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Uma análise da "Parte Primeira" da obra "Sulla

Risoluzione Delle Equazioni Algebriche", de Enrico Betti. / Cesar Ricardo Peon Martins. - Rio Claro : [s.n.], 2012 74 f. : il.

Tese (doutorado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas

Orientador: Marcos Vieira Teixeira

1. Álgebra. 2. Gallois. 3. Equações. 4. Grupos. 5. Permutações. I. Título.

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Prof. Dr. Marcos Vieira Teixeira (Orientador)

Prof. Dr. Ubiratan D’Ambrósio

Profa. Dra. Iris Dias

Prof. Dr. Henrique Lazari

Prof. Dr. Fábio Maia Bertato

Aluno: Cesar Ricardo Peon Martins

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Essa inserção nos leva à fatoração

(

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( )

( )

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e portanto, as quatro raízes da equação reduzida

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3 1 4 2

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19 x x x x t

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x = x−t1

)(

x−t2

)(

x−t3

)(

x−t4

)(

x−t5

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)(

x−t24

)

25

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(

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2 21 2 21 2 17 2 17 2 13 2 13 2 9 2 9 2 5 2 5 2 1 2 1 . . . . . . . . . . . t x t x t x t x t x t x t x t x t x t x t x t x + − + − + − + − + − + − =

0DVt1 =t21, t5 =t13, t9 =t17

/RJR

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4 9 4 9 4 5 4 5 4 1 4

1 . x t . x t .x t .x t .x t

t x x

g = − + − + − +

(

) (

) (

₂

)

4

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2 _{t . x t .x t}

x − − −

=

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( )

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(

)

(

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(

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9 2 5 2 1 2 2 9 2 5 2 9 2 1 2 5 2 1 4 2 9 2 5 2 1

6 _{. . . .}

t t t x t t t t t t x t t t

x − + + + + + −

=

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( )

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( )

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(

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)

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(

t₁2.t₅2+t₁2.t₉2+t₅2t₉2

) (

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)

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26 Cada função gera uma raiz para a cúbica [1]. E cada raiz de [1] gera as quatro raízes da equação proposta.

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= 4 4

9 4 4 5 4 4 1 4 3 2 1 1 4 1 t t t x x x x x

(

)

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ª _{+ + + − − +}

= 4 4

9 4 4 5 4 4 1 4 3 2 1 2 4 1 t t t x x x x x

(

)

_»¼º

«¬

ª _{+ + + + − −}

= 4 4

9 4 4 5 4 4 1 4 3 2 1 3 4 1 t t t x x x x x

(

)

_»¼º

«¬

ª _{+ + + − + −}

= 4 4

9 4 4 5 4 4 1 4 3 2 1 4 ₄ 1 t t t x x x x x SRLV

(

)

_»¼º=

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ª _{+ + + + + +}

= 4 4

9 4 4 5 4 4 1 4 3 2 1 1 ₄ 1 t t t x x x x x

(

)

(

)

(

)

(

)

»» » ¼ º « « « ¬ ª − + − + − + − + + − + − + + + + = 4 4 4 2 3 1 4 4 3 4 2 1 4 4 4 3 2 1 4 3 2 1 4 1 x x x x x x x x x x x x x x x x

(

) (

)

(

1 2 4 3

) (

1 3 2 4

)

1

4 3 2 1 4 3 2 1 4 . 4 1 4 1 x x x x x x x x x x x x x x x x x = » » ¼ º « « ¬ ª − + − + − + − + + − + − + + + + =

$QDORJDPHQWHSDUDx2, x3, x4

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20 ..., , 4 , 0 ; 4 ..., ,

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... . .

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28

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(

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(

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(

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)

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V a c b d

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a d c b

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(

a b d c

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( )

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( )

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(

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(

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( )

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(

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(

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(

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(

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(

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(

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(

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≡ f

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( )

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( )

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(

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( )

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( ) (

x = x−φ1

)(

x−φ2

)(

x−φ3

)(

x−φ4

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x−φ5

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... x−φ23

)(

x−φ24

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g

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( ) (

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)(

x−φ₂

)(

x−φ₃

)(

x−φ₄

)(

x−φ₅

) (

... x−φ₂₃

)(

x−φ₂₄

)

=

g

(

) (

) (

) (

) (

) (

)

(

) (

)

2

12 2 11 2 6 2 5 2 4 2 3 2 2 2

1 φ φ φ φ φ ... φ φ

φ − − − − − − −

−

= x x x x x x x x

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( ) ( )

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g =

( )

ii D VXEVWLWXLomR

(

2→4, 4→3, 3→2

)

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(

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(HPh

( )

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( ) (

₁

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₂

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₃

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₄

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₅

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₆

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₁₁

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₁₂

)

1 x = x−φ x−φ x−φ x−φ x−φ x−φ ... x−φ x−φ

h

(

) (

) (

) (

)

3

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φ − − −

−

= x x x x

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( )

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( )

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( )

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(

1→3, 3→1, 2→4, 4→2

)

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( )

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( ) (

1

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h

( )

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( )

2 ₄

( ) (

₁

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h

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(

V,a

) (

= x−φ1

)(

x−φ5

)(

x−φ9

)

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( ) ( )

i, ii H

( )

iii DFLPDSDUDFRQVWUXomRGRJUXSRGDHTXDomRWHPRV

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( )

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( )

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( )

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(

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(

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(

)

_»¼º

«¬

ª _{+ + + + + +}

= 4 4 4 4 4 4

4 3 2 1 4 1 k j

i t t

t x x x x

x

[

(

_{1 2 3 4}

)

₁ 2 ₂

]

4 1

θ α αθ θ+ + +

+ + +

= x x x x

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(

V,a

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= x−θ

)(

x−θ1

)(

x−θ2

)

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