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
Famous quotes containing the words differential and/or criteria:
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (18961948)
“We should have learnt by now that laws and court decisions can only point the way. They can establish criteria of right and wrong. And they can provide a basis for rooting out the evils of bigotry and racism. But they cannot wipe away centuries of oppression and injusticehowever much we might desire it.”
—Hubert H. Humphrey (19111978)