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?
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.