From: tycchow@athena.mit.edu (Timothy Y Chow)
Newsgroups: sci.math
Subject: Re: p-adic Numbers -- What are they?
Date: 30 Aug 1993 19:38:03 GMT
Message-ID: <25tkur$ctn@senator-bedfellow.MIT.EDU>
In article <25t891$hrp@kruuna.Helsinki.FI> kkettune@kruuna.Helsinki.FI (Kimmo Ke
ttunen) writes:
>Karl Hahn (hahn@newshost.lds.loral.com) wrote:
>> In the past week or so there have been several references to p-adics in
>> posts concerning algebra. This past weekend, I went through all my
>> algebra books and looked for info on p-adics. The closest thing I found
>> was a mention along with the statement that they were beyond the scope
>> of the book.
While several people have posted correct responses to this query, they
seemed to be a bit intimidating for the beginner. I would assume that if
Karl Hahn were ready to swallow and appreciate the definition of a
projective system or a discrete valuation of a field, he would already know
what the p-adics were!
I like the number theoretical approach myself; it's already been hinted at
in several articles, but a bit too tersely for my taste. Here's my version.
A big branch of number theory consists of finding *integer* solutions to
equations (this is what people are talking about when they mention
"Diophantine equations"). Fermat's Last Theorem is a famous example of
a Diophantine equation. Another example is this: find all integer solutions
to x^2 + y^2 = 4z^2 + 3.
How does one go about solving such a problem? Obviously, computer search
might give some information, but this is a rather trivial technique, since
brute force alone cannot find *all* solutions, or prove that there are none.
So what else is there? Well, one of the simplest (but still very powerful)
techniques in Diophantine equation theory is the use of *congruences*. The
idea is this: if the left hand side of an equation is always even (say) and
the right hand side is always odd, then there can't possibly be any solutions.
For example, 2^m = 3^n has no solutions in integers, for this reason.
Going back to our example x^2 + y^2 = 4z^2 + 3, we can use the fact (easily
proved) that perfect squares give a remainder of either zero or one when
divided by 4. That means that x^2 + y^2 must give a remainder of 0, 1 or 2
when divided by 4, and 4z^2 + 3 must give a remainder of 3 when divided by 4.
So x^2 + y^2 = 4z^2 + 3 has no solutions in integers. In modern jargon, the
equation "fails mod 4."
Well, this is a handy technique for proving that certain equations have
*no* solutions, but can it be adapted to *construct* solutions when they
exist? For example, suppose I find that a certain equation does not "fail
mod 3"---i.e., it's possible to find numbers that, when plugged into the
equation and divided by 3, give the same remainders on both sides. (Such
a set of numbers is called a "solution mod 3.") Does this mean that the
original equation is solvable? Clearly not. For example, it could still
fail mod 9. At the very least, for an equation to have a solution, it
must have solutions mod 3, 3^2, 3^3, 3^4, etc.
Next observation: a solution mod 3^2 is also a solution mod 3. More
generally, a solution mod 3^n is a solution mod 3^m if m <= n. So to
keep track of our progress, we really only need to keep track of the
highest power 3^n for which there is a solution mod 3^n. Intuitively,
to keep track of *all* the information about the solvability of an
equation modulo powers of 3, we just need to keep track of the solution
modulo "3^infinity"---the highest possible power of 3. Now here's the
punchline: the concept of a 3-adic number is just a way of making this
concept "3^infinity" precise. Saying that an equation has solutions
in 3-adic numbers (or "is solvable 3-adically") just means that it has
a solution modulo 3^n for *all* n.
More precisely, we can represent a 3-adic number by a nondecreasing
sequence of nonnegative integers, where the nth integer is less than 3^n
and where the nth integer is equal to the (n-1)st integer modulo 3^(n-1)
(i.e., the difference between the nth and (n-1)st integers is divisible
by 3^(n-1)). These sequences of integers can then be manipulated
according to rules similar to those one uses for decimal expansions.
You will find definitions of p-adic numbers that talk about "inverse
limits" and "projective systems." The "inverse limit" refers to the
kind of limiting process involved as I pass from 3^n to 3^infinity as
I outlined above. The "projective" in "projective system" just refers
to the fact that a solution mod 3^n "projects down" to a solution mod
3^m for m < n. Despite the fancy terminology, this is really nothing
more than the naive definition given in the previous paragraph.
Now 3 can be replaced by any integer. It turns out that the most useful
numbers to consider are primes---hence the term "p-adic number." It is
possible to define m-adic numbers for nonprimes m, but they aren't quite
as useful.
[Aside for the sophisticated: one often finds p-adic numbers defined in
terms of "valuations" and "metrics." It is arguably simpler to define
p-adic numbers this way than via projective systems, but the problem I
find is that if one introduces the metric right at the beginning, it
tempts the student to try to form a mental geometric picture of the
p-adics, and since the metric behaves so differently from the Euclidean
metric, the p-adics seem weirder than they really are. Instead I think
it is better to define p-adics in terms of sequences of integers; then
two numbers are "close" if they differ only in their high order terms,
and the valuation is the position of the first nonzero term---both fairly
intuitive concepts.]
Basic facts about p-adic numbers may be found in _Number_Theory_ by Borevich
and Shafarevich and _A_Course_in_Arithmetic_ by Serre. Serre is probably
too terse for the beginner but there's a lot of neat stuff in his book that
is hard to dig out from other texts.