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Extension field

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F is a field.

A field Efor which F E and for which the operationsofFare those ofErestricted to F.

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Fundamental Theorem of Field Theory(Kroneckers Theorem)

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Let Fbe a field and ( )f x a nonconstant polynomial in

[ ]F x . Then there is an extension field EofF in which

( )f x has a zero.

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( )f x splits in E

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E is an extension field ofF.

( )f x can be factored as a product of linear factors in[ ]E x .

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Splitting field for ( )f x overF

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F is a field.

An extension field EofF in which ( )f x splits, but forwhich ( )f x does not split in any proper subfield ofE.

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Existence of Splitting Fields

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Let Fbe a field and let ( )f x be a nonconstant element

of [ ]F x . Then there exists a splitting field E for ( )f x

overF.

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( ) [ ]/ ( )F a F x p x

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Let Fbe a field and ( ) [ ]p x F x be irreducible overF.

Ifa is a zero of ( )p x in some extension EofF, then

( )F a is isomorphic to [ ]/ ( )F x p x . Furthermore, if

deg ( )p x n , then every member of ( )F a can be

uniquely expressed in the form

1 21 2 1 0

...n nn n

c a c a c a c

where0 1 1

, ,...,n

c c c F

.

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( ) ( )F a F b

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Let Fbe a field and ( ) [ ]p x F x be irreducible overF.Ifa is a zero in some extension EofFand b is a zero in

some extension EofF, then the fields ( )F a and ( )F b

are isomorphic.

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Lemma, p. 351

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Let Fbe a field, let ( ) [ ]p x F x be irreducible overF,

and let a be a zero of ( )p x in some extension ofF. If

is a field isomorphism from Fto Fand b is a zero of

( ( ) )p x in some extension ofF, then there is an

isomorphism from ( )F a to '( )F b that agrees with on

Fand carries a to b.

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Extending : 'F F

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Let be an isomorphism from a field Fto a field Fand

let ( ) [ ]f x F x . IfE is a splitting field for ( )f x overF

and E is a splitting field for ( ( ))f x overF, then there

is an isomorphism from E to E that agrees with on F.

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Splitting Fields Are Unique

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Let Fbe a field and let ( ) [ ]f x F x . Then any two

splitting fields of ( )f x overFare isomorphic.

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Derivative

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Let 11 1 0

( ) ...n nn nf x a x a x a x a

belong to [ ]F x .

The polynomial 1 21 1

'( ) ( 1) ...n nn nf x na x n a x a

in [ ]F x .

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Properties of the Derivative

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Let ( ), ( ) [ ]f x g x F x and let a F . Then

1. ( ( ) ( ))' '( ) '( )f x g x f x g x .

2. ( ( ))' '( )af x af x .

3. ( ( ) ( ))' ( ) '( ) '( ) ( )f x g x f x g x f x g x

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Criterion for Multiple Zeros

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A polynomial ( )f x over a field Fhas a multiple zero in

some extension E if and only if ( )f x and '( )f x have a

common factor of positive degree in [ ]F x .

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Zeros of an Irreducible

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Let ( )f x be an irreducible polynomial over a field F. If

Fhas characteristic 0, then ( )f x has no multiple zeros.

IfFhas characteristic 0p , then ( )f x has a multiple

zero only if it is of the form ( ) ( )pf x g x for some

( ) [ ]g x F x .

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Perfect field

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A field Fwith characteristic 0 or with characteristic p

and { | }p pF a a F F .

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Finite Fields Are Perfect

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Every finite field is perfect.

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Criterion for No Multiple Zeros

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If ( )f x is an irreducible polynomial over a perfect field

F, then ( )f x has no multiple zeros.

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Zeros of an Irreducible over a Splitting Field

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Let ( )f x be an irreducible polynomial over a field Fand

let Ebe a splitting field of ( )f x overF. Then all the

zeros of ( )f x in Ehave the same multiplicity.

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Factorization of an Irreducible over a Splitting Field

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Let ( )f x be an irreducible polynomial over a field Fand

let Ebe a splitting field of ( )f x . Then ( )f x has the

form1 2

( ) ( ) ...( )n n nta x a x a x a , where 1 2, ,..., ta a a are

distinct elements ofEand a F .