Place of a field
with values in a field
,
-valued place of a field
A mapping satisfying the conditions
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(provided that the expressions on the right-hand sides are defined). The following conventions are made:
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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
.
References
[1] | S. Lang, "Algebra" , Addison-Wesley (1984) |
Comments
References
[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