Let $p$ be a prime and $I=\mathbb{Z^+}$. Let $A_i=\mathbb{Z_{p^i}}$ 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 $\mathbb{Z_{p^i}}$ in its base p expansion and then describe tuhe maps $\gamma_{ji}$. (Note in particular that $\mathbb{Z_p}$ is uncountable.
(b) Prove that $\mathbb{Z_p}$ is an integral domain that contains a copy of the integers.
(c) prove that $b_0+b_1p+b_2p^2+....$ as in (a) is a unit in $\mathbb{Z_p}$ if and only if $b_o \neq 0$.
(d) Prove that $p\mathbb{Z_p}$ is the unique maximal ideal of $\mathbb{Z_p}$ and $\mathbb{Z_p}/p\mathbb{Z_p} \cong \mathbb{Z}/p\mathbb{Z}$ (where $p=0+1p+0p^2+0p^3+...$). Prove that every nonzero ideal of $\mathbb{Z_p}$ is of the form $p^n\mathbb{Z_p}$ for some integer $n \geq 0$.
(e) Show that if $a_1 \neq 0$ (mod p) then there is an element $a = (a_i)$ in the inverse limit $\mathbb{Z_p}$ satisfying $a_j^{p-1} \cong 1$ mod $p^j$ and $\gamma_{j1}(a_j)=a_1$ $\forall j$. Deduce that $\mathbb{Z_p}$ contains $p-1$ and $(p-1)^{st}$ roots of 1.
Okay so i've been turning this book sidways and upside down trying to make sense of this but I'm having a hard time getting my feet on the ground. I just did a long problem regarding inverse limits so I have a decent handle on that, but i'm having trouble applying it to this case. Even with part (a), it's interesting to me to write every element of $\mathbb{Z_p}$ as a formal sum, when in my mind I think of $\mathbb{Z_p}$ as a infinite ordered tuple... But I suppose there is a cannonical mapping between the two. Anyway, if any smart guy reads this and can help me on these I would be forever appreciative! Thanks!
Let’s try:
Start with your consistent sequence $(a_1,a_2,\cdots)$, $a_i\in\Bbb Z/(p^i)$. Suppose inductively that you have $a_n=d_0+pd_1+p^2d_2+\cdots+p^{n-1}d_{n-1}\in\Bbb Z/(p^n)$. Next, $a_{n+1}-(d_0+pd_1+p^2d_2+\cdots+p^{n-1}d_{n-1})=p^n\delta\in\Bbb Z/(p^{n+1})$, so you just set $d_{n}=\delta$.
The rest I’ll have to let you hack out.