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7/28/2019 Gallian Ch 20
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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 .