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# P-adic numbers

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The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers. However, the definition of a Cauchy sequence relies on the metric? chosen, and by choosing a different metric for the Cauchy sequences numbers other than the real numbers can be constructed. Most of the different metrics are equivalent? to the Euclidean metric; those that are not give the p-adic numbers. There are an infinite number of such metrics, and each gives a different set of numbers, one for each prime number; hence the name p-adic numbers.

Once the p-adic numbers have been defined, for some p, then [p-adic analysis]? can be developed based on them. (Just as [real analysis]? can be developed once the real numbers have been defined.)

The real numbers have only a single [algebraic extension]?, the complex numbers; p-adic numbers, for some p, have an infinite number of algebraic extensions.

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Last edited August 16, 2001 8:09 pm (diff)
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