valuation determined by valuation domain
Proof. Let be a valuation domain, its field of fractions and the group of units of . Then is a normal subgroup of the multiplicative group . So we can form the factor group , consisting of all cosets where , and attach to it the additional “coset” getting thus a multiplicative group equipped with zero. If is the maximal ideal of (any valuation domain has a unique maximal ideal — cf. valuation domain is local), then we denote and . Then the subsemigroup of makes an ordered group equipped with zero. It is not hard to check that the mapping
from to is a Krull valuation of the field .
|Title||valuation determined by valuation domain|
|Date of creation||2013-03-22 14:54:58|
|Last modified on||2013-03-22 14:54:58|
|Last modified by||pahio (2872)|