Editing Field extension (section)

==Transcendental extension==
{{main|Transcendental extension}}
Given a field extension <math>L/K</math>, a subset ''S'' of ''L'' is called [[algebraically independent]] over ''K'' if no non-trivial polynomial relation with coefficients in ''K'' exists among the elements of ''S''. The largest cardinality of an algebraically independent set is called the [[transcendence degree]] of ''L''/''K''. It is always possible to find a set ''S'', algebraically independent over ''K'', such that ''L''/''K''(''S'') is algebraic. Such a set ''S'' is called a [[transcendence basis]] of ''L''/''K''. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension <math>L/K</math> is said to be '''{{visible anchor|purely transcendental}}''' if and only if there exists a transcendence basis ''S'' of <math>L/K</math> such that ''L'' = ''K''(''S''). Such an extension has the property that all elements of ''L'' except those of ''K'' are transcendental over ''K'', but, however, there are extensions with this property which are not purely transcendental—a class of such extensions take the form ''L''/''K'' where both ''L'' and ''K'' are algebraically closed. 

If ''L''/''K'' is purely transcendental and ''S'' is a transcendence basis of the extension, it doesn't necessarily follow that ''L'' = ''K''(''S''). On the opposite, even when one knows a transcendence basis, it may be difficult to decide whether the extension is purely separable, and if it is so, it may be difficult to find a transcendence basis ''S'' such that ''L'' = ''K''(''S'').

For example, consider the extension <math>\Q(x, y)/\Q,</math> where <math>x</math> is transcendental over <math>\Q,</math> and <math>y</math> is a [[polynomial root|root]] of the equation <math>y^2-x^3=0.</math> Such an extension can be defined as <math>\Q(X)[Y]/\langle Y^2-X^3\rangle,</math> in which <math>x</math> and <math>y</math> are the [[equivalence class]]es of <math>X</math> and <math>Y.</math> Obviously, the singleton set <math>\{x\}</math> is transcendental over <math>\Q</math> and the extension <math>\Q(x, y)/\Q(x)</math> is algebraic; hence <math>\{x\}</math> is a transcendence basis that does not generates the extension <math>\Q(x, y)/\Q(x)</math>. Similarly, <math>\{y\}</math> is a transcendence basis that does not generates the whole extension. However the extension is purely transcendental since, if one set <math>t=y/x,</math> one has <math>x=t^2</math> and <math>y=t^3,</math> and thus <math>t</math> generates the whole extension. 

Purely transcendental extensions of an algebraically closed field occur as [[function field of an algebraic variety|function fields]] of [[rational varieties]]. The problem of finding a [[rational parametrization]] of a rational variety is equivalent with the problem of finding a transcendence basis that generates the whole extension.