Editing Algebraic function field (section)

{{short description|Finitely generated extension field of positive transcendence degree}}
{{refimprove|date=December 2021}}
In [[mathematics]], an '''algebraic function field''' (often abbreviated as '''function field''') of ''n'' variables over a [[field (mathematics)|field]] ''k'' is a finitely generated [[field extension]] ''K''/''k'' which has [[transcendence degree]] ''n'' over ''k''.<ref>{{cite book |author=Gabriel Daniel |author2=Villa Salvador |name-list-style=amp|title=Topics in the Theory of Algebraic Function Fields|publisher=Springer |year= 2007|isbn=9780817645151|url=https://books.google.com/books?id=RmKpEUltmQIC}}</ref> Equivalently, an algebraic function field of ''n'' variables over ''k'' may be defined as a [[finite field extension]] of the field ''K'' = ''k''(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) of [[rational functions]] in ''n'' variables over ''k''.