your answer, but I am still not able to show ii)->i) – Mathoman Mar 26 '13 at 15:06

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