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7/28/2019 Gallian Ch 21
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a is algebraic overF
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Eis an extension of a field F, a E .
a is the zero of some nonzero polynomial in [ ]F x .
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a is transcendental overF
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a is not algebraic overF.
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algebraic extension of a field F
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An extension Eof a field Ffor which every element ofEis algebraic overF.
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Transcendental extension of a field F
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An extension Eof a field Fthat is not algebraic.
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Simple extension of the field F
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An extension of a field Fof the form ( )F a .
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Characterization of Extensions
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Let Ebe an extension of the field F, and let a E . Ifais transcendental overF, then ( ) ( )F a F x . Ifa is
algebraic overF, then ( ) [ ]/ ( )F a F x p x , where
( )p x is a polynomial in [ ]F x of minimum degree such
that ( ) 0p a . Moreover, ( )p x is irreducible overF.
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Uniqueness Property
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Ifa is algebraic over a field F, then there is a unique
monic irreducible polynomial ( )p x in [ ]F x such that
( ) 0p a .
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Divisibility Property
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Let a be algebraic overF, and let ( )p x be the minimal
polynomial fora overF. If ( ) [ ]f x F x and ( ) 0f a ,
then ( )p x divides ( )f x in [ ]F x .
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Ehas degree n overF
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Eis an extension over a field F.
Ehas dimension n as a vector space overF.Notation: [ : ]E F n .
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Finite extension ofF
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An extension Eof a field Ffor which [ : ]E F is finite.
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Infinite extension ofF
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An extension Eof a field Ffor which [ : ]E F is infinite.
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Finite Implies Algebraic
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IfEis a finite extension ofF, then Eis an algebraicextension ofF.
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[ : ] [ : ][ : ]K F K E E F
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Let Kbe a finite extension field of the field Eand let Ebe a finite extension of the field F. Then Kis a finite
extension field ofFand [ : ] [ : ][ : ]K F K E E F .
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Primitive Element Theorem
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IfFis a field of characteristic 0, and a and b are
algebraic overF, then there is an element cin ( , )F a b
such that ( , ) ( )F a b F c .
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Primitive element ofE
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Eis an extension of a field F.
An element a with the property that ( )E F a .
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Algebraic Over Algebraic Is Algebraic
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IfKis an algebraic extension ofEand Eis an algebraicextension ofF, then Kis an algebraic extension ofF.
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Subfield of Algebraic Elements
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Let Ebe an extension field of the field F. Then the setof all elements ofEthat are algebraic overFis a
subfield ofE.
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Algebraic closure ofFin E
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Eis an extension of the field F.
The subfield ofEof the elements that are algebraicoverF.
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Algebraically closed field
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A field that has no proper algebraic extension.
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Algebraic Closure ofF
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The unique (up to isomorphism) algebraic extension ofa field Fthat is algebraically closed.