place of field
where is a field, is called a place of the field , if it satisfies the following conditions.
If , then .
It is easy to see that the subring of the field is a valuation domain; so any place of a field determines a unique valuation domain in the field. Conversely, every valuation domain with field of fractions determines a place of :
Let be a valuation domain with field of fractions and the maximal ideal of , consisting of the non-units of . Then the mapping
is a place of the field .
Proof. Apparently, and the restriction is the canonical homomorphism from the ring onto the residue-class ring . Moreover, if , then does not belong to the valuation domain and thus the inverse element must belong to it without being its unit. Hence belongs to the ideal which is the kernel of the homomorphism . So we see that .
- 1 Emil Artin: . Lecture notes. Mathematisches Institut, Göttingen (1959).
|Title||place of field|
|Date of creation||2013-03-22 14:56:51|
|Last modified on||2013-03-22 14:56:51|
|Last modified by||pahio (2872)|
|Synonym||spot of field|
|Defines||place of field|