rt trt t tts r - PowerPoint PPT Presentation

❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ✽✳✺✳✷✵✶✾ ▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✽✳✺✳✷✵✶✾▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✶ ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ✴ ✺✸

❘❡❢❡r❡♥❝❡s ❚❤❡ r❡s✉❧ts t❤❛t ❛r❡ ♣r❡s❡♥t❡❞ ❤❡r❡ ❡✐t❤❡r ❝❧❛ss✐❝❛❧ ♦r ❛r❡ ❞♦♥❡ ❜② ♠❡ ❛❧♦♥❡ ♦r ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ▲❛✉r❛ ▲✉③③✐ ❛♥❞ ❋r❛♥❝✐s ▲✉✳ ✽✳✺✳✷✵✶✾▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✷ ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ✴ ✺✸

❆ ❧❛tt✐❝❡ ❆ ❧❛tt✐❝❡ L ✐s ❛ ❞✐s❝r❡t❡ ❛❞❞✐t✐✈❡ ❣r♦✉♣ ✐♥ R n ✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t ✇✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t t❤❡r❡ ❡①✐sts ❛ s❡t ♦❢ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ❡❧❡♠❡♥ts { a ✶ , . . . , a k } t❤❛t ❣❡♥❡r❛t❡ L ✳ ■❢ L = a ✶ Z + a ✷ Z + · · · + a k Z ✱ ✇❡ s❛② t❤❛t L ❤❛s ❞❡❣r❡❡ k ✳ ✽✳✺✳✷✵✶✾▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✸ ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ✴ ✺✸

▼❛tr✐① ❧❛tt✐❝❡s ▲❛tt✐❝❡s ✇❡ ❝♦♥s✐❞❡r ✐♥ t❤✐s ♣r❡s❡♥t❛t✐♦♥ ❛r❡ ❜❛s❡❞ ♦♥ ❛❞❞✐t✐✈❡ ❣r♦✉♣s ✐♥ M n × n ( C ) ✳ ❉❡❢✐♥✐t✐♦♥ ❆ ♠❛tr✐① ❧❛tt✐❝❡ L ⊆ M n × n ( C ) ❤❛s t❤❡ ❢♦r♠ L = Z B ✶ ⊕ Z B ✷ ⊕ · · · ⊕ Z B k , ✇❤❡r❡ t❤❡ ♠❛tr✐❝❡s B ✶ , . . . , B k ❛r❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ♦✈❡r R ✱ ✐✳❡✳✱ ❢♦r♠ ❛ ❧❛tt✐❝❡ ❜❛s✐s✱ ❛♥❞ k ✐s ❝❛❧❧❡❞ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❧❛tt✐❝❡✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t X , Y ∈ M n ( C ) ✳ ❚❤❡ ♥❛t✉r❛❧ ✐♥♥❡r✲♣r♦❞✉❝t ✐s ♥♦✇ � X , Y � = ℜ ( Tr ( XY † ) . ❲✐t❤ r❡s♣❡❝t t♦ t❤✐s ✐♥♥❡r✲♣r♦❞✉❝t M n ( C ) ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ s♣❛❝❡ R ✷ n ✷ ✳ ▼❛tr✐① ❢♦r♠ ✐s ❥✉st ❝♦♥✈❡♥✐❡♥t ✇❛② ♦❢ ♣r❡s❡♥t✐♥❣ ♦✉r ✈❡❝t♦rs✳ ✽✳✺✳✷✵✶✾▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✹ ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ✴ ✺✸

▼❛tr✐① ❧❛tt✐❝❡s ❲❡ ❞❡♥♦t❡ t❤❡ ♠❡❛s✉r❡ ✭♦r ❤②♣❡r✈♦❧✉♠❡✮ ♦❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛❧❧❡❧♦t♦♣❡ ♦❢ ❛ ❧❛tt✐❝❡ L ⊂ M n ( C ) ❜② ❱♦❧ ( L ) ❛♥❞ ❝❛❧❧ ✐t t❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛❧❧❡❧♦t♦♣❡ ♦❢ t❤❡ ❧❛tt✐❝❡ L ✳ ■❢ x ✶ , . . . , x k ✐s ❛ ❜❛s✐s ♦❢ L ✱ ✇❡ ❝❛♥ ❢♦r♠ t❤❡ ●r❛♠ ♠❛tr✐① ♦❢ t❤❡ ❧❛tt✐❝❡ L � � ℜ tr ( x i x † j ) ✶ ≤ i , j ≤ k . ❚❤❡ ●r❛♠ ♠❛tr✐① ❤❛s ❛ ♣♦s✐t✐✈❡ ❞❡t❡r♠✐♥❛♥t ❡q✉❛❧ t♦ ❱♦❧ ( L ) ✷ . ✽✳✺✳✷✵✶✾▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✺ ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ✴ ✺✸

▲❛tt✐❝❡s ❢r♦♠ ♥✉♠❜❡r ❢✐❡❧❞s ▲❡t ✉s ❜❡❣✐♥ ✇✐t❤ ❛ ❞❡❣r❡❡ n ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r a ✳ ▲❡t f a ( x ) = x n + c n − ✶ x n − ✶ + · · · + c ✶ x + c ✵ ❜❡ t❤❡ ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧ ♦❢ a ✭❤❡r❡ c i ∈ Z ) ✳ ❲❡ ✉s❡ ♥♦t❛t✐♦♥ K = Q ( a ) = Q ⊕ Q a ⊕ · · · ⊕ Q a n − ✶ ✳ ❚❤❡ s❡t K ✐s ❛ ✜❡❧❞ ✳ ❚❤✐s ♠❡❛♥s t❤❛t K ✐s ❛❞❞✐t✐✈❡❧② ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐✈❡❧② ❝❧♦s❡❞ ❛♥❞ ❢♦r ❡✈❡r② ❡❧❡♠❡♥t x ∈ K ✱ x � = ✵✱ t❤❡r❡ ❡①✐sts y ∈ K s✉❝❤ t❤❛t xy = ✶✳ ✽✳✺✳✷✵✶✾▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✻ ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ✴ ✺✸

▲❛tt✐❝❡s ❢r♦♠ ♥✉♠❜❡r ❢✐❡❧❞s ❲❡ ❛❧s♦ ❤❛✈❡ t❤❛t R K = Z [ a ] = Z ⊕ Z a ⊕ · · · ⊕ Z a n − ✶ . ✐s ❛ r✐♥❣ ❛♥❞ ❛ ❞❡❣r❡❡ n ❢r❡❡ Z ✲♠♦❞✉❧❡✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ s❡❡♥ ❛s ❛ s✉❜s❡t ✐♥ C ✐t ✐s ❛ ❞❡♥s❡ s❡t✳ ❙♦ ✐t ✐s ❛♥ ❛❞❞✐t✐✈❡ ❣r♦✉♣✱ ❜✉t ✐t ✐s ♥♦t ❞✐s❝r❡t❡ ✐♥ t❤❡ ♥❛t✉r❛❧ ❛♠❜✐❡♥t s♣❛❝❡✳ ✽✳✺✳✷✵✶✾▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✼ ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ✴ ✺✸

▲❛tt✐❝❡s ❢r♦♠ ♥✉♠❜❡r ❢✐❡❧❞s ❲❡ ✇✐❧❧ ❞❡♥♦t❡ ✇✐t❤ σ i ( a ) t❤❡ ❝♦♠♣❧❡① r♦♦ts ♦❢ ♣♦❧②♥♦♠✐❛❧ f a ( x ) ✳ f a ( x ) = ( x − σ ✶ ( a ))( x − σ ✷ ( x )) · · · ( x − σ n ( a )) , ❤❡r❡ σ ✶ ( a ) = a ✳ ❚❤❡s❡ ③❡r♦s ❛❧❧♦✇s ✉s t♦ ❞❡✜♥❡ n ♠❛♣♣✐♥❣s ❢r♦♠ K t♦ C ✳ ❘❡♠❡♠❜❡r t❤❛t ❡❛❝❤ x ∈ K ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s x = d ✵ + d ✶ a + · · · + d n − ✶ a n − ✶ , ✇❤❡r❡ d i ∈ Q ✳ ◆♦✇ ✇❡ ❝❛♥ ❞❡✜♥❡ σ i ( x ) = d ✵ + d ✶ σ i ( a ) + · · · + d n − ✶ σ i ( a ) n − ✶ . ✽✳✺✳✷✵✶✾▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✽ ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ✴ ✺✸

▲❛tt✐❝❡s ❢r♦♠ ♥✉♠❜❡r ❢✐❡❧❞s ❙♦ ❞❡✜♥❡❞ ♠❛♣♣✐♥❣s s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✳ ❚❤❡ ♠❛♣♣✐♥❣s σ i ❛r❡ Q ❛❧❣❡❜r❛ ❡♠❜❡❞❞✐♥❣s✳ ❲❡ ❤❛✈❡ σ i ( x + y ) = σ i ( x ) + σ i ( y ) ✳ ❆♥❞ σ i ( xy ) = σ i ( x ) σ i ( y ) ✳ ■❢ x ∈ K ✱ t❤❡♥ � n i = ✶ σ i ( x ) ∈ Q ✳ ❲❡ ❛❧s♦ ❤❛✈❡ t❤❛t σ i ( x ) = ✵ ♦♥❧② ✐❢ x = ✵✳ ■❢ x ∈ R K t❤❡♥ � n i = ✶ σ i ( x ) ∈ Z . ✽✳✺✳✷✵✶✾▲❛tt✐❝❡ ♠❡❡t✐♥❣ ▲♦♥❞♦♥ ✾ ❘♦♦♣❡ ❱❡❤❦❛❧❛❤t✐ ❆❛❧t♦ ❯♥✐✈❡rs✐t②✱ ❋✐♥❧❛♥❞ ❆ ❙❤♦rt ■♥tr♦❞✉❝t✐♦♥ t♦ ▲❛tt✐❝❡s ❢r♦♠ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❋✐❡❧❞s ✴ ✺✸