Editing Algebraic function field (section)

==Valuations and places==
Key tools to study algebraic function fields are [[absolute value (algebra)|absolute values, valuations, places]] and their completions.

Given an algebraic function field ''K''/''k'' of one variable, we define the notion of a ''valuation ring'' of ''K''/''k'': this is a [[subring]] ''O'' of ''K'' that contains ''k'' and is different from ''k'' and ''K'', and such that for any ''x'' in ''K'' we have ''x''&thinsp;&isin;&thinsp;''O'' or ''x''<sup>&nbsp;-1</sup>&thinsp;&isin;&thinsp;''O''. Each such valuation ring is a [[discrete valuation ring]] and its maximal ideal is called a ''place'' of ''K''/''k''.

A ''discrete valuation'' of ''K''/''k'' is a [[surjective]] function ''v'' : ''K'' → '''Z'''∪{&infin;} such that ''v''(x)&thinsp;=&thinsp;&infin; iff ''x''&thinsp;=&thinsp;0, ''v''(''xy'') = ''v''(''x'')&thinsp;+&thinsp;''v''(''y'') and ''v''(''x''&thinsp;+&thinsp;''y'') ≥ min(''v''(''x''),''v''(''y'')) for all ''x'',&thinsp;''y''&thinsp;&isin;&thinsp;''K'', and ''v''(''a'')&thinsp;=&thinsp;0 for all ''a''&thinsp;&isin;&thinsp;''k''&thinsp;\&thinsp;{0}.

There are natural bijective correspondences between the set of valuation rings of ''K''/''k'', the set of places of ''K''/''k'', and the set of discrete valuations of ''K''/''k''. These sets can be given a natural [[Topology|topological]] structure: the [[Zariski–Riemann space]] of ''K''/''k''.