Place of a field

From Encyclopedia of Mathematics
Revision as of 17:20, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

with values in a field , -valued place of a field

A mapping satisfying the conditions

(provided that the expressions on the right-hand sides are defined). The following conventions are made:

while the expressions , and are undefined.

An element in for which is called finite in the place ; the set of finite elements is a subring of , and the mapping is a ring homomorphism. The ring is a local ring, its maximal ideal is .

A place determines a valuation of with group of values (where and are, respectively, the groups of invertible elements of and ). The ring of this valuation is the same as . Conversely, any valuation of a field determines a place of with values in the residue class field of . Here, the ring of finite elements is the same as the ring of (integers of) the valuation .


[1] S. Lang, "Algebra" , Addison-Wesley (1984)



[a1] P.M. Cohn, "Algebra" , 2 , Wiley (1977) pp. Chapt. 9
How to Cite This Entry:
Place of a field. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Yu.G. Zarkhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article