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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.