separable
An irreducible polynomial f∈F[x] with coefficients
in a field F is separable if f factors into distinct linear factors over a splitting field
K of f.
A polynomial g with coefficients in F is separable if each irreducible
factor of g in F[x] is a separable polynomial.
An algebraic field extension K/F is separable if, for each a∈K, the minimal polynomial of a over F is separable. When F has characteristic zero, every algebraic extension of F is separable; examples of inseparable extensions include the quotient field K(u)[t]/(tp-u) over the field K(u) of rational functions in one variable, where K has characteristic
p>0.
More generally, an arbitrary field extension K/F is defined to be separable if every finitely generated intermediate field extension L/F has a transcendence basis S⊂L such that L is a separable algebraic extension of F(S).
Title | separable |
---|---|
Canonical name | Separable |
Date of creation | 2013-03-22 12:08:04 |
Last modified on | 2013-03-22 12:08:04 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 13 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 12F10 |
Classification | msc 11R32 |
Related topic | PerfectField |
Defines | separable |
Defines | separable polynomial |
Defines | separable extension |