The $p$-adics and Related Concepts

The $p$-adic integers, $\mathbb Z_p$ have interesting algebraic properties.

**Theorem: **

**The Ring of Integers in $\mathbb Q_p$**: $\mathbb Z_p$ is a ring under addition and multiplication.**The Unique Maximal Ideal in $\mathbb Z_p$**: The open unit ball $B(0, 1)$ is equal to $p \mathbb Z_p$ and is the unique maximal ideal in $\mathbb Z_p$.**The Residue Class Field of $\mathbb Z_p$**: $\mathbb Z_p / p \mathbb Z_p$ is a finite field isomorphic to $\mathbb F_p := \mathbb Z / p \mathbb Z$.- The Group of Units in
**$\mathbb Z_p$**: The boundary of the closed unit ball $U = \{ x \in \mathbb Z_p : |x|_p = 1\}$ is a group under multiplication.

**Exercise:** Prove the theorem.

#### Inverse Limits of Rings

Here we give a purely algebraic construction of $\mathbb Z_p$.

**Definition:** Suppose $(A_n)$ is a sequence of rings, and suppose for all $i \leq j$ we have a homomorphism $\varphi_{i,j} : A_j \rightarrow A_i$ such that

- $\varphi_{i,i}$ is the identity isomorphism on $A_i$.
- If $i \leq j \leq k$, $\varphi_{i,k} = \varphi_{i,j} \circ \varphi_{k,j}$.

The inverse limit of $(A_n)$ with respect to the homomorphisms $\varphi_{i,j}$ is given by $$\varprojlim A_n = \left\{(a_n) : a_i = f_{i,j}(a_j) \mbox{ for all } i < j \right\}.$$

This is a ring under component wise addition and multiplication.

To construct $\mathbb Z_p$ as an inverse limit, we start by setting $A_n = \mathbb Z/p^n \mathbb Z$. Given $i \leq j$, we define $\varphi_{i,j}: \mathbb Z/p^j \mathbb Z \rightarrow \mathbb Z/p^i \mathbb Z$ to be the homomorphisms given by $\varphi_{i,j}(a + p^j \mathbb Z) = a + p^i \mathbb Z$. If we are using the representatives $\{1, 2, \ldots, p^j – 1\}$ for $\mathbb Z/p^j \mathbb Z$ then $\varphi_{i,j}(x)$ is the remainder of the representative when dividing by $p^i$.

**Theorem:** The inverse limit of $(\mathbb Z/p^n \mathbb Z)$ with respect to the $\varphi_{i,j}(a + p^j \mathbb Z) = a + p^i \mathbb Z$ is isomorphic to $\mathbb Z_p$.

We will give an explicit isomorphism between $\mathbb Z_p$ and this inverse limit after we talk about a canonical series representation for elements of $\mathbb Z_p$.

#### Cosets and Balls

We have already observed that $\mathbb Z_p$ is the closed unit ball in $\mathbb Q_p$ and the maximal ideal $p \mathbb Z_p$ is the open unit ball. Because balls are clopen, it is sometimes more useful to think of $p \mathbb Z_p$ as the closed ball of radius $p^{-1}$. The cosets of $p \mathbb Z_p$ are $1 + p \mathbb Z_p, 2 + p \mathbb Z_p, \ldots, p-1 + p \mathbb Z_p$. If $x$ and $y$ are in the same coset, then $|x – y| \leq 1/p$ and hence the cosets of $p \mathbb Z_p$ correspond to the closed balls of radius $1/p$.

This correspondence goes even further. We note that because $p \mathbb Z_p$ is the unique maximal ideal in $\mathbb Z_p$, all other ideals are of the form $p^n \mathbb Z_p$. It turns out that every closed ball is some coset of one of these ideals.

**Theorem:** If $B$ is a closed ball of radius $p^{-n}$ in $\mathbb Z_p$, then $B$ is a coset of $p^n \mathbb Z_p$. That is, $B = r + p^n \mathbb Z_p$ for some representative $r \in \{0,1,\ldots,p^n-1\}$