Editing Separable extension (section)

== Separability of transcendental extensions ==

Separability problems may arise when dealing with [[transcendental extension]]s. This is typically the case for [[algebraic geometry]] over a field of prime characteristic, where the [[function field of an algebraic variety]] has a [[transcendence degree]] over the ground field that is equal to the [[dimension of an algebraic variety|dimension]] of the variety.

For defining the separability of a transcendental extension, it is natural to use the fact that every field extension is an algebraic extension of a [[purely transcendental extension]]. This leads to the following definition.

A ''separating transcendence basis'' of an extension <math>E\supseteq F</math> is a [[transcendence basis]]  {{math|''T''}} of {{math|''E''}} such that {{math|''E''}} is a separable algebraic extension of {{math|''F''(''T'')}}.  A [[finitely generated field extension]] is ''separable'' if and only it has a separating transcendence basis; an extension that is not finitely generated is called separable if every finitely generated subextension has a separating transcendence basis.<ref name=FJ38>Fried & Jarden (2008) p.38</ref>

Let <math>E\supseteq F</math> be a field extension of [[characteristic exponent of a field|characteristic exponent]] {{math|''p''}} (that is {{math|1=''p'' = 1}} in characteristic zero and, otherwise, {{math|''p''}} is the characteristic). The following properties are equivalent:
*{{math|''E''}} is a separable extension of {{math|''F''}},
*<math>E^p</math> and {{math|''F''}} are [[linearly disjoint]] over <math>F^p,</math>
*<math>F^{1/p} \otimes_F E</math> is [[reduced ring|reduced]],
*<math>L \otimes_F E</math> is reduced for every field extension {{math|''L''}} of {{math|''E''}},
where <math>\otimes_F</math> denotes the [[tensor product of fields]], <math>F^p</math> is the field of the {{math|''p''}}th powers of the elements of {{math|''F''}} (for any field {{math|''F''}}), and <math>F^{1/p}</math> is the field obtained by [[Adjunction (field theory)|adjoining]] to {{math|''F''}} the {{math|''p''}}th root of all its elements (see [[Separable algebra]] for details).