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How to prove that if $\mathbb{Z}_p$ is the set of $p$-adic integers then $\displaystyle{\mathbb{Z}_p=\varprojlim\mathbb{Z}/p^n\mathbb{Z}}$ where the limit denotes the inverse limit?

$\mathbb{Z}_p$ is the inverse limit of the inverse system $(\mathbb{Z}/p^n\mathbb{Z}, f_{mn})_{\mathbb{N}}$, but I don't know what the $f_{mn}$ are.

Can someone help me?

  • 2
    The $f_{mn}$ are the "obvious" maps $\mathbb{Z}/p^m\mathbb{Z}\to\mathbb{Z}/p^n\mathbb{Z}$, where $m\gt n$. Note that $p^m\mathbb{Z}\subseteq p^n\mathbb{Z}$, so that $\mathbb{Z}/p^n\mathbb{Z}$ is a quotient of $\mathbb{Z}/p^n\mathbb{Z}$. The connective maps are the quotient maps.2012-06-11
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    How are you defining the $p$-adics, if not as the inverse limit? I'm asking because for many, the $p$-adics **are** the inverse limit (by definition), so asking how to prove that they are the same would be asking how to "prove" a definition. So obviously, you must have a different definition of $\mathbb{Z}_p$, and you probably should specify it.2012-06-11
  • 1
    Now that I think about it, after you state your definition of $\Bbb Z_p$ (I guess it is with power series expansions), you can also mention if you want to show isomorphisms as additive groups, as rings, and if you want topology in the mix as well. I was hasty adding the (topological-groups) tag; I imagine you just want an isomorphism as bare groups.2012-06-11
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    You can view a $p$-adic integer $a_0 + a_1 p + ...$ as a sequence $(..., a_2 p^2 + a_1p + a_0, a_1 p + a_0, a_0)$, which is an element of the inverse limit. This is how the isomorphism goes I believe.2012-06-11

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