Vipul: /* Weaker properties */

2009-05-10T16:28:45Z

Weaker properties

← Older revision		Revision as of 16:28, 10 May 2009
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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]]

Vipul: Created page with '{{field extension property}} ==Definition== Suppose $L$ is a field extension of a field $K$ . In other words, $K$ is a subfield of $L$
2009-05-10T16:28:27Z

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</...'

New page
{{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</math>. We say that <math>L</math> is a '''normal extension''' of <math>K</math> if it satisfies the following equivalent conditions:

# <math>L</math> is an [[algebraic extension]] of <math>K</math>, with the property that for any <math>\alpha \in L</math>, the minimal polynomial of <math>\alpha</math> over <math>K</math> splits completely over <math>L</math> (i.e., it can be written as a product of linear factors in <math>L[x]</math>).
# <math>L</math> is an [[algebraic extension]] of <math>K</math>, with the property that if <math>p(x) \in K[x]</math> is an [[irreducible polynomial]] having a root in <math>L</math>, then <math>p(x)</math> splits completely into linear factors in <math>L[x]</math>.
# <math>L</math> is the splitting field over <math>K</math> of a (possibly infinite) collection of polynomials in <math>K[x]</math>. (When the collection of polynomials is finite, then we get the stronger notion of [[finite normal extension]]).

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Galois extension]]
* [[Weaker than::Finite normal extension]]

===Weaker properties===

* [[Stronger than::Algebraic extension]]
* [[Stronger than::Extension for which every intermediate isomorphism extends to an automorphism]]
* [[Stronger than::Extension for which every intermediate automorphism extends to an automorphism]]