example of nonperfect field

In this entry, we exhibit an example of a field that is not a perfect fieldMathworldPlanetmath.

Let F=𝔽p(t), where 𝔽p is the field with p elements and t transcendentalPlanetmathPlanetmath over 𝔽p. The splitting fieldMathworldPlanetmath E of the irreducible polynomialMathworldPlanetmath f=xp-t is not separablePlanetmathPlanetmath over F. Indeed, if α is an element of E such that αp=t, we have


which shows that f has one root of multiplicity p.

Title example of nonperfect field
Canonical name ExampleOfNonperfectField
Date of creation 2013-03-22 13:08:31
Last modified on 2013-03-22 13:08:31
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Example
Classification msc 12F10