Finite normal extension - Revision history

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
2009-05-10T16:34:05Z

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...'

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{{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>. <math>L</math> is termed a '''finite normal extension''' of <math>K</math> if it is both a [[defining ingredient::finite extension]] and a [[defining ingredient::normal extension]] of <math>K</math>. More explicitly, it satisfies the following equivalent conditions:

# <math>L</math> is a finite extension of <math>K</math>, and if <math>p(x) \in K[x]</math> is an [[irreducible polynomial]] having a root in <math>L</math>, then <math>p</math> splits completely into linear factors over <math>L[x]</math>.
# <math>L</math> is a finite extension of <math>K</math>, and the minimal polynomial of any element of <math>L</math> splits completely over <math>K</math>.
# <math>L</math> is the [[splitting field]] over <math>K</math> of a finite collection of irreducible polynomials.
# <math>L</math> is the splitting field over <math>K</math> of a single polynomial (not necessarily irreducible).
# <math>L</math> is the splitting field over <math>K</math> of a single [[irreducible polynomial]].

==Relation with other properties==

===Stronger properties===

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

===Weaker properties===

* [[Stronger than::Finite extension]]
* [[Stronger than::Normal extension]]