A generalization of the *p*-adic Number first proposed by Kürschák in 1913. A valuation
on a Field is a Function from to the Real Numbers such that the
following properties hold for all :

- 1. ,
- 2. Iff ,
- 3. ,
- 4. Implies for some constant (independent of ).

If (4) is satisfied for , then satisfies the Triangle Inequality,

- 4a. for all .

- 4b. .

The simplest valuation is the Absolute Value for Real Numbers. A valuation satisfying (4b) is called non-Archimedean Valuation; otherwise, it is called Archimedean.

If is a valuation on and
, then we can define a new valuation by

(1) |

If two valuations are equivalent, then they are both non-Archimedean
or both Archimedean. , , and with the usual Euclidean norms
are Archimedean valuated fields. For any Prime , the *p*-adic Number with
the -adic valuation is a non-Archimedean valuated field.

If is any Field, we can define the trivial valuation on by for all and , which is a non-Archimedean valuation. If is a Finite Field, then the only possible valuation over is the trivial one. It can be shown that any valuation on is equivalent to one of the following: the trivial valuation, Euclidean absolute norm , or -adic valuation .

The equivalence of any nontrivial valuation of
to either the usual Absolute Value or to a *p*-adic Number absolute value was proved by Ostrowski
(1935). Equivalent valuations give rise to the same topology. Conversely, if two valuations have the same topology, then
they are equivalent. A stronger result is the following: Let , , ..., be valuations
over which are pairwise inequivalent and let , , ..., be elements of . Then there exists an
infinite sequence (, , ...) of elements of such that

(2) |

(3) |

(4) |

A discrete valuation is a valuation for which the Valuation Group is a discrete subset of the Real
Numbers . Equivalently, a valuation (on a Field ) is discrete if there exists a Real
Number such that

(5) |

If is a valuation on , then it induces a metric

(6) |

**References**

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

Ostrowski, A. ``Untersuchungen zur aritmetischen Theorie der Körper.'' *Math. Zeit.* **39**, 269-404, 1935.

© 1996-9

1999-05-26