Perfect field - Revision history

Vipul at 22:28, 14 May 2009

2009-05-14T22:28:43Z

@@ Line 17: / Line 17: @@
 * [[Weaker than::Algebraically closed field]]
-* [[Weaker than::Field of characteristic zero]]
+* [[Weaker than::Field of characteristic zero]]: {{proofofstrictimplicationat|[[Characteristic zero implies perfect]]|[[Perfect not implies characteristic zero]]}}
-* [[Weaker than::Finite field]]
+* [[Weaker than::Finite field]]: {{proofofstrictimplicationat|[[Finite implies perfect]]|[[Perfect of prime characteristic not implies finite]]}}

Vipul at 22:07, 14 May 2009

2009-05-14T22:07:27Z

← Older revision		Revision as of 22:07, 14 May 2009
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	# Every [[irreducible polynomial]] over the field is a [[defining ingredient::separable polynomial]].		# Every [[irreducible polynomial]] over the field is a [[defining ingredient::separable polynomial]].

			===Equivalence of definitions===

			{{further\|[[Equivalence of definitions of perfect field]]}}
	==Relation with other properties==		==Relation with other properties==

Vipul: Created page with '{{field property}} ==Definition== A field is termed '''perfect''' if it satisfies the following equivalent conditions: # It is either a field of characteristic zero or...'

2009-05-10T16:13:46Z

Created page with '{{field property}} ==Definition== A field is termed '''perfect''' if it satisfies the following equivalent conditions: # It is either a field of characteristic zero or...'

New page

{{field property}}

==Definition==

A [[field]] is termed '''perfect''' if it satisfies the following equivalent conditions:

# It is either a [[field of characteristic zero]] or it has characteristic <math>p</math> for some prime <math>p</math>, and the [[defining ingredient::Frobenius map]] is surjective.
# Every [[finite extension]] of the field is a [[defining ingredient::separable extension]].
# Every [[irreducible polynomial]] over the field is a [[defining ingredient::separable polynomial]].

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Algebraically closed field]]
* [[Weaker than::Field of characteristic zero]]
* [[Weaker than::Finite field]]