As usual, sorry for the delay. Most of the time when I write this blog, I'm writing about stuff that I know about. I'm learning about the p-adic numbers as I write these posts, so it's a lot more work for me. But I think many of you will be happy to hear about the other reason for the delay: I've been working on a book based on some of my favorite posts from the history of this blog. It's nearly ready! I'll have more news about getting pre-release versions of it later this week!

But now, back to p-adic numbers!

In the last post, we looked a bit at the definition of the p-adic numbers, and how p-adic arithmetic works. But what makes p-adic numbers really interesting and valuable is *metrics*.

Metrics are one of those ideas that are simultaneously simple and astonishingly complicated. The basic concept of a metric is straightforward: I've got two numbers, and I want to know how far apart they are. But it becomes complicated because it turns out that there are many different ways of defining metrics. We tend to think of metrics in terms of euclidean geometry and distance - the words that I to describe metrics "how far apart" come from our geometric intuition. In math, though, you can't ever just rely on intuition: you need to be able to define things precisely. And precisely defining a metric is difficult. It's also fascinating: you can create the real numbers from the integers and rationals by defining a metric, and the metric will reveal the gaps between the rationals. Completing the metric - filling in those gaps - gives you the real numbers. Or, in fact, if you fill them in differently, the p-adic numbers.

To define just what a metric is, we need to start with fields and norms. A field is an abstract algebraic structure which describes the behavior of numbers. It's an abstract way of talking about the basic structure of numbers with addition and multiplication operations. I've talked about fields before, most recently when I was debunking the crackpottery of E. E. Escultura here.

A norm is a generalization of the concept of absolute value. If you've got a field \(F\), then a norm of \(F\) is a function, the norm of is a function \(| cdot |\) from values in \(F\) to non-negative numbers.

- \(| x | = 0\) if and only if \(x = 0\).
- \( |x y| = |x| |y|\)
- \(|x + y| le |x| + |y|\)

A norm on \(F\) can be used to define a distance metric \(d(x, y)\) between \(x\) and \(y\) in \(F\) as \(| x - y|\).

For example, the absolute value is clearly a norm over the real numbers, and it defines the euclidean distance between them.

So where do the gaps come from?

You can define a sequence \(a\) of values in \(F\) as \(a = { a_i }\)

for some set of values \(i\). There's a special kind of sequence called

a *Cauchy sequence*, which is a sequence where \(lim_{i,j rightarrow infty} |a_n - a_m| = 0\).

You can show that any Cauchy sequence converges to a real number. But even if every element of a Cauchy sequence is a rational number, it's pretty easy to show that many (in fact, most!) Cauchy sequences do *not* converge to rational numbers. There's something in between the rational numbers which Cauchy sequences of rational numbers can converge to, but it's not a rational number. When we talk about the gaps in the rational numbers, that's what we mean. (Yes, I'm hand-waving a bit, but getting into the details

would be a distraction, and this is the basic idea!)

When you're playing with number fields, the fundamental choice that you get is just how to fill in those gaps. If you fill them in using a metric based on a Euclidean norm, you get the real numbers. What makes the p-adic numbers is just a different norm, which defines a different metric.

The idea of the p-adic metric is that there's another way of describing the distance between numbers. We're used to thinking about distance measured like a ruler on a numberline, which is what gives us the reals. For the p-adics, we're going to define distance in a different way, based on the structure of numbers. The way that the p-adic metric works is based on how a number is built relative to the prime-number base.

We define the p-adic metric in terms of the p-adic norm exactly the way that we defined Euclidean distance in terms of the absolute value norm. For the p-adic number, we start off with a norm on the integers, and then generalize it. In the P-adic integers, the norm of a number is based around the largest power of the base that's a factor of that number: for an integer \(x\), if \(p^n\) is the largest power of \(p\) that's a factor of \(x\), then the the p-adic norm of \(x\) (written \(|x|_p\)) is \(p^{-n}\). So the more times you multiply a number by the p-adic base, the *smaller* the p-adic norm of that number is.

The way we apply that to the rationals is to extend the definition of p-factoring: if \(p\) is our p-adic base, then we can define the p-adic norm of a rational number as:

- \(|0|_p = 0\)
- For other rational numbers \(x\): \(|x|_p = p^{-text{ord}_p(x)}\) where:
- If \(x\) is a natural number, then \(text{ord}_p(x)\) is the exponent of the largest power of \(p\) that divides \(x\).
- If \(x\) is a rational number \(a/b\), then \(text{ord}(a/b) = ord(a) - ord(b)\).

Another way of saying that is based on a property of rational numbers and primes. For any prime number \(p\), you can take any rational number \(x\), and represent it as a p-based ratio \(p^nfrac{a}{b}\), where neither \(a\) nor \(b\) is divisible by \(p\). That representation is unique - there's only one possible set of values for \(a\), \(b\), and \(n\) where that's true. In that case, p-adic norm of \(x\),

\(|x|_p == p^{-n}\).

Ok, that's a nice definition, but what on earth does it mean?

Two p-adic numbers \(x\) and \(y\) are close together if \(x - y\) is divisible by a large power of \(p\).

In effect, this is the exact opposite of what we're used to. In the real numbers written out in decimal for as a series of digits, the metric says that the more digits numbers have in common moving from left to right, the closer together they are. So 9999 and 9998 are closer than 9999 and 9988.

But with P-adic numbers, it doesn't work that way. The P-adic numbers are closer together if, moving right to left, they have a common prefix. The distance ends up looking very strange. In 7-adic, the distance between 56666 and 66666 is smaller than the distance between 66665 and 66666!

As strange as it looks, it does make a peculiar kind of sense. The p-adic distance is measuring a valuable and meaningful kind of distance between numbers - their distance in terms of

their relationship to the base prime number p. That leads to a lot of interesting stuff, much of which is, to be honest, well beyond my comprehension! For example, the Wiles proof of Fermat's last theorem uses properties of the P-adic metric!

Without getting into anything as hairy as FLT, there are still ways of seeing why the p-adic metric is valuable. Next post, we'll look at something called Hensel's lemma, which both shows how something like Newton's method for root-finding works in the p-adic numbers, and also shows some nice algebraic properties of root-finding that aren't nearly as clear for the real numbers.

"To define just what a metric is, we need to start with fields and norms."

Important Nitpick:

Metrics are much more general than number fields; the notion of a metric makes sense for any topological space, but (extra-important) not every topological space admits any metric functions at all! Those that are are called "metrizable", and it turns out to be a very interesting subject to determine exactly which topological spaces are metrizable.

On top of that: every norm does give rise to a (distance) metric, as you say, but

notevery metric comes from a norm, even in topological torsors (like vector spaces over topological fields), where the distance-metric construction makes sense.I'm not sure I'm following this correctly. if p is 7, it means numbers 1 to 6 and 11 to 16 (that is 8 to 13 in base 7) have ord(x) 0, so their norm is 1, which is maximal norm. numbers 10 (7), 20 (14)... have norm 1/10 (1/7) etc?

Yes (assuming 1/10 means the multiplicative inverse of 10 in base 7). And if you restrict yourself to the p-adic _integers_, you're statement about having maximal norm is correct. However with p-adic _numbers_, you can have norm larger than 1. For example, in base 10, 1/7 has norm 7.

An amusing way to remember the relationship between the p-adic valuation (what you denote $mathrm{ord}_p$) and the p-adic norm is "The bigger the p-ness, the closer to zero."

I got this from Ben McReynolds, one of my profs, who said he got it from a professor of his from grad school.

My ultimate favorite statement about p-adic norm: Every triangle in the p-adic numbers is isosceles. That is, if a, b, and c are three p-adic numbers, then ||a-b|| = ||a - c||, ||a-b|| = ||b - c||, or ||a-c|| = ||b -c||.

I feel that what is really important to the idea of norms and what makes p-adic numbers special is that the p-adic norm is "non-archemedean", i.e. there is an extra property to the norm that n(x+y)<=max(n(x),n(y)). I don't know if you included that.