purely inseparable

Let F be a field of characteristicPlanetmathPlanetmathPlanetmath p>0 and let α be an elementMathworldMathworld which is algebraic over F. Then α is said to be purely inseparable over F if αpnF for some n0.

An algebraic field extension K/F is purely inseparable if each element of K is purely inseparable over F.

Purely inseparable extensions have the following property: if K/F is purely inseparable, and A is an algebraic closureMathworldPlanetmath of F which contains K, then any homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath KA which fixes F necessarily fixes K.

Let K/F be an arbitrary algebraic extension. Then there is an intermediate field E such that K/E is purely inseparable, and E/F is separablePlanetmathPlanetmath.


Let s be an indeterminate, and let K=𝔽3(s) where 𝔽3 is the finite fieldMathworldPlanetmath with 3 elements. Let F=𝔽3(s6). Then K/F is neither separable, nor purely inseparable. Let E=𝔽3(s3). Then E/F is separable, and K/E is purely inseparable.

Title purely inseparable
Canonical name PurelyInseparable
Date of creation 2013-03-22 14:49:08
Last modified on 2013-03-22 14:49:08
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Definition
Classification msc 12F15