p-adic Number

A -adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime are related to proximity in the so called "-adic metric."

Any nonzero rational number can be represented by

(1)

where is a prime number, and are integers not divisible by , and is a unique integer. Then define the p-adic norm of by

(2)

Also define the -adic norm

(3)

The -adics were probably first introduced by Hensel (1897) in a paper which was concerned with the development of algebraic numbers in power series. -adic numbers were then generalized to valuations by Kűrschák in 1913. Hasse (1923) subsequently formulated the Hasse principle, which is one of the chief applications of local field theory. Skolem's -adic method, which is used in attacking certain Diophantine equations, is another powerful application of -adic numbers. Another application is the theorem that the harmonic numbers are never integers (except for ). A similar application is the proof of the von Staudt-Clausen theorem using the -adic valuation, although the technical details are somewhat difficult. Yet another application is provided by the Mahler-Lech theorem.

Every rational has an "essentially" unique -adic expansion ("essentially" since zero terms can always be added at the beginning)

(4)

with an integer, the integers between 0 and inclusive, and where the sum is convergent with respect to -adic valuation. If and , then the expansion is unique. Burger and Struppeck (1996) show that for a prime and a positive integer,

(5)

where the -adic expansion of is

(6)

and

(7)

For sufficiently large ,

(8)

The -adic valuation on gives rise to the -adic metric

(9)

which in turn gives rise to the -adic topology. It can be shown that the rationals, together with the -adic metric, do not form a complete metric space. The completion of this space can therefore be constructed, and the set of -adic numbers is defined to be this completed space.

Just as the real numbers are the completion of the rationals with respect to the usual absolute valuation , the -adic numbers are the completion of with respect to the -adic valuation . The -adic numbers are useful in solving Diophantine equations. For example, the equation can easily be shown to have no solutions in the field of 2-adic numbers (we simply take the valuation of both sides). Because the 2-adic numbers contain the rationals as a subset, we can immediately see that the equation has no solutions in the rationals. So we have an immediate proof of the irrationality of .

This is a common argument that is used in solving these types of equations: in order to show that an equation has no solutions in , we show that it has no solutions in an extension field. For another example, consider . This equation has no solutions in because it has no solutions in the reals , and is a subset of .

Now consider the converse. Suppose we have an equation that does have solutions in and in all the for every prime . Can we conclude that the equation has a solution in ? Unfortunately, in general, the answer is no, but there are classes of equations for which the answer is yes. Such equations are said to satisfy the Hasse principle.