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Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers, $\mathfrak{P}$ a non-zero prime ideal of $\mathcal{O}_K$, and $p\in \Z\cap \mathfrak{P}$. There are a couple of ways to construct $K_\mathfrak{P}$, the $p$-adic completion of $K$ at $\mathfrak{P}$.

First, we can take the inverse limit $$ \lim_{\leftarrow} \mathcal{O}_K/\mathfrak{P}^n $$ which is an integral domain. Its field of fractions is $K_\mathfrak{P}$.

Second, since $\mathcal{O}_K$ is a Dedekind domain, if $\alpha\in K^*$ the fractional ideal $$\langle \alpha\rangle = \prod_{\mathfrak{Q}} \mathfrak{Q}^{e_\mathfrak{Q}}$$ where the product is over all non-zero prime ideals $\mathfrak{Q}$, all $e_{\mathfrak{Q}}\in\Z$, and all but finitely many $e_\mathfrak{Q}=0$. Then we define $v_\mathfrak{P}(\alpha)=e_\mathfrak{P}$, and then the metric $d$ on $K$ by $d(\alpha, \beta) = p^{-v_{\mathfrak{P}}(\alpha-\beta)}$ if $\alpha\neq \beta$ and $d(\alpha,\alpha)=0$. Then the completion of $K$ with respect to this metric is $K_\mathfrak{P}$.

If $K=\Q(a)$, and $f\in\Q[x]$ is the monic irreducible polynomial for $a$ over $\Q$, then adjoining the roots of $f$ to $\Q_p$ provide another means of constructing the completions.

Finally, the local algebra of $K$, $\prod_{j=1}^g K_j$ is a product of the $p$-adic completions of $K$. The $p$-adic completions of $K$ correspond to the nonarchimedian places of $K$.

Knowl status:
  • Review status: beta
  • Last edited by John Jones on 2021-07-06 12:19:22
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