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Q:
Question on the units in $\Bbb Z_p$ with $p$ not a prime
I am reviewing chapter 4 of the book ‘Algebra’.
The authors of the book claim that every element in $\Bbb Z_p$ can be written as $\sum_{j=0}^{p-1}a_jp^j$ and every unit in $\Bbb Z_p$ can be written as $\frac{1}{p^t}$
My question is: Why the author chose $\Bbb Z_p$ in the first place and not $\Bbb Z$ or $\Bbb Z_{p^n}$ where $n\in \Bbb N$?
A:
The reason they chose $\Bbb Z_p$ is because it has $(p-1)p^{t-1}$ elements (subsequently proved to be a multiple of $p^t$), namely, the $\sum_{j=0}^{p-1}j = \frac{p(p-1)}{2}$ $p$-adic integers. You might want to think of $\Bbb Z_p$ as an (infinite) cyclic group $\Bbb Z_p = \{0, 1, 2,…, p-1\}$.
For example, $\Bbb Z_{30}$ would have $29$ elements ($30 \mid 60$), and they were chosen to form a group isomorphic to $\Bbb Z$. However, $\Bbb Z$ has $\infty$ elements. For example, $\frac{1}{3} \in \Bbb Z$.
Likewise, $\Bbb Z_p$ has $p$ elements.
In general, $\Bbb Z_n$ has $n$ elements for $n \mid m$, and has $0$ elements for $n
mid m$.