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One way to define the p-adic integers is as the $p$-adic completion of $\mathbb{Z}$. With some additional work, it can be shown that this is isomorphic to $\mathbb{Z}[[x]]/(x-p)$.

Now, I know that another approach is to define $\mathbb{Z}_p$ as the ring of power series with powers of $p$ and coefficients from ${0, 1,..., p-1}$.

My question is: How can we see that the first definition of $\mathbb{Z}_p$ coincides with the second definition? and how can we find an explicit isomorphism from $\mathbb{Z}[[x]]/(x-p)$ to the ring of power series described above?

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    The second definition is incomplete. You haven't specified either the addition or the multiplication.2011-07-24
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    Is it obvious how to define the ring structure for your second construction (without referencing the usual tacks of completing the valued field or taking an inverse limit)? I've heard that this is possible, but have never seen it done.2011-07-24
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    @Qiaochu: The arithmetic operations are simply defined as in the case of formal power series, when p replaces x.2011-07-24
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    @Alexander: that is incorrect. You need to take into account carrying or else you get either something that is not well-defined or $\mathbb{F}_p[[x]]$, which has characteristic $p$ rather than $0$.2011-07-24
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    @Alexander : Well it's not exactly like formal power series in $x$ because of carry over (the rule looks more like the usual rule for addition and multiplication in base $p$). For a more precise description, you might have a look at http://en.wikipedia.org/wiki/Witt_vector .2011-07-24
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    Even though the analogy with power series was apparently part of Hensel's original motivation, as Qiaochu Y. notes above, and as in Gerry M's answer below, this representation has problems with multiplication. It is most definitely _not_ just sending $x$ to $p$, etc. This picture of p-adic things is at best a very temporary thing, I think, because of such issues and others.2011-07-25
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    Related: https://math.stackexchange.com/questions/527052/ring-of-p-adic-integers-mathbb-z-p?rq=12018-02-24

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