The absolute Frobenius morphism is a purely inseparable morphism of degree.
2.
Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known.
3.
Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry over fields of prime characteristic.
4.
It is clear that it is enough to show the assertion when " L " / " K " is either separable or purely inseparable.
5.
Introduced a variation of Galois theory for purely inseparable extensions of exponent 1, where the Galois groups of field automorphisms in Galois theory are replaced by restricted Lie algebras of derivations.
6.
The Jacobson Bourbaki theorem implies both the usual Galois correspondence for subfields of a Galois extension, and Jacobson's Galois correspondence for subfields of a purely inseparable extension of exponent at most 1.
7.
On the other hand, an arbitrary algebraic extension E \ supseteq F may not possess an intermediate extension that is " purely inseparable " over and over which is " separable ".
8.
In characteristic " p ", an isogeny of degree " p " of abelian varieties must, for their function fields, give either an Artin Schreier extension or a purely inseparable extension.
9.
The extreme opposite of the concept of separable extension, namely the concept of purely inseparable extension, also occurs quite naturally, as every algebraic extension may be decomposed in a unique way as a purely inseparable extension of separable extension.
10.
The extreme opposite of the concept of separable extension, namely the concept of purely inseparable extension, also occurs quite naturally, as every algebraic extension may be decomposed in a unique way as a purely inseparable extension of separable extension.
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