A function field over a field 
is a finitely generated field extension 
. Its transcendence degree
is the dimension of the associated algebraic
variety. Function fields are also called algebraic function fields.
For example, a one-variable function field over 
is a finite extension 
of the field 
of rational
functions, where 
is transcendental and 
is a root of a nonzero polynomial 
with 
. A one-variable function field over 
is a finite extension
of 
,
whereas higher-dimensional function fields allow several algebraically
independent variables. Birational reconstruction questions over algebraically
closed constant fields use transcendence
degree at least 2 in the announced Bogomolov-Pop and Milnor
K-theory results.
