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Algebraic closure (view source)

Revision as of 15:11, August 25, 2020

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<li> The only irreducible polynomials in F[x] have degree 1.
 
<li> The only irreducible polynomials in F[x] have degree 1.
 
<li> F has no finite [[Field extension|extensions]] other than itself.
 
<li> F has no finite [[Field extension|extensions]] other than itself.
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<li> F is a maximal field of its [[transcendence degree]].
+
<li> F is a maximal field of its transcendence degree.
−
<li> F is the union of all its [[finitely-generated]] [[Localization|localized]] [[Ring (mathematics)|subrings]].
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<li> F is the union of all its finitely-generated [[Localization|localized]] [[Ring (mathematics)|subrings]].
 
<li> Every [[algebraic variety]] defined over F has a [[point]].
 
<li> Every [[algebraic variety]] defined over F has a [[point]].
 
<li> F is not finite and the [[automorphism]] group of F has [[cardinality]] strictly greater than the cardinality of F.
 
<li> F is not finite and the [[automorphism]] group of F has [[cardinality]] strictly greater than the cardinality of F.
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