Editing Transcendental extension (section)

{{Short description|Field extension that is not algebraic}}
{{Use American English|date = January 2019}}

In [[mathematics]], a '''transcendental extension''' <math>L/K</math> is a [[field extension]] such that there exists an element in the field <math>L</math> that is [[transcendental element|transcendental]] over the field <math>K</math>; that is, an element that is not a root of any [[univariate polynomial]] with coefficients in <math>K</math>. In other words, a transcendental extension is a field extension that is not [[Algebraic extension|algebraic]]. For example, <math>\mathbb{C}</math> and <math>\mathbb{R}</math> are both transcendental extensions of <math>\mathbb{Q}.</math>

A '''transcendence basis''' of a field extension <math>L/K</math> (or a transcendence basis of <math>L</math> over <math>K</math>) is a maximal [[algebraically independent]] [[subset]] of <math>L</math> over <math>K.</math> Transcendence bases share many properties with [[basis (linear algebra)|bases]] of [[vector space]]s. In particular, all transcendence bases of a field extension have the same [[cardinality]], called the '''transcendence degree''' of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero.

Transcendental extensions are widely used in [[algebraic geometry]]. For example, the [[dimension of an algebraic variety|dimension]] of an [[algebraic variety]] is the transcendence degree of its [[function field of an algebraic variety|function field]]. Also, [[global function field]]s are transcendental extensions of degree one of a [[finite field]], and play in [[number theory]] in [[positive characteristic]] a role that is very similar to the role of [[algebraic number field]]s in characteristic zero.