Differential Criteria
The separability can be studied with the aid of derivations and Kähler differentials. Let be a finitely generated field extension of a field . Then
where the equality holds if and only if F is separable over k.
In particular, if is an algebraic extension, then if and only if is separable.
Let be a basis of and . Then is separable algebraic over if and only if the matrix is invertible. In particular, when, above is called the separating transcendence basis.
Read more about this topic: Separable Extension
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