ring of S-integers


Let K be a number fieldMathworldPlanetmath and let S be a finite set of absolute valuesMathworldPlanetmathPlanetmathPlanetmath of K, containing all archimedean valuations. The ring of S-integers of K, usually denoted by RS, is the ring:

RS={kK:ν(k)0 for all valuations νS}.

Notice that, for any set S as above, the ring of integersMathworldPlanetmath of K, 𝒪K, is always contained in RS.


Let K= and let S={νp,||} where p is a prime and νp is the usual p-adic valuation, and || is the usual absolute value. Then


, i.e. RS is the result of adjoining (as a new ring element) 1/p to (i.e. we allow to invert p).

Title ring of S-integers
Canonical name RingOfSintegers
Date of creation 2013-03-22 15:57:27
Last modified on 2013-03-22 15:57:27
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Definition
Classification msc 13B22
Synonym ring of S-integers