Place of a field
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 .
|||S. Lang, "Algebra" , Addison-Wesley (1984)|
|[a1]||P.M. Cohn, "Algebra" , 2 , Wiley (1977) pp. Chapt. 9|
Place of a field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Place_of_a_field&oldid=17112