separable
An irreducible polynomial with coefficients in a field is separable if factors into distinct linear factors over a splitting field of .
A polynomial with coefficients in is separable if each irreducible factor of in is a separable polynomial.
An algebraic field extension is separable if, for each , the minimal polynomial of over is separable. When has characteristic zero, every algebraic extension of is separable; examples of inseparable extensions include the quotient field over the field of rational functions in one variable, where has characteristic .
More generally, an arbitrary field extension is defined to be separable if every finitely generated intermediate field extension has a transcendence basis such that is a separable algebraic extension of .
Title | separable |
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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 |