Maths Society Speaker - Dr Honnor: The p-adic Numbers: Another Light to Shine on the Rational Numbers
We were very lucky at Maths society to have Dr Honnor talk about The p-adic Numbers. He has summarised the talk below:
"The 
-adic numbers, first
defined by Kurt Hensel in 1897 are a cornerstone of modern algebraic number
theory. The in question here will be a prime number and in
the maths society talk we looked at a way of defining these fantastic numbers.
We begin by considering the standard decimal expansions of the real numbers.
It is worth noting that even at this early stage there is
something a little remarkable happening. On the left hand side of our
calculation we have 
which is clearly finite where as on the right
hand side we have a summation involving an infinite number of terms. It is the
fact that 
gets small very quickly, as 
tends to infinity, that makes this infinite
sum possible. A consequence of this is that we can write any real number in the
following way:
Here 
is an integer greater than or equal to 
and the coefficients 
and 
are 
or 
. Rather than writing a
real number in base 
, as we have done above
we now consider writing positive integers in terms of powers of 
. For example:
It is possible to write any positive integer in this way. As one further
example we have.
We now consider an infinite sum which doesn’t converge.
Of course, writing the ‘
’ at the start of this
is absurd, as the summation on the right hand side doesn’t converge. For now
though, let’s pretend that this nonsense makes sense and consider what happens
when we add 
to the above.
Continuing this process we can see that we get an infinite
numbers of ’s in the summation and
hence deduce:
The remainder of the talk was spent trying to give some
justification to this nonsensical calculation. We begin by returning to our
example with . The reason this
summation was valid was due to the fact that the terms in the sum become very
small. The way we are used to defining the size of a number is with the
absolute value. This gives the distance of the number from .
For any rational number 
we define the absolute value as follows.
In mathematics we like to generalise constructions to distil the
key ideas. In this case, the question is: What properties would we like to have
for a function that measures the size of a number? After thinking for a while
hopefully we can agree that the following would be a good list of properties to
have.
It is straightforward to see that the absolute value satisfies these properties. The third property is referred to as the triangle inequality due to the following diagram:
-adic norm, though this
could be done for any prime number , we work with to give ourselves a concrete example.
The point here is that for any integer 
, we can write is as 
where 
is coprime to . The -adic norm of is then defined in the following way: 
. We then take the -adic norm of a
fraction as shown above. With this norm, the numbers that are highly divisible
by 5 are ‘small’ and a rational number with a large power of in the denominator is ‘large’.
Using the -adic norm, the sum
from the start now makes sense since the large powers of become small very quickly in the -adic norm.
Furthermore, since 
, does indeed represent 
in the -adic numbers!
In our definition before, the number can be replaced with any other prime to give the -adic norm. It is
straightforward to check that the -adic norm satisfies
the properties we want to have for a distance measure. In fact, instead of the
triangle inequality, an even stronger result holds, which is given below.
This stronger property means that the -adic numbers are much
nicer to work with then the rational numbers, though we won’t go into detail
about this. We should also ask if there are any more ways of measuring the size
of a number other than the absolute value and the -adic norm. It was
proved by Ostrowski in 1916, that indeed these are all the ways of measuring
the size of a rational number.
We are now ready to define the -adic numbers and do
this in a similar way to the description of the real numbers we gave at the
start. Note that the signs of the powers are swapped as it is now the large
powers of that are small. The -adic numbers are then
all the numbers which can be written as
Here, is an integer greater than or equal to and the coefficients and are 
. We write 
for the collection of all -adic numbers.
-adic numbers give us
another way to study the rational numbers since the rational numbers are
contained in for every prime. But each one is different
from each other and from the real numbers so can provide different information
for each prime. As an example of some of the differences, 
is in 
but not in for any prime . Also, the imaginary
number 
isn’t in or 
but it is in 
!
One use for -adic numbers, is in
trying to solve Diophantine equations, these are multi-variable polynomial
equations with integer coefficients where we only want to find integer
solutions. Equations of this kind are famously difficult to solve, for example
Fermat’s Last Theorem which concerns a specific collection of Diophantine
equations took over 350 years to prove!
If an equation has an integer solution
then is must be possible to solve it in every -adic field as well as
the real numbers. While the reverse implication is not true, it is true that if
an equation cannot be solved in the -adic numbers for some
prime then it is not possible to solve it in the
rational numbers and hence also the integers. For example, it is quick to find two
solutions to the equation
These are 
and 
. Proving these are the
only two solutions is more difficult. One way to study this problem is to look
for solutions in the 
-adic numbers where the
structure of these numbers show these are the only two possible solutions."