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I have been reading a bit about unramified extensions. If $K$ is $\mathbb{Q}$ or $\mathbb{Q}_p$ (p-adics), then there is a maximal unramified extension $K^{nr}$ of $K$. Then I have read in some notes that $$ {\rm Gal}(K^{nr} / K) \cong \hat{\mathbb{Z}}. $$ My question is: what is $\hat{\mathbb{Z}}$?

I have been told that this is $$\varprojlim \mathbb{Z} / n\mathbb{Z},$$

But I am not sure what this is. I think it is called an inverse limit. Is there a concrete way to think about this?

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  • $\begingroup$ I think this answer gives a clear idea of what an inverse limit is (at least in some restricted sense). Using the hat notation is a rather unfortunate overuse of notation in my opinion since $\widehat{\Bbb Z}$ has a completely different meaning in harmonic analysis (which somewhat intersects $p$-adic analysis). $\endgroup$ – Cameron Williams Aug 22 '14 at 22:17
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    $\begingroup$ If you are happy with $p$-adics, then one description of $\widehat{\mathbb{Z}}$ (the profinite completion of the integers, see groupprops.subwiki.org/wiki/…) is as the product $\prod_p \mathbb{Z}_p$, via the Chinese remainder theorem. $\endgroup$ – Qiaochu Yuan Aug 23 '14 at 6:18
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The ring $\widehat{\mathbb{Z}}$, called the profinite completion of $\mathbb{Z}$, can be thought of as a subring of the infinite product $\prod_{n = 1}^{\infty} \mathbb{Z} / n\mathbb{Z}$, which satisfies a `coherent sequence' condition. That is, a tuple $(x_1,x_2,x_3,\ldots ) \in \prod_{n=1}^{\infty} \mathbb{Z} / n \mathbb{Z}$ is in $\widehat{\mathbb{Z}}$ iff $x_n \equiv x_m \bmod m$ whenever $m$ divides $n$ (you can also think of this condition as saying that $x_n$ maps to $x_m$ under the canonical map $\mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/ m \mathbb{Z}$ whenever $m$ divides $n$).

Similarly, the $p$-adic integers $\mathbb{Z}_p$ are the tuples in $\prod_{n=1}^{\infty} \mathbb{Z} / p^n \mathbb{Z}$ that satisfy the analogous condition for the canonical maps $\mathbb{Z} / p^n \mathbb{Z} \to \mathbb{Z} / p^m \mathbb{Z}$ for $n \geq m$. The more general construction, called an inverse limit, is summarized here.

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