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) |

(2) |

(3) |

As an example, consider the Fraction

(4) |

(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).

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) |

(11) |

(12) |

(13) |

(14) |

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

(15) |

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.

**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.

© 1996-9

1999-05-26