p-adic absolute value

English

Noun

1. Template:Lb A norm for the rational numbers, with some prime number p as parameter, such that any rational number of the form p^k(a/b) — where a, b and k are integers and a, b and p are coprime — is mapped to the rational number p^-k and 0 is mapped to 0. (Note: any nonzero rational number can be reduced to such a form.)

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Usage notes

• A notation for the p-adic absolute value of rational number x is $|x{|}_p$.
• The function is actually from the set of rational numbers to the set of real numbers, because it is used to construct/define a completion of the set of real numbers, namely, the field of p-adic numbers, and this field inherits this p-adic absolute value and extends it to apply to p-adic irrationals, which could well be mapped to real numbers in general (not merely rationals).

Synonyms

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See also

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Further reading

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