Witt Vector Motivation for Teichmüller representatives I'm quite comfortable with the motivation for Teichmüller representatives given in this MO post, but I'm curious about something said in the wikipedia article:

The elements of $\mathbb{Z}_p$ can be expanded as (formal) power series in $p$:
$$a_0 + a_1 p^1 + a_2 p^2 + \cdots,$$
where $a_i$ are usually taken from the integer interval $[0,p-1]=\{0, 1, \ldots, p-1\}.$ Of course, this power series usually will not converge in $\mathbb{R}$ using the standard metric on the reals, but it will converge in $\mathbb{Z}_p,$ with the $p$-adic metric. We will sketch a method of defining ring operations for such power series.
Letting $a+b$ be denoted by $c$, one might consider the following definition for addition:
$$\begin{align}
c_0 &\equiv a_0+b_0 && \bmod p \\
c_0+c_1 p &\equiv (a_0+b_0) + (a_1 +b_1)p && \bmod p^2 \\
c_0+c_1 p+c_2 p^2 &\equiv (a_0+b_0) + (a_1 +b_1)p+(a_2+b_2) p^2 &&\bmod p^3
\end{align}$$
and one could make a similar definition for multiplication. However, this is not a closed formula, since the new coefficients are not in the allowed set $[0,p-1]$.

I've bolded the last line above to emphasize it. This example is used to motivate choosing a new set of representatives, but I don't see the problem. If I carry out the process of solving those equations for the sum $c=a+b$ with, say, $p=7, a = 55 = 5\cdot 7 + 5$ and $b = 46 = 4\cdot 7 + 6$, I obtain values of $c_0 = 4, c_1 = 3, c_2=1$ which are within the allowed set $\{0, \ldots, 6\}$. I get the same thing if I use addition with carrying. I don't see how any other example can fail to be closed in the sense above. What am I doing wrong or misunderstanding here?
 A: The equations given there for $c_i$ are more naive than you think: in your example, they say that $(5 \cdot 7 + 5) + (4 \cdot 7 + 6) = 9\cdot 7 + 11$.  Converting this into $1\cdot 7^2 + 3 \cdot 7 + 4$ requires carrying, which is a finicky piecewise-defined operation that we want to avoid.
Perhaps this will better explain how non-algebraic the carrying operation is: given two two-digit numbers $a_1 a_0$ and $b_1 b_0$ written in base $p$, you know how to express their sum $c_2 c_1 c_0$ in base $p$ by carrying. This process involves working in $\mathbb Z$ and checking various inequalities, e.g. whether $a_0 + b_0 \geq p$. Can you find a way to calculate the digits $c_0, c_1, c_2 \in \mathbb F_p$ in terms of the digits $a_0, a_1, b_0, b_1 \in \mathbb F_p$ without making any reference to $\mathbb Z$?
The boldfaced sentence seems to be mixing up two senses of the word "closed".  On the one hand, base-$p$ expansions are not closed under the operation of naive addition (without carrying), since for example $9 \cdot 7 + 11$ is not a valid base-$p$ expansion.  (This is of course just a reflection of the fact that the subset $\{0, \dots, p-1\} \subset \mathbb Z$ is not closed under addition.)  On the other hand, it is difficult to give an explicit closed formula for the operation of addition with carrying (as opposed to an algorithm for computing it).
