Joe Gurr

Intro

"... in mathematics you don't understand things. You just get used to them"- John Von Neumann

I wanted to write a simple introduction to the p-adic numbers, however in my research I came across this youtube video, which does a better job than I ever could have.

I had started writing this up and had most of the same ideas in place before finding this video, however I found the structure and approach of this video to be exemplary. I restructured what I had to be in roughly the same order and I found that the content was almost identical.

I decided to continue writing this even though there is already an excellent resource on this in order to practice my technical writing a little.

In any case, all credit to the merits of this post should go to the video's creator Rokker815, while any defects are surely my own.

I am going to walk you through a simplified construction of the p-adic numbers by showing the similarities to how the reals are constructed. Then we will go through some properties of the p-adic numbers.

This post is structured in the following order:

• Tools
• p-adic Absolute Value and Metric
• Geometry of \(( \mathbb{Q}, d_p )\)
• Motivation for \( \mathbb{Q}_p \)
• Properties of \( \mathbb{Q}_p \)

Tools

First, some definitions:

Rational Numbers: \(\mathbb{Q} = \{ \frac{a}{b} | a, b \in \mathbb{Z}, b \ne 0 \}\)

Field: a set \( \mathbb{F} \) on which \( +, -, \times, \div \) are defined and behave as expected (e.g. \(\mathbb{Q}\) with normal operations forms a field, but \(\mathbb{N}\) with normal operations is NOT a field)

Absolute Value: First define \( \mathbb{Q}_{\geq 0} = \{ x \in \mathbb{Q} | x \geq 0\}\) (known as the nonnegative rationals). Then any function \( | \cdot | : \mathbb{F} \rightarrow \mathbb{Q}_{\geq 0}\) that satisfies

• \(|a| \geq 0\)
• \(|a| = 0 \iff a = 0\)
• \(|ab| = |a||b|\)
• \(|a + b| \leq |a| + |b|\)

is known as an absolute value on the field (absolute values are not unique. For example there is always the trivial one where it takes value 0 for some value of \(\mathbb{F}\) and 1 for all other values).

The classic absolute value on \(\mathbb{Q}\) is that \(|x|\) is just \(x\) when \(x \in \mathbb{Q}_{\geq 0}\) and \(-x\) otherwise.

Metric Space: given a set \(X\), a metric \(d\), is a "distance" function \(d: X \times X \rightarrow \mathbb{Q}_+\) which has the following properties:

• \(d(x,y) \geq 0\)
• \(d(x,y) = 0 \iff x = y\)
• \(d(x,y) = d(y,x)\)
• \(d(x,y) \leq d(x,z) + d(y,z)\)

A metric space is a set \(X\), with a metric on it \((X, d)\).

e.g. a set \(\mathbb{Q}\), and a metric \(d(x,y) = |x - y|\), give us a metric space \((\mathbb{Q}, d)\).

Open Ball: \(B(a, r) = \{x \in X | d(a, x) < r\}\)

e.g.

One Way to Express a Rational Number

Fix \(p\) prime. Let \(x \in \mathbb{Q}\). The we can always express \(x\) as

\(x = p^ n \frac{g}{h}\)

where \(\frac{g}{h}\) is a fraction in its lowest terms (i.e. neither \(g\) or \(h\) is divisible by \(p\).)

p-adic Absolute Value on Q

For \(x\in\mathbb{Q}, x \neq 0\) we define the p-adic absolute value from the rationals to the non-negative rationals as \(|\cdot|_p : \mathbb{Q} \rightarrow \mathbb{Q}_+\), as \(|x|_p = |p^n\frac{g}{h}| = p^{-n}\) and \(|0|_p = 0\)

e.g.let \(p=7, x = \frac{147}{5}\) then you have \(|\frac{147}{5}|_7 = |7^2\frac{3}{5}|_7 = 7^{-2}\)