Let $p$ be a prime and $I=\mathbb{Z^+}$. Let $A_i=\mathbb{Z}/p^i\mathbb{Z}$ and let $\gamma_{ji}$ be the natural projection maps $\gamma_{ji}:$ $a$ (mod $p^j$) $\rightarrow a$ (mod $p^i$). The Inverse limit is called the ring of p-adic integers and is denoted by $\mathbb{Z_p}$.
(a) show that every element of $\mathbb{Z_p}$ can be written uniquely as an infinite formal sum $bo+b_1p+b_2p^2+b_3p^3+...$ with each $b_i \in \{0,1,2,....p-1\}$. Describe the rules for adding and multiplying such formal sums corresponding to addition and multiplication. [write a least residue in each $\mathbb{Z}/p^i\mathbb{Z}$ in its base p expansion and then describe the maps $\gamma_{ji}$. (Note in particular that $\mathbb{Z_p}$ is uncountable.
Can someone help me with this problem. I came up with the following:
Suppose $(x_1,x_2, \cdots)$ is in $\mathbb{Z_p}$. Then, by using the definition of an inverse limit, we can define the $b_i's$ recursively as $b_0=x_1$ and $b_{i+1}=\frac{x_{i+2}-x_{i+1}}{p^{i+1}}$. Note that since $x_{i+2}$ and $x_{i+1}$ are the same mod $p^{i+1}$, this is indeed an integer.
The unique part of the problem is very confusing. What does the book mean by unique? It has not given us a notion of equality in this set of formal prime power series. Am I supposed to define when two series are equal on my own? Maybe the notion of equality is that two series are equal iff they have the same coefficient for every prime power. If yes, how do I go about proving that the representation is unique.
Your notion of equality of the two series is fine. Using this and a simple induction argument you can show the representation is unique. Write down first few terms perhaps.
You can also find a map from the infinite power series to $Z_p$ and show this is the inverse of the map from $Z_p$ to the power series ring. This may not be required for your problem but it's useful computation wise later on.