# Place of a field

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff) 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 .

How to Cite This Entry:
Place of a field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Place_of_a_field&oldid=17112
This article was adapted from an original article by Yu.G. Zarkhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article