splitting field
Let f∈F[x] be a polynomial over a field F. A splitting field
for f is a field extension K of F such that
-
1.
f splits (factors into a product of linear factors) in K[x],
-
2.
K is the smallest field with this property (any sub-extension field of K which satisfies the first property is equal to K).
Theorem: Any polynomial over any field has a splitting field, and any two such splitting fields are isomorphic. A splitting field is always a normal extension of the ground field.
Title | splitting field |
---|---|
Canonical name | SplittingField |
Date of creation | 2013-03-22 12:08:01 |
Last modified on | 2013-03-22 12:08:01 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 7 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 12F05 |
Related topic | NormalExtension |