An element of a field is an integral element of the field , iff
for every non-archimedean valuation of this field.
. The only non-archimedean valuations of are the -adic valuations (where is a rational prime) and the trivial valuation (all values are 1 except the value of 0). The valuation ring of consists of all so-called p-integral rational numbers whose denominators are not divisible by . The valuation ring of the trivial valuation is, generally, the whole field. Thus, is, by definition, the intersection of the ’s for all ; this is the set of rationals whose denominators are not divisible by any prime, which is exactly the set of ordinary integers.
If is the field of the -adic numbers (http://planetmath.org/NonIsomorphicCompletionsOfMathbbQ), it has only one non-trivial valuation, the -adic valuation, and now the ring is its valuation ring, which is the ring of -adic integers (http://planetmath.org/PAdicIntegers); this is visualized in the 2-adic (dyadic) case in the article “-adic canonical form”.
|Date of creation||2013-03-22 14:15:56|
|Last modified on||2013-03-22 14:15:56|
|Last modified by||pahio (2872)|