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It's known that the ring of p-adic integers $\mathbb{Z}_p$ can be characterized as the I-adic completion of $\mathbb{Z}$ for $I=(p)$. Is there any similar characterization for $\mathbb{Q}_p$ (i.e. an ideal $I$ of $\mathbb{Q}$ such that $\mathbb{Q}_p$ is the $I$-adic completion of $\mathbb{Q}$)?

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$\mathbb{Q}$, being a field, has no non-trivial ideals. Instead, one can think about $\mathbb{Q}_p$ simply as the fraction field of $\mathbb{Z}_p$. Alternately, the $p$-adic metric on $\mathbb{Z}$ naturally extends to $\mathbb{Q}$, and completing $\mathbb{Q}$ with respect to this metric gives $\mathbb{Q}_p$.

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