A -adic number is an extension of the Field of Rational Numbers 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 -adic absolute value of by
 (2)

 (3)

As an example, consider the Fraction

 (4)

It has -adic absolute values given by
 (5) (6) (7) (8) (9)

The -adic absolute value satisfies the relations

1. for all ,

2. Iff ,

3. for all and ,

4. for all and (the Triangle Inequality), and

5. for all and (the Strong Triangle Inequality).

In the above, relation 4 follows trivially from relation 5, but relations 4 and 5 are relevant in the more general Valuation Theory.

The -adics were probably first introduced by Hensel in 1902 in a paper which was concerned with the development of algebraic numbers in Power Series. -adic numbers were then generalized to Valuations by Kürschák in 1913. In the early 1920s, Hasse formulated the Local-Global Principle (now usually called 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)

 (10)

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,
 (11)

where the -adic expansion of is
 (12)

and
 (13)

For sufficiently large ,
 (14)

 (15)

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 a Field Extension. 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 . 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.

See also Ax-Kochen Isomorphism Theorem, Diophantine Equation, Harmonic Number, Hasse Principle, Local Field, Local-Global Principle, Mahler-Lech Theorem, Product Formula, Valuation, Valuation Theory, von Staudt-Clausen Theorem

References

Burger, E. B. and Struppeck, T. Does Really Converge? Infinite Series and -adic Analysis.'' Amer. Math. Monthly 103, 565-577, 1996.

Cassels, J. W. S. and Scott, J. W. Local Fields. Cambridge, England: Cambridge University Press, 1986.

Gouvêa, F. Q. -adic Numbers: An Introduction, 2nd ed. New York: Springer-Verlag, 1997.

Koblitz, N. -adic Numbers, -adic Analysis, and Zeta-Functions, 2nd ed. New York: Springer-Verlag, 1984.

Mahler, K. -adic Numbers and Their Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1981.