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Let $K/\mathbf{Q}$ be a number field with ring of integers $O_K$.

Is $O_K\cap K^\ast = O_K^\ast$?

I can't show that the inverse of an element in $O_K\cap K^\ast$ lies in $O_K^\ast$...

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    $O_K \cap K^\times = O_K \cap (K\setminus 0) = O_K \setminus 0$. It might be helpful to think of $K = \mathbf{Q}$, then $O_K = \mathbf{Z}$ but $\mathbf{Z}^\times = \pm 1$.2011-10-18
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    Can't possibly be true, since $K^* = K-\{0\}$. The intersection is just all nonzero elements of $\mathcal{O}_K$, and those are never all units (since $\mathcal{O}_K$ always includes $\mathbb{Z}$, but no other rationals).2011-10-18

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