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Automorphism ofE

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E is an extension of field F.

A ring isomorphism from Eonto E.

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Galois Group ofEOverF

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The set of all automorphisms ofE that take everyelement ofF to itself.

Denoted ( / )Gal E F .

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Fixed field ofH

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H is a subgroup of ( / )Gal E F .

The set { | ( ) }H

E x E x x H .

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Fundamental Theorem of Galois Theory(Part 1 of 2)

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Let Fbe a field of characteristic 0 or a finite field. IfE isthe splitting field overF for some polynomial in [ ]F x ,

then the mapping from the set of subfields ofE

containing F to the set of subgroups of ( / )Gal E F

given by ( / )K Gal E K is a one-to-onecorrespondence. Furthermore, for any subfield KofE

containing F,

1. [ : ] | ( / )|E K Gal E K and

[ : ] | ( / )| / | ( / ) |K F Gal E F Gal E K . [The index of

( / )Gal E K in ( / )Gal E F equals the degree ofKoverF.]

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Fundamental Theorem of Galois Theory

(Part 2 of 2)

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2. IfK is the splitting field of some polynomial in

[ ]F x , then ( / )Gal E K is a normal subgroup of

( / )Gal E F and ( / )Gal K F is isomorphic to

( / ) / ( / )Gal E F Gal E K .

3.( / )Gal E K

K E . [The fixed field of ( / )Gal E K is

K.]4. IfH is a subgroup of ( / )Gal E F , then

( / )H

H Gal E E . [The automorphism group ofE

fixingH

E is H.]

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Solvable by Radicals OverF

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Let Fbe a field, and ( ) [ ]f x F x .

( )f x splits in some extension1 2

( , ,..., )nF a a a ofFand

there exist positive integers1 2, ,..., nk k k such that

11

ka F and 1 1,...,( )

iki i

a F a a

for 2,...,i n .

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Solvable Group

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A group G has a series of subgroups

0 1{ } ...

ke H H H G , where for each 0 i k ,

iH is normal in

1iH

and

1/ iiH H is Abelian.

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Splitting Field of nx a

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Let Fbe a field of characteristic 0 and let a F . IfE isthe splitting field of nx a overF, then the Galois group

( / )Gal E F is solvable.

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Factor Group of a Solvable Group Is Solvable

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A factor group of a solvable group is solvable.

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Nand G/NSolvable Implies G Is Solvable

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Let Nbe a normal subgroup of a group G. If both NandG/Nare solvable, then G is solvable.

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Solvable by Radicals Implies Solvable Group

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Let Fbe a field of characteristic 0 and let ( ) [ ]f x F x .

Suppose that ( )f x splits in1 2

( , ,..., )t

F a a a , where

11

na F and 1 1,...,( )

ini i

a F a a

for 2,...,i t . Let Ebe

the splitting field for ( )f x overF in1 2

( , ,..., )nF a a a .

Then the Galois group ( / )Gal E F is solvable.