# Further reading on the $p$-adic metric and related theory.

In his book Introduction to Topology, Bert Mendelson asks to prove that

$$(\Bbb Z,d_p)$$

is a metric space, where $p$ is a fixed prime and

$$d_p(m,n)=\begin{cases} 0 \;,\text{ if }m=n \cr {p^{-t}}\;,\text{ if } m\neq n\end{cases}$$

where $t$ is the multiplicty with which $p$ divides $m-n$. Now, it is almost trivial to check the first three properties, namely, that

$$d(m,n) \geq 0$$

$$d(m,n) =0 \iff m=n$$

$$d(m,n)=d(n,m)$$

and the only laborious was to check the last property (the triangle inequality). I proceeded as follows:

Let $a,b,c$ be integers, and let

$$a-b=p^s \cdot k$$

$$b-c=p^r \cdot l$$

where $l,k$ aren't divisible by $p$.

Then $$a-c=(a-b)+(b-c)=p^s \cdot k+p^r \cdot l$$

Now we have three cases, $s>r$, $r>s$ and $r=s$. We have respectively:

$$a-c=(a-b)+(b-c)=p^r \cdot(p^{s-r} \cdot k+ l)=p^r \cdot Q$$ $$a-c=(a-b)+(b-c)=p^{s} \cdot( k+p^{r-s} \cdot l)=p^s \cdot R$$ $$a-c=(a-b)+(b-c)=p^s \cdot (k+l)=p^s \cdot T$$

In any case,

$$d\left( {a,c} \right) \leqslant d\left( {a,b} \right) + d\left( {b,c} \right)$$

since

\eqalign{ & \frac{1}{{{p^r}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^r}}} \cr & \frac{1}{{{p^s}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^r}}} \cr & \frac{1}{{{p^s}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^s}}} \cr}

It might also be the case $k+l=p^u$ for some $u$ so that the last inequality is

$$\frac{1}{{{p^{s + u}}}} \leqslant \frac{1}{{{p^s}}} + \frac{1}{{{p^s}}}$$

$(1)$ Am I missing something in the above? The author asks to prove that in fact, if $t=t_p(m,n)$ is the exponent of $p$, that

$$t\left( {a,c} \right) \geqslant \min \left\{ {t\left( {a,b} \right),t\left( {b,c} \right)} \right\}$$

That seems to follow from the above arguement, since if $s \neq r$ then

$$t\left( {a,c} \right) = t\left( {a,b} \right){\text{ or }}t\left( {a,c} \right) = t\left( {b,c} \right)$$

and if $s=r$ then

$$t\left( {a,c} \right) \geqslant t\left( {a,b} \right){\text{ or }}t\left( {a,c} \right) \geqslant t\left( {b,c} \right)$$

$(2)$ Is there any further reading you can suggest on $p$-adicity?

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What do you want to know? –  Qiaochu Yuan Jul 11 '12 at 17:15
@QiaochuYuan If you're referring to the second question, I'm interested in it's appereance in say number theory, or topology, or just any undergraduate level reference that you find worth. –  Pedro Tamaroff Jul 11 '12 at 17:27

I am currently trying to learn about $p$-adic numbers and analysis too, so I too would be really interested to hear the opinions of people who know more than I do about this. I am currently using the following three texts, but don't intend to work through them fully; just enough to get something useful for a better understanding of how they can be used in number theory:

1. Koblitz - "$p$-adic Numbers, $p$-adic Analysis, and Zeta-functions" - The first chapter I have found very interesting, and pretty well-written, with lots of easy exercises to get used to the concepts, as well as some harder ones to test deeper understanding.
2. Robert - "A Course in $p$-adic Analysis" - covers much more material at a more advanced level than Koblitz, but isn't (quite) as off-putting as it seems at first, and it possible to pick out quite a few bits from the first two chapters which are illuminating.
3. Borevich and Shafarevich - "Number Theory" - This one has some relatively understandable stuff on $p$-adic numbers in the first chapter which I have found really useful as it gives a different feel to the topic in a rather more old-fashioned approach.
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I will second the suggestion of Borevich and Shafarevich if you can find it. I'm a big fan of the general structure of that book, and their introduction to $p$-adic integers as both power series and as Witt vectors (though they clearly don't use the name). –  Brandon Carter Jul 11 '12 at 22:10

What you have done seems correct. Also notice that from what you have done you get that $$d(a,c) \leq \max \{d(a,b), d(b,c) \},$$ which is stronger than the triangle inequality.

For an introduction to $p$-adic numbers, I would suggest Fernando Gouvea's $p$-adic Numbers: An introduction. It should be easy for an undergraduate to understand the book, and I think it is a very nice introduction.

Edit: I should add that it's not easy for any undergraduate. One should've had a few courses which have a lot of proofs, and not just the standard calculus courses.

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Your argument looks fine to me.

A very good online introduction to $p$-adics are these notes; it covers modular arithmetic leading up to Hensel's, basic analysis with the numbers, the very strange metric topology of $\Bbb Q_p$ (every point inside of a ball is a center, there are locally but not globally constant functions, etc.), and a bit of field theory and algebraic number theory.

I don't really have any recommendation for number-theoretically in-depth stuff. Also, I tend to stick to online documents because the web is where I spend all my time anyway.

In actuality, my favorite discussion is p-adic integration and the theory of groups, which involves category theory and abstract algebra (the $p$-adics are constructed as an inverse limit of topological rings, for example), as well as measure theory and group theory. This source perhaps takes more background to digest its contents satisfactorily, but it hits my buttons well.

Just for fun, I also suggest Pictures of Ultrametric Spaces. Related entertainment: (working with adeles) the character group of $\Bbb Q$ and (working with profinite integers) profinite Fibonacci numbers.

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