with some integers and indivisible by . We define
of the field .
The value group of the -adic valuation consists of all integer-powers of the prime number . The valuation ring of the valuation is called the ring of the p-integral rational numbers; their denominators, when reduced (http://planetmath.org/Fraction) to lowest terms, are not divisible by .
The field of rationals has the 2-adic, 3-adic, 5-adic, 7-adic and so on valuations (which may be called, according to Greek, dyadic, triadic, pentadic, heptadic and so on). They all are non-equivalent (http://planetmath.org/EquivalentValuations) with each other.
If one replaces the number by any positive less than 1, one obtains an equivalent (http://planetmath.org/EquivalentValuations) -adic valuation; among these the valuation with is sometimes called the normed -adic valuation. Analogously we can say that the absolute value is the normed archimedean valuation of which corresponds the infinite prime of .
|Date of creation||2013-03-22 14:55:50|
|Last modified on||2013-03-22 14:55:50|
|Last modified by||pahio (2872)|
|Defines||-integral rational number|
|Defines||normed -adic valuation|
|Defines||normed archimedean valuation|