Normal extension: Difference between revisions
(Created page with '{{field extension property}} ==Definition== Suppose <math>L</math> is a field extension of a field <math>K</math>. In other words, <math>K</math> is a subfield of <math>L</...') |
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* [[Stronger than::Extension for which every intermediate isomorphism extends to an automorphism]] | * [[Stronger than::Extension for which every intermediate isomorphism extends to an automorphism]] | ||
* [[Stronger than::Extension for which every intermediate automorphism extends to an automorphism]] | * [[Stronger than::Extension for which every intermediate automorphism extends to an automorphism]] | ||
* [[Stronger than::Subnormal extension]] |
Latest revision as of 16:28, 10 May 2009
This article defines a field extension property: a property that can be evaluated to true/false for any field extension.
View a complete list of field extension properties|View a complete list of field properties
Definition
Suppose is a field extension of a field . In other words, is a subfield of . We say that is a normal extension of if it satisfies the following equivalent conditions:
- is an algebraic extension of , with the property that for any , the minimal polynomial of over splits completely over (i.e., it can be written as a product of linear factors in ).
- is an algebraic extension of , with the property that if is an irreducible polynomial having a root in , then splits completely into linear factors in .
- is the splitting field over of a (possibly infinite) collection of polynomials in . (When the collection of polynomials is finite, then we get the stronger notion of finite normal extension).