§ A motivation for p-adic analysis
I've seen the definitions of p-adic numbers scattered around on the internet,
but this analogy as motivated by the book
p-adic numbers by Fernando Gouvea
really made me understand why one would study the p-adics, and why the
definitions are natural. So I'm going to recapitulate the material, with the
aim of having somoene who reads this post be left with a sense of why it's
profitable to study the p-adics, and what sorts of analogies are fruitful when
thinking about them.
We wish to draw an analogy between the ring , where
are the prime ideals, and where are the prime ideals. We wish
to take all operations one can perform with polynomials, such as generating
functions ( ),
taylor expansions (expanding aronund ),
and see what their analogous objects will look like in
relative to a prime .
§ Perspective: Taylor series as writing in base :
Now, for example, given a prime , we can write any positive integer
in base , as where .
For example, consider . The expansion of 72 is
This shows us that 72 is divisible by .
This perspective to take is that this us the information local to prime ,
about what order the number is divisible by ,
just as the taylor expansion tells us around of a polynomial
tells us to what order vanishes at a point .
§ Perspective: rational numbers and rational functions as infinite series:
Now, we investigate the behaviour of expressions such as
We know that the above formula is correct formally from the theory of
generating functions. Hence, we take inspiration to define values for
Let's take , and we know that .
We now calculate as:
However, we don't really know how to interpret , since we assumed
the coefficients are always non-negative. What we can do is to rewrite ,
and then use this to make the coefficient positive. Performing this transformation
for every negative coefficient, we arrive at:
We can verify that this is indeed correct, by multiplying with
and checking that the result is :
What winds up happening is that all the numbers after end up being cleared
due to the carrying of .
This little calculation indicates that we can also define take the -adic
expansion of rational numbers.
§ Perspective: -1 as a p-adic number
We next want to find a p-adic expansion of -1, since we can then expand
out theory to work out "in general". The core idea is to "borrow" , so
that we can write -1 as , and then we fix , just like we fixed
. This eventually leads us to an infinite series expansion for . Written
down formally, the calculation proceeds as:
This now gives us access to negative numbers, since we can formally multiply
the series of two numbers, to write .
Notice that this definition of also curiously matches the 2s complement
definition, where we have . In this case, the expansion is
infinite, while in the 2s complement case, it is finite. I would be very
interested to explore this connection more fully.
§ What have we achieved so far?
We've now managed to completely reinterpret all the numbers we care about in
the rationals as power series in base . This is pretty neat. We're next
going to try to complete this, just as we complete the rationals to get
the reals. We're going to show that we get a different number system on
completion, called .
To perform this, we first look at how the -adic numbers help us solve
congruences mod p, and how this gives rise to completions to equations such
as , which in the reals give us , and in
give us a different answer!
Let's start by solving an equation we already know how to solve:
We already know the solutions to in are
Explicitly, the solutions are:
This was somewhat predictable. We move to a slightly more interesting case.
- At this point, the answer remains constant.
The solution sets are:
This gives us the infinite 3-adic expansion:
Note that we can't really predict the digits in the 3-adic sequence of -5,
but we can keep expanding and finding more digits.
Also see that the solutions are "coherent". In that, if we look at the
solution mod 9, which is , and then consider it mod 3, we get . So,
we can say that given a sequence of integers ,
is p-adically coherent sequence iff:
§ Viewpoint: Solution sets of
Since our solution sets are coherent, we can view the solutions as a tree,
with the expansions of and then continuing onwards
from there. That is, the sequences are
We now construct a solution to the equation in the 7-adic system,
thereby showing that is indeed strictly larger than ,
since this equation does not have rational roots.
For , we have the solutions as , .
To find solutions for , we recall that we need our solutions to be consistent
with those for . So, we solve for:
Solving the first of these:
- , .
This gives the solution . The other branch ()
gives us .
We can continue this process indefinitely (exercise), giving us the sequences:
We can show that the sequences of solutions we get satisfy the equation
. This is so by construction. Hence, contains
a solution that does not, and is therefore strictly bigger, since
we can already represent every rational in in .
§ Use case: Solving as a recurrence
Let's use the tools we have built so far to solve the equation .
Instead of solving it using algebra, we look at it as a recurrence .
This gives us the terms:
In , this is a divergent sequence. However, we know that the
solution so , at least as a generating function.
Plugging this in, we get that the answer should be:
which is indeed the correct answer.
Now this required some really shady stuff in . However, with a change
of viewpoint, we can explain what's going on. We can look at the above series
as being a series in . Now, this series does really converge,
and by the same argument as above, it converges to .
The nice thing about this is that a dubious computation becomes a legal one
by changing one's perspective on where the above series lives.
§ Viewpoint: 'Evaluation' for p-adics
The last thing that we need to import from the theory of polynomials
is the ability to evaluate them: Given a rational function ,
where are polynomials, we can
evaluate it at some arbitrary point , as long as is not a zero
of the polynomial .
We would like a similar function, such that for a fixed prime , we obtain
a ring homomorphism from , which we will
denote as , where we are imagining that we are "evaluating" the prime
against the rational .
We define the value of at the prime to be equal to
, where . That is, we compute the
usual to evaluate , except we do this , to stay with
Note that if , then we cannot evaluate
the rational , and we say that has a pole at . The order
of the pole is the number of times occurs in the prime factorization of .
I'm not sure how profitable this viewpoint is, so I
asked on math.se,
and I'll update this post when I recieve a good answer.
§ Perspective: Forcing the formal sum to converge by imposing a new norm:
So far, we have dealt with infinite series in base , which have terms
Clearly, these sums are divergent as per the usual topology on .
However, we would enjoy assigning analytic meaning to these series. Hence, we
wish to consider a new notion of the absolute value of a number, which makes it
such that with large are considered small.
We define the absolute value for a field as a function
. It obeys the axioms:
We want the triangle inequality so it's metric-like, and the norm to be
multiplicative so it measures the size of elements.
The usual absolute value satisfies
Now, we create a new absolute value that measures primeness. We first introduce
a gadget known as a valuation, which measures the -ness of a number. We use
this to create a norm that makes number smaller as their -ness increases.
This will allow infinite series in to converge.
- for all
- , for all .
§ p-adic valuation: Definition
First, we introduce
a valuation , where is
the power of the prime in the prime factorization of . More formally,
is the unique number such that:
The valuation gets larger as we have larger powers of in the prime
factorization of a number. However, we want the norm to get smaller. Also,
we need the norm to be multiplicative, while , which
To fix both of these, we create a norm by exponentiating .
This converts the additive property into a multiplicative property. We
exponentiate with a negative sign so that higher values of lead to
smaller values of the norm.
- , where .
- We extend the valuation to the rationals by defining .
- We set . The intuition is that can be divided by an infinite number of times.
§ p-adic abosolute value: Definition
Now, we define the p-adic absolute value of a number as
So is indeed a norm, which measures -ness, and is smaller as
gets larger in the power of the factorization of , causing our
infinite series to converge.
There is a question of why we chose a base for . It would
appear that any choice of would be legal.
I asked this on
- the norm of is .
- If , then , and hence .
- The norm is multiplicative since is additive.
- Since .Hence, the triangle inequality is also satisfied.
and the answer is that this choosing a base gives us the nice formula
That is, the product of all norms and the usual norm
(denoted by )
give us the number 1. The reason is that the give us
while the usual norm contains a multiple
, thereby cancelling each other out.
What we've done in this whirlwind tour is to try and draw analogies between
the ring of polynomials and the ring , by trying
to draw analogies between their prime ideals: and . So,
we imported the notions of generating functions, polynomial evaluation, and
completions (of ) to gain a picture of what is like.
We also tried out the theory we've built against some toy problems, that shows
us that this point of view maybe profitable. If you found this interesting,
I highly recommend the book
p-adic numbers by Fernando Gouvea.