A brief tour of the basic structure of the $p$-adic rational numbers and their generalizations.

- Absolute Values on $\mathbb Q$
- Completions of $\mathbb Q$
- The Topology of $\mathbb Q_p$
- The Algebra of $\mathbb Z_p$
- Series Expansions in $\mathbb Q_p$
- Measures on $\mathbb Q_p$
- The Geometry of $\mathbb Q_p$ and $\mathbb Z_p$

### Absolute Values on $\mathbb Q$

Definition: An absolute value $| \cdot |$ on $\mathbb Q$ is a real valued function such that for any $x, y \in \mathbb Q$, Positive Definite: $|x| \geq 0$ and is equal to zero if and only if $x = 0$. Multiplicative: $| x y | = | x | \cdot | y |$ Triangle…

### Completions of $\mathbb Q$

Recall how we construct the real numbers from $\mathbb Q$. Here $| \cdot |$ is the usual absolute value, but the whole point of this section is that this construction generalizes for other absolute values. The basic idea is that we may make (new) real numbers by including limits of convergent sequences of rational numbers.…

### The Topology of $\mathbb Q_p$

We may extend $| \cdot |_p$ to $\mathbb Q_p$ by setting $| [a_n] |_p = \lim |a_n |_p$. This limit necessarily converges, and provides an absolute value on $\mathbb Q_p$ (which we still denote by $|\cdot|_p$). This absolute value produces a metric $d_p(x, y) = | x – y |_p$ and from this we can…

### The Algebra of $\mathbb Z_p$

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…

### Series Expansions in $\mathbb Q_p$

Here we give a practical representation of $p$-adic numbers in terms of a power series in powers of $p.$ This perspective will connect our understanding of the $p$_adics as a Cauchy completion and as an inverse limit. We first do this for the $p$-adic integers and then explain how to extend it to all of…

### Measures on $\mathbb Q_p$

Because $\mathbb Q_p$ is a locally compact abelian group it has a natural translation invariant measure, Haar measure, which allows us to formulate a natural theory of integration of (certain) real valued functions over subsets of the $p$-adics. Here we talk about this Haar measure and it’s basic properties. The following may be useful readings…

### The Geometry of $\mathbb Q_p$ and $\mathbb Z_p$

Once we define distance appropriately, we’ll find a distance preserving embedding of $\mathbb Z_p$ into $\mathbb R^2$â€”that is we’ll be able to draw a picture of $\mathbb Z_p$ and determine the distance between points by looking at that picture. This in turn will provide a related geometric picture for $\mathbb Q_p$. The $5$-adic integers in…