Editing Transcendental extension (section)

== Transcendence basis ==
[[Zorn's lemma]] shows there exists a maximal [[linearly independent]] subset of a vector space (i.e., a basis). A similar argument with Zorn's lemma shows that, given a field extension ''L'' / ''K'', there exists a maximal algebraically independent subset of ''L'' over ''K''.<ref>{{harvnb|Milne|loc=Theorem 9.13.}}</ref> It is then called a '''transcendence basis'''. By maximality, an algebraically independent subset ''S'' of ''L'' over ''K'' is a transcendence basis if and only if ''L'' is an [[algebraic extension]] of ''K''(''S''), the field obtained by [[adjoining (field theory)|adjoining]] the elements of ''S'' to ''K''.

The [[exchange lemma]] (a version for algebraically independent sets<ref>{{harvnb|Milne|loc=Lemma 9.6.}}</ref>) implies that if ''S'' and ''S<nowiki>'</nowiki>'' are transcendence bases, then ''S'' and ''S<nowiki>'</nowiki>'' have the same [[cardinality]]. Then the common cardinality of transcendence bases is called the '''transcendence degree''' of ''L'' over ''K'' and is denoted as <math>\operatorname{tr.deg.}_K L</math> or <math>\operatorname{tr.deg.}(L/K)</math>. There is thus an analogy: a transcendence basis and transcendence degree, on the one hand, and a basis and dimension on the other hand. This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions both form examples of [[Matroid|finitary matroids]] ([[Pregeometry (model theory)|pregeometries]]). Any finitary matroid has a basis, and all bases have the same cardinality.<ref>{{citation|title=Applied Discrete Structures|first=K. D.|last=Joshi|publisher=New Age International|year=1997|isbn=9788122408263|page=909|url=https://books.google.com/books?id=lxIgGGJXacoC&pg=PA909}}.</ref>

If ''G'' is a generating set of ''L'' (i.e., ''L'' = ''K''(''G'')), then a transcendence basis for ''L'' can be taken as a subset of ''G''. Thus, <math>\operatorname{tr.deg.}_K L \le </math> the minimum cardinality of generating sets of ''L'' over ''K''. In particular, a [[finitely generated field extension]] admits a finite transcendence basis.

If no field ''K'' is specified, the transcendence degree of a field ''L'' is its degree relative to some fixed base field; for example, the [[prime field]] of the same [[characteristic (algebra)|characteristic]], or ''K'', if ''L'' is an [[algebraic function field]] over ''K''.

The field extension ''L'' / ''K'' is '''purely transcendental''' if there is a subset ''S'' of ''L'' that is algebraically independent over ''K'' and such that ''L'' = ''K''(''S'').

A '''separating transcendence basis''' of ''L'' / ''K'' is a transcendence basis ''S'' such that ''L'' is a [[separable algebraic extension]] over ''K''(''S''). A field extension ''L'' / ''K'' is said to be '''separably generated''' if it admits a separating transcendence basis.<ref>{{harvnb|Hartshorne|1977|loc=Ch I, § 4, just before Theorem 4.7.A}}</ref> If a field extension is finitely generated and it is also separably generated, then each generating set of the field extension contains a separating transcendence basis.<ref>{{harvnb|Hartshorne|1977|loc=Ch I, Theorem 4.7.A}}</ref> Over a [[perfect field]], every finitely generated field extension is separably generated; i.e., it admits a finite separating transcendence basis.<ref>{{harvnb|Milne|loc=Theorem 9.27.}}</ref>