CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s

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n✲❣r❛❞✉❛❞❛s ❞❛ ➪❧❣❡❜r❛

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❊✈❛♥❞r♦ ❘✐✈❛

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s

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n✲❣r❛❞✉❛❞❛s ❞❛ ➪❧❣❡❜r❛

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❊✈❛♥❞r♦ ❘✐✈❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❍✉♠❜❡rt♦ ▲✉✐③ ❚❛❧♣♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥✲ çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

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Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária UFSCar Processamento Técnico

com os dados fornecidos pelo(a) autor(a)

R616i

Riva, Evandro

Identidades polinomiais Zn-graduadas da álgebra Mn(F) / Evandro Riva. -- São Carlos : UFSCar, 2016. 54 p.

Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2016.

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s ♣♦r ♠❡ ❞❛r ❢♦rç❛ ✐♥t❡r✐♦r ♣❛r❛ s✉♣♦rt❛r ❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ♠♦str❛r ♦ ❝❛♠✐♥❤♦ ♥❛s ❤♦r❛s ✐♥❝❡rt❛s✳

❆ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦ ♠❡✉ ♣❛✐ ❙❛❞✐ ❡ ♠✐♥❤❛ ♠ã❡ ▲✉❝✐❛✱ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐✲ t❛r❡♠ q✉❡ ❡✉ s❡r✐❛ ❝❛♣❛③✱ s❡♠♣r❡ ♣❛ss❛r❡♠ ♦s ♠❛✐s ♣✉r♦s ♣❡♥s❛♠❡♥t♦s ❡ ♦r❛çõ❡s ❞✉r❛♥t❡ t♦❞❛ ♠✐♥❤❛ ✈✐❞❛✳ ❆ ♠✐♥❤❛ ✐r♠ã ❊❞✐♥❛✱ ♠❡✉ ❝✉♥❤❛❞♦ ❱♦❧♠❛r ❡ ♦s ♠❡✉s s♦❜r✐♥❤♦s ❘❛❢❛❡❧ ❡ ■s❛❞♦r❛ ♣❡❧♦ ✐♠❡♥s♦ ❛♣♦✐♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ✈♦❝ê ❊❞✐♥❛✱ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐♦✉ ♠❡s♠♦ ❞✐st❛♥t❡✱ s❡♠♣r❡ r❡③♦✉ ❡ t♦r❝❡✉ ♣♦r ♠✐♠✳ ❚❛♠❜é♠ ♠❡✉s ❛✈ós✱ t✐♦s ❡ ♣r✐♠♦s✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ♣r♦❢❡ss♦r ❍✉♠❜❡rt♦✱ ❛❣r❛❞❡ç♦ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❛♠✐③❛❞❡✱ ❝♦♥s❡❧❤♦s ❡ ♣♦r t♦❞♦ s✉♣♦rt❡ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❈❛r❧♦s✱ ❉❛✐❛♥❛ ❡ ▼❛t❡✉s q✉❡ ♣❛ss❛r❛♠ ♣♦r t♦❞♦s ❡ss❡s ♠♦♠❡♥t♦s ❛♦ ♠❡✉ ❧❛❞♦✱ q✉❡✱ ✐♥ú♠❡r❛s ✈❡③❡s ♠❡ ❛✉①✐❧✐❛r❛♠ ❡ ♠❡ ✐♥❝❡♥t✐✈❛r❛♠✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ✈♦❝ê ▼❛t❡✉s✱ ♣❡❧❛s q✉❛s❡ ✐♥✜♥✐t❛s ❤♦r❛s ❞❡ ❡st✉❞♦ ❡ ❛❥✉❞❛✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ❯❚❋P❘ ❡ ❞♦ ❉▼✳

❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞❛ ❯❚❋P❘ ❡ ❞❛ ❯❋❙❈❛r ♣❡❧❛ ❡①❝❡❧❡♥t❡ ❢♦r♠❛çã♦ q✉❡ ♠❡ ♣r♦♣♦r❝✐✲ ♦♥❛r❛♠✳

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❘❡s✉♠♦

◆❡st❛ ❞✐ss❡rt❛çã♦ ❡st✉❞❛r❡♠♦s á❧❣❡❜r❛sG✲❣r❛❞✉❛❞❛s ❡ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐sG✲❣r❛❞✉❛❞❛s✱ ♦♥❞❡Gé ✉♠ ❣r✉♣♦ ❛❞✐t✐✈♦✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❡s❝r❡✈❡r❡♠♦s ✉♠❛ ❜❛s❡ ✜♥✐t❛ ♣❛r❛ ❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐sZn✲❣r❛❞✉❛❞❛s ❞❛ á❧❣❡❜r❛ ❞❛s ♠❛tr✐③❡s n×n✱ ❝♦♠ ❡♥tr❛❞❛s ❡♠

✉♠ ❝♦r♣♦ F✳ ❊st❡ ❡st✉❞♦ s❡rá s✉❜❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ❡t❛♣❛s✿ q✉❛♥❞♦ ♦ ❝♦r♣♦ F ❢♦r ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ❡ q✉❛♥❞♦ ♦ ❝♦r♣♦ F ❢♦r ✐♥✜♥✐t♦✳ ❊st❡s r❡s✉❧t❛❞♦s ❢♦r❛♠ ❞❡s❝r✐t♦s ♣♦r ❱❛s✐❧♦✈s❦② ❬✶✽❪ ❡♠ ✶✾✾✾ ❡ ♣♦r ❆③❡✈❡❞♦ ❬✷❪ ❡♠ ✷✵✵✻✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦s ✇❡ ✇✐❧❧ st✉❞②G✲❣r❛❞❡❞ ❛❧❣❡❜r❛s ❛♥❞G✲❣r❛❞❡❞ ♣♦❧②♥♦♠✐❛❧ ✐❞❡♥t✐t✐❡s✱ ✇❤❡r❡ G ✐s ❛♥ ❛❞❞✐t✐✈❡ ❣r♦✉♣✳ ❋♦r ♠❛✐♥ r❡s✉❧t ✇❡ ✇✐❧❧ ❞❡s❝r✐❜❡ ❛ ✜♥✐t❡ ❜❛s✐s ❢♦r Zn✲❣r❛❞❡❞

♣♦❧②♥♦♠✐❛❧ ✐❞❡♥t✐t✐❡s ♦❢ t❤❡ ♠❛tr✐① ❛❧❣❡❜r❛ ♦❢ ♦r❞❡rn×n✱ ✇✐t❤ ❡♥tr✐❡s ✐♥ ❛ ✜❡❧❞F✳ ❚❤✐s st✉❞② ✇✐❧❧ ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ t✇♦ st❛❣❡s✿ ✇❤❡♥ t❤❡ ✜❡❧❞F ❤❛s ❝❤❛r❛❝t❡r✐st✐❝ ③❡r♦ ❛♥❞ ✇❤❡♥ t❤❡ ✜❡❧❞F ✐s ✐♥✜♥✐t❡✳ ❚❤❡s❡ r❡s✉❧ts ✇❡r❡ ❞❡s❝r✐❜❡❞ ❜② ❱❛s✐❧♦✈s❦② ❬✶✽❪ ✐♥ ✶✾✾✾ ❛♥❞ ❆③❡✈❡❞♦ ❬✷❪ ✐♥ ✷✵✵✻✳

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❙✉♠ár✐♦

✶ Pr❡❧✐♠✐♥❛r❡s ✺

✶✳✶ ➪❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ➪❧❣❡❜r❛ ▲✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ P♦❧✐♥ô♠✐♦s ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✺ ❖ ❚❡♦r❡♠❛ ❞❡ ❆♠✐ts✉r✲▲❡✈✐t③❦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✻ ▼❛tr✐③❡s ●❡♥ér✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷ ●r❛❞✉❛çõ❡s ✷✸

✷✳✶ G✲❣r❛❞✉❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶✳✶ ❆ Zn✲❣r❛❞✉❛çã♦ ❞❡ Mn(F) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✷✳✶✳✷ ❆ Zn✲❣r❛❞✉❛çã♦ ❞❡ FhXi ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✷✳✷ ■❞❡♥t✐❞❛❞❡s G✲❣r❛❞✉❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸ ■❞❡♥t✐❞❛❞❡s Zn✲❣r❛❞✉❛❞❛s ❞❡ Mn(F) ✷✾

✸✳✶ ❈♦r♣♦ ❝♦♠ ❈❛r❛❝t❡ríst✐❝❛ ❩❡r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ❈♦r♣♦ ■♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

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■♥tr♦❞✉çã♦

❊♠ ♠❛t❡♠át✐❝❛✱ ✉♠❛ ✐♠♣♦rt❛♥t❡ ár❡❛ ♥❛ t❡♦r✐❛ ❞❡ ❛♥é✐s é ❛ t❡♦r✐❛ ❞❡ á❧❣❡❜r❛s q✉❡ s❛t✐s❢❛③❡♠ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s✱ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ P■✲á❧❣❡❜r❛s ✭❞♦ ✐♥❣❧ês P♦❧②♥♦♠✐❛❧ ■❞❡♥t✐t✐❡s✮✳ ❊st❛ ❝❧❛ss❡ ❞❡ á❧❣❡❜r❛s é ♠✉✐t♦ ❛♠♣❧❛ ❡ ❡♥❣❧♦❜❛ ✈ár✐❛s ❝❧❛ss❡s ❞❡ á❧❣❡❜r❛s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❡①✐st❡♠ ❞✐✈❡rs❛s ❛❜♦r❞❛❣❡♥s ❡ t❡♦r✐❛s s♦❜r❡ P■✲á❧❣❡❜r❛s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥♦ ❞❡❝♦rr❡r ❞❡st❡ tr❛❜❛❧❤♦✱ ❡♥❝♦♥tr❛r❡♠♦s ✉♠❛ ❜❛s❡ ✜♥✐t❛ ♣❛r❛ ❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❛ á❧❣❡❜r❛ ❞❛s ♠❛tr✐③❡s n×n ❝♦♠ ❡♥tr❛❞❛s ❡♠ ✉♠ ❝♦r♣♦ F✱ ❞❡♥♦t❛❞❛ ♣♦r Mn(F)✳ ❆

té❝♥✐❝❛ q✉❡ s❡rá ✉t✐❧✐③❛❞❛ ♣❛r❛ s✐♠♣❧✐✜❝❛r ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥❝♦♥tr❛r ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❡♠ Mn(F) ❝♦♥s✐st❡ ❡♠ ✐♥tr♦❞✉③✐r ♥❡ss❛ á❧❣❡❜r❛ ✉♠❛ Zn✲❣r❛❞✉❛çã♦✱ ♣♦✐s✱ ❝♦♠ ✐ss♦✱ s❡rá

♣♦ssí✈❡❧ ❡①✐❜✐r t❛❧ ❜❛s❡ ✜♥✐t❛ ♣❛r❛ ❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❛ á❧❣❡❜r❛ Mn(F)✳ P❛r❛

✐ss♦ ♣r❡❝✐s❛r❡♠♦s ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧ ✐♥✜♥✐t♦✳ ❉❡♥♦t❡♠♦s ♣♦rFhXi ♦F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❜❛s❡ ❢♦r♠❛❞❛ ♣♦r ✶ ❡ ♣❡❧❛s ♣❛❧❛✈r❛sxi1xi2· · ·xin ❝♦♠xi1, xi2,· · · , xin ∈X❡n≥0✳ ❉❡✜♥✐r❡♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♥tr❡ ❞✉❛s ♣❛❧❛✈r❛s ♣♦r ❝♦♥❝❛t❡♥❛çã♦✱ ♦✉ s❡❥❛✿

(xi1xi2· · ·xin)(xj1xj2· · ·xjn) = xi1xi2· · ·xinxj1xj2· · ·xjn,

♦F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ FhXi é ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❝♦♠ ✉♥✐❞❛❞❡ ✭♣❛❧❛✈r❛ ✈❛③✐❛✮ ❣❡r❛❞❛

♣♦r X✳ ❉✐③❡♠♦s q✉❡ ✉♠ ♣♦❧✐♥ô♠✐♦ f(x1, ..., xn) ♥❛s ✈❛r✐á✈❡✐s ♥ã♦ ❝♦♠✉t❛t✐✈❛s x1, ..., xn

é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ ✉♠❛ á❧❣❡❜r❛ A s❡✱ ♣❛r❛ q✉❛✐sq✉❡r a1, ..., an ∈ A✱

t❡♠✲s❡ f(a1, ..., an) = 0 ❡♠ A✳ ❙❡ ❡①✐st✐r ✉♠ ♣♦❧✐♥ô♠✐♦ ♥ã♦ ♥✉❧♦ ❝♦♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡

♣❛r❛ ✉♠❛ á❧❣❡❜r❛A ❡♥tã♦ ❞✐③✲s❡ q✉❡A é ✉♠❛ á❧❣❡❜r❛ ❝♦♠ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♦✉ q✉❡ A é ✉♠❛ P■✲á❧❣❡❜r❛ ❡ f é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ A✳ P♦r ❡①❡♠♣❧♦✱ s❡ A ❢♦r ✉♠❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛✱ ❡♥tã♦ f(x1, x2) = x1x2 −x2x1 é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧

(18)

■♥tr♦❞✉çã♦

q✉❛❧q✉❡r ❡♥❞♦♠♦r✜s♠♦ ❞❡st❛ á❧❣❡❜r❛✳ ■❞❡❛✐s ❝♦♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ sã♦ ❝❤❛♠❛❞♦s T✲ ✐❞❡❛✐s✳ ❉❡s❝r❡✈❡r ❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ A s✐❣♥✐✜❝❛ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ ❣❡r❛❞♦r ♣❛r❛ T(A)❝♦♠♦T✲✐❞❡❛❧✳ ❖s ♣r✐♠❡✐r♦s tr❛❜❛❧❤♦s ❡♥✈♦❧✈❡♥❞♦ P■✲á❧❣❡❜r❛✱ ❛♣❛r❡❝❡r❛♠✱ ❞❡ ❢♦r♠❛ ✐♠♣❧í❝✐t❛✱ ♣♦r ✈♦❧t❛ ❞❛ ❞é❝❛❞❛ ❞❡ ✶✾✸✵ ❝♦♠ ❛s ♣❡sq✉✐s❛s ❞❡ ❉ë❤♥ ❬✸❪ ❡ ❲❛❣♥❡r ❬✶✾❪✱ ♠❛s s♦♠❡♥t❡ ❛ ♣❛rt✐r ❞❡ ✶✾✹✽✱ ❛♣ós ♦ ❛rt✐❣♦ ❞❡ ❑❛♣❧❛♥s❦② ❬✾❪ q✉❡ ❡ss❛ t❡♦r✐❛ r❡❛❧♠❡♥t❡ s❡ ❞❡s❡♥✈♦❧✈❡✉✳ ❉❡♥tr❡ ❛s q✉❡stõ❡s ❧❡✈❛♥t❛❞❛s ♣♦r ❑❛♣❧❛♥s❦②✱ ✉♠❛ ❡r❛ s♦❜r❡ q✉❛❧ s❡r✐❛ ♦ ♠❡♥♦r ❣r❛✉ ❞❡ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ s❛t✐s❢❡✐t❛ ♣❡❧❛ á❧❣❡❜r❛ ❞❛ ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ n × n✱ s♦❜r❡ ✉♠ ❝♦r♣♦✳ ❆ r❡s♣♦st❛ ❢♦✐ ❞❛❞❛ ❡♠ ✶✾✺✵ ❝♦♠ ♦ ❝❡❧❡❜r❛❞♦ r❡s✉❧t❛❞♦ ❞❡

❆♠✐ts✉r✲▲❡✈✐t③❦✐ ❬✶❪ ❡♠ q✉❡ ♦ ♣♦❧✐♥ô♠✐♦

St2n(x1, ..., xn) =

X

σ∈S2n

(−1)σxσ(1)· · ·xσ(2n),

♦♥❞❡S2nr❡♣r❡s❡♥t❛ ♦ ❣r✉♣♦ s✐♠étr✐❝♦ ❞❡ ❣r❛✉2n❡(−1)σ r❡♣r❡s❡♥t❛ ♦ s✐♥❛❧ ❞❛ ♣❡r♠✉t❛çã♦

σ ∈ S2n✱ ❝❤❛♠❛❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦ st❛♥❞❛r❞ ❞❡ ❣r❛✉ 2n✱ é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧

♣❛r❛ ❡st❛ á❧❣❡❜r❛ ❡ 2n é ♦ ♠❡♥♦r ❣r❛✉✳ ❊st❡ r❡s✉❧t❛❞♦ ♠❛r❝♦✉ ♦ ✐♥í❝✐♦ ❞❡ ✉♠❛ ♥♦✈❛ ❛❜♦r❞❛❣❡♠ à t❡♦r✐❛ ❞❛s P■✲á❧❣❡❜r❛s✱ ❢♦❝❛❞❛s ❡♠ ❞❡s❝r❡✈❡r ❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ✉♠❛ ❞❛❞❛ á❧❣❡❜r❛✳ ❆s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❛ á❧❣❡❜r❛ Mn(F) ❞❛s ♠❛tr✐③❡s n ×n

❝♦♠ ❡♥tr❛❞❛s ❡♠ ✉♠ ❝♦r♣♦ F sã♦ ✐♠♣♦rt❛♥t❡s ♥❛ t❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s✱ ❝♦♥t✉❞♦✱ s♦❜r❡ ❝♦r♣♦s ✐♥✜♥✐t♦s✱ ❜❛s❡s ✜♥✐t❛s ♣❛r❛ T(Mn(F)) sã♦ ❞❡t❡r♠✐♥❛❞❛s s♦♠❡♥t❡ q✉❛♥❞♦ n = 2 ❡

❝❛r❛❝t❡ríst✐❝❛ ❞❡ F é ❞✐❢❡r❡♥t❡ ❞❡ ✷✳ ❖ ❝❛s♦ ❡♠ q✉❡n = 3 ❛✐♥❞❛ ❡stá ❡♠ ❛❜❡rt♦✳ ❊♠ ❬✶✸❪

❘❛③♠②s❧♦✈ ❞❡t❡r♠✐♥♦✉ ✉♠❛ ❜❛s❡ ♣❛r❛ ❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ M2(F)❝♦♠ ✾ ❡❧❡♠❡♥t♦s ♥♦ ❝❛s♦

❡♠ q✉❡ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡F é ✐❣✉❛❧ ❛ ③❡r♦ ❡ ❡♠ ❬✻❪ ❉r❡♥s❦② ♠❡❧❤♦r♦✉ ❡st❡ r❡s✉❧t❛❞♦ ❞❛♥❞♦ ✉♠❛ ❜❛s❡ ♠✐♥✐♠❛❧ ❝♦♠ ❞✉❛s ✐❞❡♥t✐❞❛❞❡s✳ ❯♠❛ ❜❛s❡ ♣❛r❛M2(F)s♦❜r❡ ❝♦r♣♦s ✐♥✜♥✐t♦s ❞❡

❝❛r❛❝t❡ríst✐❝❛ p > 2 ❢♦✐ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ❑♦s❤❧✉❦♦✈ ❡♠ ❬✶✶❪✳ ❖✉tr♦ ❝♦♥❝❡✐t♦ ✐♠♣♦rt❛♥t❡

♥❛ ár❡❛ ❞❡ P■✲á❧❣❡❜r❛s é ♦ ❞❡ ✐❞❡♥t✐❞❛❞❡s ❣r❛❞✉❛❞❛s✱ ✐st♦ é✱ s❡❥❛ G ✉♠ ❣r✉♣♦ ❛❞✐t✐✈♦✱ ❡♥tã♦ ✉♠❛ á❧❣❡❜r❛ éG✲❣r❛❞✉❛❞❛ s❡ ♣✉❞❡r s❡r ❡s❝r✐t❛ ❝♦♠♦ s♦♠❛ ❞✐r❡t❛ ❞❡ s✉❜❡s♣❛ç♦s

A=X

g∈G

M

A(g),

❞❡ ♠♦❞♦ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r g, h∈ G t❡♠✲s❡ A(g)_A(h) _⊆ _A(g+h)_{✳ ❊st❛ ár❡❛ ♣❡r♠✐t✐✉ q✉❡}

❑❡♠❡r ❬✶✵❪ ❞❡ss❡ ✉♠❛ r❡s♣♦st❛ ♣♦s✐t✐✈❛ ♣❛r❛ ♦ ❢❛♠♦s♦ ♣r♦❜❧❡♠❛ ❞❡ ❙♣❡❝❤t ❬✶✻❪ q✉❡ é ♦ s❡❣✉✐♥t❡✿ ✏❙❡rá q✉❡ t♦❞❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ♣♦ss✉✐ ✉♠❛ ❜❛s❡ ✜♥✐t❛ ♣❛r❛ s✉❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s❄✑✳ ▲♦❣♦ ❛♣ós ♦s tr❛❜❛❧❤♦s ❞❡ ❑❡♠❡r✱

(19)

♣♦r ✈♦❧t❛ ❞❡ ✶✾✽✼✱ ❛s ✐❞❡♥t✐❞❛❞❡s ❣r❛❞✉❛❞❛s t♦r♥❛r❡♠✲s❡ ♦❜❥❡t♦ ❞❡ ♣❡sq✉✐s❛ ♠✉✐t♦ ❛t✐✈❛✳ ❈♦♥s✐❞❡r❡♠♦s ❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❧✐✈r❡FhXi❝♦♠X = [

α∈G

X(α)_{✱ ❡♠ q✉❡}_{_X(α)_;_α _∈_●_}_é

✉♠❛ ❢❛♠✐❧✐❛ ❞❡ ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s✱ ❡♥✉♠❡rá✈❡✐s ✐♥✜♥✐t♦s ❡ ❞✐s❥✉♥t♦sX(α)_{✳ ❖❜s❡r✈❡♠♦s}

q✉❡ ♦s ♠♦♥ô♠✐♦s✿ {xi1xi2. . . xik :k = 1,2, ...;xi1, xi2, ..., xik ∈X} ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ FhXi ❝♦♠♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ✈❛r✐á✈❡❧x∈X ♣♦ss✉✐ ❣r❛✉ ❤♦♠♦❣ê♥❡♦ α✱ ❞❡♥♦t❛❞♦ ♣♦rα(x) =α✱ s❡x∈X(α)_{✳ ❖ ❣r❛✉ ❤♦♠♦❣ê♥❡♦ ❞❡ ✉♠ ♠♦♥ô♠✐♦}_n₌_x

i1xi2. . . xik é ❞❡✜♥✐❞♦ ♣♦r✿ α(n) = α(xi1) +α(xi2) +. . .+α(xik)✳ P❛r❛ α ∈ G✱ ❞❡♥♦t❡ ♣♦r FhXi

(α)

♦ s✉❜❡s♣❛ç♦ ❞❡ FhXi ❣❡r❛❞♦ ♣♦r t♦❞♦s ♦s ♠♦♥ô♠✐♦s q✉❡ t❡♠ ❣r❛✉ ❤♦♠♦❣ê♥❡♦ α✳ ◆♦t❡ q✉❡✿ FhXi(α)_F_h_X_i(β) _⊆_F_h_X_i(α+β)_,_∀_{α, β} _∈_{G✳ ❆ss✐♠✱}

FhXi=X

α∈G

M

FhXi(α)_,

t♦r♥❛✲s❡ ✉♠❛ á❧❣❡❜r❛G✲❣r❛❞✉❛❞❛✳ ❖s ❡❧❡♠❡♥t♦s ❞❛ á❧❣❡❜r❛G✲❣r❛❞✉❛❞❛FhXisã♦ ❝❤❛♠❛✲

❞♦s ❞❡ ♣♦❧✐♥ô♠✐♦sG✲❣r❛❞✉❛❞♦s ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣♦❧✐♥ô♠✐♦s ❣r❛❞✉❛❞♦s✳ ❉✐③❡♠♦s q✉❡ ✉♠ ♣♦❧✐♥ô♠✐♦G✲❣r❛❞✉❛❞♦ f(x1, x2, ..., xk) é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ❣r❛❞✉❛❞❛ ♣❛r❛ ❛

á❧❣❡❜r❛ G✲❣r❛❞✉❛❞❛ A s❡ f(a1, a2, ..., ak) = 0 ♣❛r❛ q✉❛✐sq✉❡r ai ∈ A(α(xi))✱ i = 1,2, ..., k✳

❉❡♥♦t❛r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❛s ✐❞❡♥t✐❞❛❞❡sG✲❣r❛❞✉❛❞❛s ♣♦rT(A)gr_{✳ ❆q✉✐ ✈❛❧❡ r❡ss❛❧t❛r q✉❡}

T(A)gr _{é ✉♠} _T

G✲✐❞❡❛❧✱ ♦✉ s❡❥❛✱ ✉♠ ✐❞❡❛❧ ❞❡FhXi q✉❡ é ❢❡❝❤❛❞♦ ♣❛r❛ t♦❞♦ ❡♥❞♦♠♦r✜s♠♦

G✲❣r❛❞✉❛❞♦ ❞❡ FhXi✳ ◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛r❡♠♦s ❜❛s❡s ✜♥✐t❛s ♣❛r❛ ❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐✲

♥♦♠✐❛✐s Zn✲❣r❛❞✉❛❞❛s ❞❛ á❧❣❡❜r❛ ♠❛tr✐❝✐❛❧ Mn(F) q✉❛♥❞♦ ♦ ❝♦r♣♦ F t❡♠ ❝❛r❛❝t❡ríst✐❝❛

③❡r♦ ❡ q✉❛♥❞♦ ♦ ❝♦r♣♦ F ❢♦r ✐♥✜♥✐t♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ p ❛r❜✐trár✐❛✱ t❡♥❞♦ ❝♦♠♦ r❡❢❡rê♥✲ ❝✐❛ ♣r✐♥❝✐♣❛❧ ♦s ❛rt✐❣♦s ❬✶✽❪ ❡ ❬✷❪✱ ❛ ❞✐ss❡rt❛çã♦ ❡stá ♦r❣❛♥✐③❛❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❖ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ tr❛t❛rá ❞♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ♣❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ r❡st❛♥t❡ ❞♦ tr❛❜❛❧❤♦✳ ■♥✐❝✐❛r❡♠♦s ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ á❧❣❡❜r❛ ❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s ✐♠♣♦rt❛♥t❡s✱ ❧♦❣♦ ❛♣ós✱ ❞❡✜♥✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ á❧❣❡❜r❛ ❧✐✈r❡✱ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s✱ ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐✲ ❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s ❡ ♣♦r ✜♠ ♠❛tr✐③❡s ❣❡♥ér✐❝❛s✳ ❆ ❛♣r❡s❡♥t❛çã♦ ❞❡ss❡s ❝♦♥❝❡✐t♦s s❡rá ❛❝♦♠♣❛♥❤❛❞❛ ♣❡❧♦s r❡s✉❧t❛❞♦s ❡ ❡①❡♠♣❧♦s r❡❧❡✈❛♥t❡s ❡ ❡ss❡♥❝✐❛✐s ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞❛ t❡♦r✐❛✳ ◆♦ ❝❛♣ít✉❧♦ ✷✱ ✐♥tr♦❞✉③✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ á❧❣❡❜r❛G✲❣r❛❞✉❛❞❛ ❡ ❞❛r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s✳ ❆♣ós ❡ss❛ ❞❡✜♥✐çã♦✱ ❞❡s❝r❡✈❡r❡♠♦s ❛Zn✲❣r❛❞✉❛çã♦ ❞❡ Mn(F)

❡ ❛Zn✲❣r❛❞✉❛çã♦ ❞❡ FhXi✱ q✉❡ s❡rã♦ ♦ ❢♦❝♦ ❞❡ ❛t❡♥çã♦ ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✱ ♣♦r ✜♠✱

❞❡✜♥✐r❡♠♦s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s G✲❣r❛❞✉❛❞❛s✱ ❞❛♥❞♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s✳ ◆♦ t❡r❝❡✐r♦ ❡ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❝❛♣ít✉❧♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ❞❡s✲

(20)

❝r❡✈❡♠♦s ❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s Zn✲❣r❛❞✉❛❞❛s ❞❛ á❧❣❡❜r❛ Mn(F)✱ q✉❛♥❞♦ ♦ ❝♦r♣♦

F t❡♠ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ❡ q✉❛♥❞♦ ♦ ❝♦r♣♦ F é ✐♥✜♥✐t♦✳ ❙❡❣✉✐♥❞♦ ♦s ♣❛ss♦s ❞❛❞♦s ♣♦r ❱❛s✐❧♦✈s❦② ❬✶✽❪ ❡ ♣♦r ❆③❡✈❡❞♦ ❬✷❪✱ ❡♠ ❝❛❞❛ ❝❛s♦✱ t♦r♥❛r✲s❡✲á ❡✈✐❞❡♥t❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❞♦✐s tr❛❜❛❧❤♦s✱ q✉❡ ❡♠♣r❡❣❛♠ ♠ét♦❞♦s ❡ss❡❝✐❛❧♠❡♥t❡ ❞✐❢❡r❡♥t❡s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ♠❡s♠♦ r❡s✉❧t❛❞♦✳

(21)

❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡s❝r❡✈❡r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s✱ r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s q✉❡ ✉s❛r❡♠♦s ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✳ ❆♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✱ F ❞❡♥♦t❛rá ✉♠ ❝♦r♣♦ q✉❛❧q✉❡r✱ ❡①❝❡t♦ q✉❛♥❞♦ ❡s♣❡❝✐✜❝❛❞♦ ♦ ❝♦♥trár✐♦✳

✶✳✶ ➪❧❣❡❜r❛s

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ á❧❣❡❜r❛✱ s✉❜á❧❣❡❜r❛✱ ✐❞❡❛❧ ❡ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❡ r❡❧❛çõ❡s ❡♥tr❡ ❡st❛s ❡str✉t✉r❛s✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❙❡❥❛ A ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ ✉♠ ❝♦r♣♦ F✳ ❉✐③❡♠♦s q✉❡ A é ✉♠❛ ❋✲á❧❣❡❜r❛ s❡ A é ♠✉♥✐❞♦ ❝♦♠ ✉♠❛ ♦♣❡r❛çã♦ ❜✐♥ár✐❛∗: (A, A)−→A✱ ❝❤❛♠❛❞❛ ♣r♦❞✉t♦✱ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦s✱ a, b, c∈A ❡ t♦❞♦ λ ∈F✱ t❡♠✲s❡✿

• (a+b)∗c=a∗c+b∗c;

• a∗(b+c) = a∗b+a∗c;

• λ(a∗b) = (λa)∗b=a∗(λb)✳

❯s✉❛❧♠❡♥t❡ ❞❡♥♦t❛♠♦sa∗b=a·b =ab✱ ❡ ❛♦ ✐♥✈és ❞❡ ❡s❝r❡✈❡r♠♦sF✲á❧❣❡❜r❛✱ ❡s❝r❡✈❡r❡♠♦s ❛♣❡♥❛s á❧❣❡❜r❛ ❞❡✐①❛♥❞♦ ✐♠♣❧í❝✐t♦ ♦ ❝♦r♣♦ ❋✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ➪❧❣❡❜r❛ é✿

• ❆ss♦❝✐❛t✐✈❛ s❡ ♦ ♣r♦❞✉t♦ ❞❡ Aé ❛ss♦❝✐❛t✐✈♦✱ ✐st♦ é✱ s❡(ab)c=a(bc)✱ ♣❛r❛ q✉❛✐sq✉❡r

a, b, c∈A✳

(22)

✶✳✶✳ ➪❧❣❡❜r❛s

• ❈♦♠✉t❛t✐✈❛ s❡ ♦ ♣r♦❞✉t♦ é ❝♦♠✉t❛t✐✈♦✱ ✐st♦ é✱ s❡ab=ba✱ ♣❛r❛ q✉❛✐sq✉❡r a, b∈A✳ • ❯♥✐tár✐❛ ✭♦✉ ❝♦♠ ✉♥✐❞❛❞❡✮ s❡ ❆ ♣♦ss✉✐ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ♣r♦❞✉t♦✱

✐st♦ é✱ s❡ ❡①✐st❡ 1∈A t❛❧ q✉❡ 1a=a1 =a✱ ♣❛r❛ q✉❛❧q✉❡r a∈A✳

❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❯♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ S ❞❡ ✉♠❛ á❧❣❡❜r❛ A é ❝❤❛♠❛❞♦ ❞❡ s✉❜á❧❣❡❜r❛ s❡ s1s2 ∈S ♣❛r❛ t♦❞♦s s1, s2 ∈S✳

❯♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ I ❞❡ A é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡ A s❡ AI ⊆I ❡ IA ⊆I✳

❊①❡♠♣❧♦ ✶✳✶✳✹✳ ❙❡ L é ✉♠❛ ❡①t❡♥sã♦ ❞♦ ❝♦r♣♦ F✱ ❡♥tã♦ Lé ✉♠❛ á❧❣❡❜r❛ ❝♦♠ ❛s ♦♣❡r❛✲ çõ❡s ❞♦ ❝♦r♣♦✳

❉❡ ❢❛t♦✱ ❜❛st❛ ❝♦♥s✐❞❡r❛r ❡♠ L ❛ ♦♣❡r❛çã♦ ✐♥❞✉③✐❞❛ ❡♠ F✳

❊①❡♠♣❧♦ ✶✳✶✳✺✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦❧✐♥ô♠✐♦s ❡♠n✈❛r✐á✈❡✐s ❝♦♠✉t❛t✐✈❛sx1, ..., xn✱

❞❡♥♦t❛❞♦ ♣♦r F[x1, ..., xn]✳ ❊st❡ ❝♦♥❥✉♥t♦ é ❝❧❛r❛♠❡♥t❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ r❡❧❛çã♦ ❛

s♦♠❛ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡ ♦ ♣r♦❞✉t♦ ♣♦r ❡s❝❛❧❛r❡s ❞♦ ❝♦r♣♦ ❡✱ ❝♦♠ ♦ ♣r♦❞✉t♦ ✉s✉❛❧ ❞❡ ♣♦❧✐♥ô✲ ♠✐♦s✱ é ✉♠❛ á❧❣❡❜r❛✳

❊①❡♠♣❧♦ ✶✳✶✳✻✳ P❛r❛ n∈N✱ ♦ ❡s♣❛ç♦ Mn(F) ❞❡ t♦❞❛s ❛s ♠❛tr✐③❡s n×n ❝♦♠ ❡♥tr❛❞❛s

❡♠ F é ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❝♦♠ ✉♥✐❞❛❞❡✱ ♠✉♥✐❞❛ ❞❛s ♦♣❡r❛çõ❡s ✉s✉❛✐s ❞❡ s♦♠❛ ❡ ♣r♦❞✉t♦✳ ◆❡st❛ á❧❣❡❜r❛ ❞❡st❛❝❛r❡♠♦s ❛s ♠❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s Ei,j✱ ♣❛r❛ 1 ≤ i, j ≤n✱

♦♥❞❡ Ei,j é ❛ ♠❛tr✐③ ❝✉❥❛ ú♥✐❝❛ ❡♥tr❛❞❛ ♥ã♦ ♥✉❧❛ é ✶✱ ♥❛ i✲és✐♠❛ ❧✐♥❤❛ ❡ ♥❛ j✲és✐♠❛

❝♦❧✉♥❛✳ ❊ss❛s ♠❛tr✐③❡s ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ Mn(F) ❝♦♠♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

❊①❡♠♣❧♦ ✶✳✶✳✼✳ ❈♦♥s✐❞❡r❡ ❛s ♠❛tr✐③❡s tr✐❛♥❣✉❧❛r❡s s✉♣❡r✐♦r n×n ❝♦♠ ❡♥tr❛❞❛s ❡♠ F✱ ❞❡♥♦t❛❞❛ ♣♦r Un(F)✱ Un(F) é ✉♠❛ s✉❜á❧❣❡❜r❛ ❞❡ Mn(F)✱ ❝✉❥❛ ❜❛s❡ é ❞❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦

{Ei,j; 1≤i≤j ≤n}✳

❉❡✜♥✐çã♦ ✶✳✶✳✽✳ ❉✐③❡♠♦s q✉❡ ♦ ♣♦❧✐♥ô♠✐♦

x1x2−x2x1 := [x1, x2],

é ♦ ❝♦♠✉t❛❞♦r ❞❡ x1 ❡ x2 ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✷✳ P♦r ✐♥❞✉çã♦✱ ❞❡✜♥✐r❡♠♦s ♦ ❝♦♠✉t❛❞♦r

❞❡ ❝♦♠♣r✐♠❡♥t♦ n ♣♦r

[x1, x2, ..., xn] = [[x1, x2, ..., xn−1], xn].

(23)

✶✳✶✳ ➪❧❣❡❜r❛s

❊①❡♠♣❧♦ ✶✳✶✳✾✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ á❧❣❡❜r❛ A é ✉♠❛ ➪❧❣❡❜r❛ ❞❡ ▲✐❡ s❡ ♣❛r❛ t♦❞♦s a, b, c∈A✱ ✈❛❧❡✿

  

 

a2 ₌_aa_{= 0} _{✭❛♥t✐✲❝♦♠✉t❛t✐✈❛✮}

(ab)c+ (bc)a+ (ca)b = 0 ✭✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐✮

❊①❡♠♣❧♦ ✶✳✶✳✶✵✳ ❙❡A é ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛✱ ♦ ♣r♦❞✉t♦ ❞❛❞♦ ♣♦r [a, b] ❞❡✜♥❡ ❡♠ A ✉♠❛ ♥♦✈❛ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛✱ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r A(−)_{✱ ❡ ❝♦♠♦} _[_{a, a}_{] = 0} _❡ _[_{a, b, c}_{] +}

[b, c, a] + [c, a, b] = 0 ♣❛r❛ q✉❛✐sq✉❡r a, b, c∈A✱ s❡❣✉❡ q✉❡ A(−) _{é ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳}

❉❡✜♥✐çã♦ ✶✳✶✳✶✶✳ ❙❡❥❛♠ A ❡ B ❞✉❛s á❧❣❡❜r❛s✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r f :A−→B é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s s❡f(ab) =f(a)f(b)✱ ♣❛r❛ t♦❞♦sa, b∈A✳

❉✐③❡♠♦s q✉❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s f : A −→ B é ✉♠ ✐s♦♠♦r✜s♠♦ s❡ ❡❧❡ ❢♦r ❜✐❥❡t✐✈♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✷✳ ❙❡❥❛ I ✉♠ ✐❞❡❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ A✳ ❙❡ a∈A✱ ❞❡♥♦t❡ ♣♦r✿

a=a+I ={a+i; i∈I}.

❖❜s❡r✈❡ q✉❡ a é ❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ a s❡❣✉♥❞♦ ❛ r❡❧❛çã♦ ∼ ❞❡✜♥✐❞❛ ❡♠ A ♣♦r a∼b⇔(a−b)∈I✳

❖ ❝♦♥❥✉♥t♦ A

I ={a; a ∈ A} é ♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ ❞❡ A ♣♦r I✳ ❘❡❧❡♠❜r❛♠♦s q✉❡ ❝♦♠ ❛s ♦♣❡r❛çõ❡s✿ _

 

 

a1+a2 =a1+a2

αa=αa, ♦♥❞❡ a1, a2 ∈A ❡ α∈F✱ t❡♠♦s q✉❡

A

I é ✉♠ F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❆❣♦r❛ ❞❡✜♥✐♥❞♦ a1∗a2 =a1∗a2,

t❡♠♦s q✉❡ A

I é ✉♠❛ á❧❣❡❜r❛✱ ❝❤❛♠❛❞❛ á❧❣❡❜r❛ q✉♦❝✐❡♥t❡✳ ❯♠ ❝♦♥❝❡✐t♦ q✉❡ s❡rá ❢r❡✲ q✉❡♥t❡♠❡♥t❡ ✉s❛❞♦ ♥♦ ❞❡❝♦rr❡r ❞♦ ❝❛♣ít✉❧♦ ✸ é ♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ ♦✉ s❡❥❛✱ ❞❛❞♦s A ✉♠❛ á❧❣❡❜r❛ ❡ I ✉♠ ✐❞❡❛❧ ❞❡ A s❡ a, b ∈ A✱ ❞✐③❡♠♦s q✉❡ a é ❝♦♥❣r✉❡♥t❡ ❛ b ♠ó❞✉❧♦ ■✱ s❡ a−b∈I✱ ✉s✉❛❧♠❡♥t❡ ❡s❝r❡✈❡♠♦s a≡b mod I✳

❚❡♦r❡♠❛ ✶✳✶✳✶✸✳ ❙❡❥❛ f : A1 −→ A2 ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✳ ❖ ♥ú❝❧❡♦ ❞❡ f✱

❞❡♥♦t❛❞♦ ♣♦r ❑❡r(f) = {a∈A1;f(a) = 0} é ✉♠ ✐❞❡❛❧ ❞❡ A1 ❡

A1

Ker(f) ∼=Im(f)✳

(24)

✶✳✷✳ ➪❧❣❡❜r❛ ▲✐✈r❡

❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ é ❛♥á❧♦❣❛ ❛♦ ❝❛s♦ ❞❡ ❛♥é✐s✳

✶✳✷ ➪❧❣❡❜r❛ ▲✐✈r❡

◆❡st❡ s❡çã♦✱ ❡st✉❞❛r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ á❧❣❡❜r❛ ❧✐✈r❡✱ ♣❛r❛ ✐st♦✱ s❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ❡ X ⊆ A✳ ❈♦♥s✐❞❡r❡ P al(X) ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❛❧❛✈r❛s ❢♦r♠❛❞❛s ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❡ X✳ ❙❡

t♦❞♦ ❡❧❡♠❡♥t♦ ❡♠ A é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ❡❧❡♠❡♥t♦s ❡♠ P al(X)✱ ❡♥tã♦ ❞✐③❡♠♦s

q✉❡ A é ❣❡r❛❞♦ ♣♦r X✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❉❡✜♥✐r❡♠♦s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♣❛❧❛✈r❛ xi1· · ·xin ❝♦♠♦ s❡♥❞♦ n✱ q✉❛♥❞♦ n = 0✱ ❝❤❛♠❛r❡♠♦s ❡ss❛ ♣❛❧❛✈r❛ ❞❡ ♣❛❧❛✈r❛ ✈❛③✐❛ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r ✶✳

❆❧é♠ ❞✐ss♦✱ ❞✐③❡♠♦s q✉❡ ❞✉❛s ♣❛❧❛✈r❛s xi1· · ·xin ❡ xj1· · ·xjm sã♦ ✐❣✉❛✐s s❡ n = m ❡ i1 =j1, ..., in =jm✳

❉❡✜♥✐çã♦ ✶✳✷✳✷✳ ❙❡❥❛ V ✉♠❛ ❝❧❛ss❡ ❞❡ á❧❣❡❜r❛s ❡ A ∈ V ✉♠❛ á❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ X✳ ❆ á❧❣❡❜r❛ A é ❝❤❛♠❛❞❛ ❧✐✈r❡ ♥❛ ❝❧❛ss❡ ❱✱ ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞❛ ♣♦r ❳✱ s❡ ♣❛r❛ q✉❛❧q✉❡r á❧❣❡❜r❛ R ∈ V✱ t♦❞❛ ❢✉♥çã♦ g : X −→ R ♣♦❞❡ s❡r ❡st❡♥❞✐❞❛ ❛ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s G:A−→R✳

❉❡✜♥✐çã♦ ✶✳✷✳✸✳ ❙❡❥❛X ✉♠ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧ ✐♥✜♥✐t♦✳ ❉❡♥♦t❡ ♣♦rFhXi ♦F✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❜❛s❡ ❢♦r♠❛❞❛ ♣♦r ✶ ❡ ♣❡❧❛s ♣❛❧❛✈r❛s

xi1xi2· · ·xin,

♦♥❞❡ xi1, xi2, ..., xin ∈ X ❡ n ≥ 0✳ ❉❡✜♥✐r❡♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♥tr❡ ❞✉❛s ♣❛❧❛✈r❛s ♣♦r ❝♦♥❝❛t❡♥❛çã♦✱ ♦✉ s❡❥❛✿

(xi1xi2· · ·xin)(xj1xj2· · ·xjn) =xi1xi2· · ·xinxj1xj2· · ·xjn.

❊st❡♥❞❡♥❞♦ ♣♦r ❧✐♥❡❛r✐❞❛❞❡ ❡st❡ ♣r♦❞✉t♦ ♣❛r❛ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ FhXi✱ ❡♥tã♦ FhXi é

á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❝♦♠ ✉♥✐❞❛❞❡ ✭♣❛❧❛✈r❛ ✈❛③✐❛✮ ❣❡r❛❞❛ ♣♦r X✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✹✳ ❆ á❧❣❡❜r❛ FhXi é ❧✐✈r❡ ♥❛ ❝❧❛ss❡ ❞❡ t♦❞❛s ❛s á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s

✉♥✐tár✐❛s✱ ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞❛ ♣♦r X✳

(25)

✶✳✸✳ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s

❉❡♠♦♥str❛çã♦✿ ❙❡ p ∈ FhXi✱ ❞❡♥♦t❛♠♦s p = p(x1, ..., xn) s❡ ❡❧❡ é ❢♦r♠❛❞♦ ♣❡❧♦s

❡❧❡♠❡♥t♦s x1, ..., xn✳ ❙❡❥❛ g : X −→ R ✉♠❛ ❢✉♥çã♦✱ ♦♥❞❡ R é ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛✲

t✐✈❛ ❝♦♠ ✉♥✐❞❛❞❡✳ ❉❡♥♦t❡ g(xi) = ri✱ ♣❛r❛ t♦❞♦ i ❡ ❞❡✜♥✐r❡♠♦s G : FhXi −→ R ♣♦r

G(p(x1, ..., xn)) =p(r1, ..., rn)✳

❊♥tã♦G é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s q✉❡ ❡st❡♥❞❡ g✳

❉❡✜♥✐çã♦ ✶✳✷✳✺✳ ❙❡❥❛ A ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❜❛s❡ ♦r❞❡♥❛❞❛

{ei, i∈I} ✭✶✳✶✮

♦♥❞❡ I é ✉♠ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧✳

❆ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥ E(A) é ❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❣❡r❛❞❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ ✶✳✶ q✉❡

s❛t✐s❢❛③✿

eiej =−ejei,

♣❛r❛ t♦❞♦i, j ∈I ❡ t❛♠❜é♠

e2i = 0,

s❡ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡F é ✷✳ ❉❡st❛❝❛♠♦s ♥❛ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥ ♦s s❡❣✉✐♥t❡s s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s✿

• E(0)_{✱ ❣❡r❛❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦} _{₁_{, e}

i1ei2· · ·eim✿ m é ♣❛r⑥

• E(1)_{✱ ❣❡r❛❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦} _{_e

i1ei2· · ·eim✿ m é í♠♣❛r⑥

❚❡♠♦s q✉❡ E = E(0)_⊕_E(1)_{✱ ❝♦♠♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❱❡r❡♠♦s ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ q✉❡ ❛}

á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥✱ ❝♦♠ ❡st❛ ❡str✉t✉r❛ ❞❡ s♦♠❛ ❞✐r❡t❛✱ é ✉♠❛ á❧❣❡❜r❛ ❣r❛❞✉❛❞❛✳

✶✳✸ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s

◆❡st❛ s❡çã♦✱ ✈❡r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s✱ ❛❧é♠ ❞❡ ❡①❡♠♣❧♦s ♣❛r❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞❡st❛ t❡♦r✐❛✳ ❉❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡✱ X ❞❡♥♦t❛rá ♦ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧ X = {x1, x2, ...} ❡ FhXi é ❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❧✐✈r❡✱ ❧✐✈r❡♠❡♥t❡

❣❡r❛❞❛ ♣♦r X✳

(26)

✶✳✸✳ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s

❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❡ f(x1, ..., xn) =f ∈ FhXi✳ ❉✐③❡♠♦s

q✉❡f é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ As❡ f(r1, ..., rn) = 0 ♣❛r❛ t♦❞♦s r1, ..., rn ∈A✳

❉❡♥♦t❛♠♦s ♣♦r T(A) ♦ ❝♦♥❥✉♥t♦ ❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ A✳ ❙❡ T(A) 6= {0}

❞✐③❡♠♦s q✉❡ A é ✉♠❛ P■ ✲ á❧❣❡❜r❛✳

❊①❡♠♣❧♦ ✶✳✸✳✷✳ ❖ ♣♦❧✐♥ô♠✐♦ ♥✉❧♦ f = 0 é s❡♠♣r❡ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛

q✉❛❧q✉❡r á❧❣❡❜r❛ A ❡ é ❝❤❛♠❛❞♦ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ tr✐✈✐❛❧ ❞❡ ❆✳

❊①❡♠♣❧♦ ✶✳✸✳✸✳ ❙❡ A ❢♦r ✉♠❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛✱ ❡♥tã♦ A é ✉♠❛ P■✲á❧❣❡❜r❛✱ ❛ s❛❜❡r✱ f(x1, x2) = [x1, x2]∈T(A)✳

❉❡✜♥✐çã♦ ✶✳✸✳✹✳ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛✱ ♦ ❝❡♥tr♦ ❞❡ A✱ ❞❡♥♦t❛❞♦ ♣♦r Z(A)✱ é ❞❛❞♦ ♣❡❧♦

❝♦♥❥✉♥t♦✿

Z(A) ={x∈A;xa=ax, ♣❛r❛ t♦❞♦ a∈A}.

❊①❡♠♣❧♦ ✶✳✸✳✺✳ ❙❡❥❛ E ❛ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥ ❞❡✜♥✐❞❛ ❡♠ ✶✳✷✳✺ ❡♥tã♦✿

f(x1, x2, x3) = [x1, x2, x3]∈T(E).

❉❡ ❢❛t♦✱ ❞❡♥♦t❡ ♣♦r B ❛ ❜❛s❡ ❞❡ E ❢♦r♠❛❞❛ ♣♦r b =ei1ei2· · ·ein✱ ♦♥❞❡ i1 < i2 <· · ·< in ❡ n ≥0✳ ◆♦t❡ q✉❡✿

(ei1ei2· · ·ein)(ej1ej2· · ·ejn) = (−1)

σ₍_e

j1ej2· · ·ejn)(ei1ei2· · ·ein), ♦♥❞❡ (−1)σ _{é ♦ s✐♥❛❧ ❞❛ ♣❡r♠✉t❛çã♦} _σ _∈_S

n ❡ Sn é ♦ ❣r✉♣♦ s✐♠étr✐❝♦ ❞❡ ❣r❛✉ n✳ ❆ss✐♠✱

s❡ n é ♣❛r✱ ❡♥tã♦ b∈ Z(E) ✭❝❡♥tr♦ ❞❡ E✮✳ ❆❣♦r❛✱ s❡❥❛♠ b1, b2, b3 ∈B t❡♠♦s✿

• [b1, b2, b3] = [[b1, b2], b3] = 0 s❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ b1, b2 ♦✉ b3 é ♣❛r✳

• [b1, b2, b3] = [[b1, b2], b3] = [2b1b2, b3] = 0 s❡ ♦s ❝♦♠♣r✐♠❡♥t♦s ❞❡ b1, b2 ❡ b3 sã♦

í♠♣❛r❡s✳

❊①❡♠♣❧♦ ✶✳✸✳✻✳ ❚♦♠❡ ❛ á❧❣❡❜r❛ M2(F) ❞❛s ♠❛tr✐③❡s 2×2 ❝♦♠ ❡♥tr❛❞❛s ♥♦ ❝♦r♣♦ F✱ ♦

♣♦❧✐♥ô♠✐♦✿

f(x1, x2, x3) = [[x1, x2]2, x3],

❝❤❛♠❛❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦ ❞❡ ❍❛❧❧✱ é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ M2(F)✳

❈♦♠ ❡❢❡✐t♦✱ s❡❥❛

(27)

✶✳✸✳ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s

a=



 a11 a12

a21 a22





❖ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ a é p(x) =x2₋_tr❛ç♦₍_a₎_x₊_❞❡t₍_a₎_{✱ ❛❣♦r❛ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛}

❞❡ ❈❛②❧❡②✲❍❛♠✐❧t♦♥✱ s❡❣✉❡ q✉❡ p(a) =a2₋_tr❛ç♦₍_a₎_a₊_❞❡t₍_a₎_Id 2 = 0✳

❙❡ b1, b2 sã♦ ♠❛tr✐③❡s q✉❛✐sq✉❡r ❡♠ M2(F) ❡ a = [b1, b2] ❡♥tã♦ tr❛ç♦(a) = 0 ❡ t❛♠❜é♠

[b1, b2]2 =−❞❡t([b1, b2])Id2 ∈ Z(M2(F))✳ ▲♦❣♦✱ ♣❛r❛ b3 ∈M2(F) t❡♠♦s [[b1, b2]2, b3] = 0

❊①❡♠♣❧♦ ✶✳✸✳✼✳ ❆ á❧❣❡❜r❛ ❞❛s ♠❛tr✐③❡s tr✐❛♥❣✉❧❛r❡s s✉♣❡r✐♦r ❞❡ ♦r❡❞❡♠ n✱ Un(F)✱ s❛✲

t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡

f(x1, x2, ..., x2n) = [x1, x2]· · ·[x2n−1, x2n].

❉❡ ❢❛t♦✱ s❡ r1, r2 ∈Un(F)✱ ❡♥tã♦ [r1, r2] ♣❡rt❡♥❝❡ ❛ Un(F) ❡ ♣♦ss✉✐ ❞✐❛❣♦♥❛❧ ♥✉❧❛✳ ❈♦♠♦

♦ ♣r♦❞✉t♦ ❞❡n ♠❛tr✐③❡s ❡♠Un(F)✱ ❝♦♠ ❞✐❛❣♦♥❛❧ ♥✉❧❛✱ é ❛ ♠❛tr✐③ ♥✉❧❛✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

❉❡✜♥✐çã♦ ✶✳✸✳✽✳ ❋✐①❛❞♦n ❞❡♥♦t❡ ♣♦r

Stn(x1, ..., xn) =

X

σ∈Sn

(−1)σxσ(1)· · ·xσ(n),

♦ ♣♦❧✐♥ô♠✐♦ st❛♥❞❛r❞ ❞❡ ❣r❛✉ ♥✱ ♦♥❞❡ Sn é ♦ ❣r✉♣♦ s✐♠étr✐❝♦ ❞❡ ❣r❛✉ n ❡ (−1)σ é ♦

s✐♥❛❧ ❞❛ ♣❡r♠✉t❛çã♦ σ✳

❊①❡♠♣❧♦ ✶✳✸✳✾✳ ❚♦❞❛ á❧❣❡❜r❛ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ é ✉♠❛ P I✲á❧❣❡❜r❛✳

❉❡ ❢❛t♦✱ s❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ B = {v1, ..., vm} ✉♠❛ ❜❛s❡ ♣❛r❛ A✳

❙✉♣♦♥❤❛ m < n ❡ t♦♠❡a1, ..., an ∈ B✱ ❡s❝r❡✈❛ ai = m

X

j=1

αijvj✳ ❊♥tã♦ t❡♠♦s q✉❡

Stn(a1, ..., an) = m

X

j1=1

· · ·

m

X

jn=1

(α1j1· · ·αnjn)Stn(vj1, ..., vjn) = 0,

♦✉ s❡❥❛✱ s❡ ✉♠❛ á❧❣❡❜r❛ t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛✱ t❡♠♦s q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ st❛♥❞❛r❞ ❞❡ ❣r❛✉n é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ ❡st❛ á❧❣❡❜r❛✳

❉❡✜♥✐çã♦ ✶✳✸✳✶✵✳ ❉✐③❡♠♦s q✉❡ ✉♠ ✐❞❡❛❧ I ❞❡ FhXi é ✉♠ ❚✲✐❞❡❛❧✱ s❡ ϕ(I) ⊆ I ♣❛r❛ t♦❞♦ ❡♥❞♦♠♦r✜s♠♦ ϕ ❞❡ FhXi✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ I é ✉♠ ❚✲✐❞❡❛❧ s❡ é ✐♥✈❛r✐❛♥t❡ s♦❜ t♦❞♦s ♦s ❡♥❞♦♠♦r✜s♠♦s ❞❡ FhXi✳

(28)

✶✳✸✳ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s

Pr♦♣♦s✐çã♦ ✶✳✸✳✶✶✳ ❖ ❝♦♥❥✉♥t♦ T(A) ❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ✉♠❛ á❧❣❡❜r❛ A é ✉♠ T✲✐❞❡❛❧ ❞❡ FhXi✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ I é ✉♠ T✲✐❞❡❛❧ ❞❡ FhXi✱ ❡♥tã♦ ❡①✐st❡ ❛❧❣✉♠❛

á❧❣❡❜r❛ R t❛❧ q✉❡ T(R) =I

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ f(x1, ..., xn)∈T(A) ❡ϕ ∈❊♥❞(FhXi)✱ s❡ ψ :FhXi −→A é ✉♠

❤♦♠♦♠♦r✜s♠♦ q✉❛❧q✉❡r✱ ❡♥tã♦ψ(ϕ(f)) = (ψ◦ϕ)(f) = 0✱ ♣♦✐s ψ◦ϕ :FhXi −→A é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❡ f ∈T(A)✳ ▲♦❣♦✱ϕ(f)∈❑❡r(ψ) ❡ ♣♦rt❛♥t♦ ϕ(f)∈T(A)✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ I ✉♠ T✲✐❞❡❛❧ ❞❡ FhXi✳ ❚♦♠❡♠♦s ❛ á❧❣❡❜r❛ q✉♦❝✐❡♥t❡ R = FhXi

I ❡ ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛ π : FhXi −→ FhXi

I ✳ ❙❡ f ∈ T(R)✱ ❡♥tã♦ f ∈ ❑❡r(π) = I✱ ❛s✲ s✐♠ t❡♠♦s✱ T(R) ⊆ I✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ f(x1, ..., xn) ∈ I ❡ g1, ..., gn ∈ FhXi✱ ❡♥tã♦✱

f(g1, ..., gn)∈I ❡ ❞❛í f(g1, ..., gn) =f(g1, ..., gn) = 0✳ ▲♦❣♦✱f ∈T(R)✳

❉❡✜♥✐çã♦ ✶✳✸✳✶✷✳ ❙❡❥❛ S ⊆ FhXi✱ ♦ T✲✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r S✱ ❞❡♥♦t❛❞♦ ♣♦r hSiT_{✱ é ♦}

❝♦♥❥✉♥t♦✿

hSiT ₌_❙♣❛♥

F {p1 ϕ(f) p2 :f ∈S, ϕ∈❊♥❞(FhXi), p1, p2 ∈FhXi}.

❊①❡♠♣❧♦ ✶✳✸✳✶✸✳ ❙❡A é ✉♠❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛ ❡ ❝♦♠ ✉♥✐❞❛❞❡ ❡F é ✉♠ ❝♦r♣♦ ✐♥✜♥✐t♦✱ ❡♥tã♦

T(A) = h[x1, x2]iT.

❊①❡♠♣❧♦ ✶✳✸✳✶✹✳ ❊♠ ❬✶✸❪ ❘❛③♠②s❧♦✈ ❞❡t❡r♠✐♥♦✉ ✉♠❛ ❜❛s❡ ♣❛r❛ ❛s ✐❞❡♥t✐❞❛❞❡s ❞❡M2(F)

❝♦♠ ✾ ❡❧❡♠❡♥t♦s ♥♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ F é ✐❣✉❛❧ ❛ ③❡r♦ ❡ ❡♠ ❬✻❪ ❉r❡♥s❦✐ ♠❡❧❤♦r♦✉ ❡st❡ r❡s✉❧t❛❞♦ ❞❛♥❞♦ ✉♠❛ ❜❛s❡ ♠✐♥✐♠❛❧ ❝♦♠ ❞✉❛s ✐❞❡♥t✐❞❛❞❡s✱ ❛ s❛❜❡r✱

T(M2(F)) =hSt4(x1, x2, x3, x4),[[x1, x2]2, x3]iT.

■st♦ é✱ ❛ ❜❛s❡ ❞❛s ✐❞❡♥t✐❞❛❞❡s é ❣❡r❛❞♦ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦ st❛♥❞❛r❞ ❞❡ ❣r❛✉ ✹ ❡ ♣❡❧♦ ♣♦❧✐♥ô♠✐♦ ❞❡ ❍❛❧❧✳

❆ ❞❡♠♦♥str❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥♦s ♣ró①✐♠♦s ❡①❡♠♣❧♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛✲ ❞❛s ❡♠ ❬✺❪ ♣á❣✐♥❛s50 ❡ 52r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❊①❡♠♣❧♦ ✶✳✸✳✶✺✳ ❙❡ F ❢♦r ✉♠ ❝♦r♣♦ ✐♥✜♥✐t♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡ ✷✱ ❡♥tã♦

T(E) =h[x1, x2, x3]iT,

(29)

✶✳✹✳ P♦❧✐♥ô♠✐♦s ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s

♦♥❞❡ E é ❛ á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥ ✐♥✜♥❛t❛♠❡♥t❡ ❣❡r❛❞❛ ♣♦r e1, e2, ...✳

❊①❡♠♣❧♦ ✶✳✸✳✶✻✳ ❙❡❥❛F ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ✵ ❡ s❡❥❛Un(F)❛ á❧❣❡❜r❛ ❞❛s ♠❛tr✐③❡s

tr✐❛♥❣✉❧❛r❡s s✉♣❡r✐♦r✳ ❊♥tã♦

T(Un(F)) =h[x1, x2][x3, x4]· · ·[x2n−1, x2n]iT.

✶✳✹ P♦❧✐♥ô♠✐♦s ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s

◆❡st❛ s❡çã♦✱ ❡st✉❞❛r❡♠♦s ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s✱ q✉❡ s❡rã♦ ❡ss❡♥✲ ❝✐❛✐s ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✱ ♣♦✐s✱ q✉❛♥❞♦ ✉♠ ❝♦r♣♦ é ✐♥✜♥✐t♦ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❣❡r❛✲ ❞♦r ❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❛ á❧❣❡❜r❛ s♦❜r❡ ❡st❡ ❝♦r♣♦ é ❢♦r♠❛❞♦ ♣♦r ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦s✱ ❡ q✉❛♥❞♦ ♦ ❝♦r♣♦ t❡♠ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❣❡r❛❞♦r ❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❛ á❧❣❡❜r❛ é ❢♦r♠❛❞♦ ♣❡❧♦s s❡✉s ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐❧✐♥❡❛r❡s✱ ❡ ❡st❡ ❢❛t♦ s❡rá ❢r❡q✉❡♥t❡♠❡♥t❡ ✉s❛❞♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❝❛♣ít✉❧♦ ✸✳

❉❡✜♥✐çã♦ ✶✳✹✳✶✳ ❙❡❥❛♠ m ∈ FhXi ✉♠ ♠♦♥ô♠✐♦✱ f ∈ FhXi ✉♠ ♣♦❧✐♥ô♠✐♦ ❡ xi ∈ X✳

❉❡✜♥✐r❡♠♦s✿

❛✮ ❖ ❣r❛✉ ❞❡ m ❡♠ xi ✭degxim✮ ❝♦♠♦ s❡♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡ ❛ ✈❛r✐á✈❡❧ xi ❛♣❛r❡❝❡ ❡♠ m✳

❜✮ ❖ ❣r❛✉ ❞❡ f ❡♠ xi ✭degxif✮ ❝♦♠♦ s❡♥❞♦ ♦ ♠❛✐♦r ❣r❛✉ ❡♠ xi ❞❡ ❛❧❣✉♠ ♠♦♥ô♠✐♦ ❞❡ f

❯♠ ♣♦❧✐♥ô♠✐♦ f ∈ FhXi é ❞✐t♦ ❤♦♠♦❣ê♥❡♦ ❡♠ xi s❡ t♦❞♦s ♦s s❡✉s ♠♦♥ô♠✐♦s tê♠ ♦

♠❡s♠♦ ❣r❛✉ ❡♠ xi✳ ❖ ♣♦❧✐♥ô♠✐♦ f é ❞✐t♦ ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦ q✉❛♥❞♦ é ❤♦♠♦❣ê♥❡♦ ❡♠

t♦❞❛s ❛s ✈❛r✐á✈❡✐s✳

❙❡m =m(x1, x2, . . . , xn) é ✉♠ ♠♦♥ô♠✐♦ ❞❡ FhXi✱ ❞❡✜♥✐r❡♠♦s ♦ ♠✉❧t✐❣r❛✉ ❞❡ m ❝♦♠♦

s❡♥❞♦ ❛n✲✉♣❧❛(a1, a2, . . . , an)♦♥❞❡ai = degxim✳ ❙❡f ∈FhXi✱ ❛ s♦♠❛ ❞❡ t♦❞♦s ♦s ♠♦♥ô✲ ♠✐♦s ❞❡f ❝♦♠ ✉♠ ❞❛❞♦ ♠✉❧t✐❣r❛✉ é ❞✐t❛ s❡r ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ♠✉❧t✐✲❤♦♠♦❣ê♥❡❛ ❞❡f✳ ❖❜s❡r✈❡ ❡♥tã♦ q✉❡f é ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ ❝♦♠♣♦♥❡♥t❡ ♠✉❧t✐✲❤♦♠♦❣ê♥❡❛✳

(30)

✶✳✹✳ P♦❧✐♥ô♠✐♦s ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s

❙❡♥❞♦ f(x1, . . . , xn)∈FhXi ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ m ❡♠ xi ❡λ ∈F✱ t❡♠♦s

f(x1, . . . , xi−1, λxi, xi+1, . . . , xn) = λmf(x1, . . . , xi−1, xi, xi+1, . . . , xn).

P❛rt✐❝✉❧❛r♠❡♥t❡✱ s❡f é ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ ✶ ❡♠ xi ✭♦✉ ❧✐♥❡❛r ❡♠ xi✮✱ t❡♠♦s

f(x1, . . . , xi−1, λxi, xi+1, . . . , xn) = λf(x1, . . . , xi−1, xi, xi+1, . . . , xn).

❚❛♠❜é♠ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ s❡ f(x1, . . . , xn)é ❧✐♥❡❛r ❡♠ xi✱ ❡♥tã♦

f(x1, . . . , xi−1, y1+. . .+ym, xi+1, . . . , xn) = m

X

j=1

f(x1, . . . , xi−1, yj, xi+1, . . . , xn).

❉✐③❡♠♦s q✉❡ ✉♠ ♣♦❧✐♥ô♠✐♦f(x1, x2, . . . , xn)∈FhXié ♠✉❧t✐❧✐♥❡❛r s❡ é ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦

❝♦♠ ♠✉❧t✐❣r❛✉ (1,1, . . . ,1)✱ ♦✉ s❡❥❛✱ s❡ ❡♠ ❝❛❞❛ ♠♦♥ô♠✐♦ ❝❛❞❛ ✈❛r✐á✈❡❧ t❡♠ ❣r❛✉ ❡①❛t❛✲

♠❡♥t❡ ✶✳ ◆❡st❡ ❝❛s♦✱ f t❡♠ ❛ ❢♦r♠❛

X

σ∈Sn

ασxσ(1)xσ(2). . . xσ(n) , ❝♦♠ ασ ∈F.

❊①❡♠♣❧♦ ✶✳✹✳✷✳ ❙❡❥❛ f(x1, x2, x3) = x1x3+ 2x2x3x2−x22x3✳ ❚❡♠♦s q✉❡ x1x3 ❡2x2x3x2−

x2

2x3 sã♦ ❛s ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ♠✉❧t✐✲❤♦♠♦❣ê♥❡❛s ❞❡f✱ s❡♥❞♦ q✉❡ ❛ ♣r✐♠❡✐r❛ t❡♠ ♠✉❧t✐❣r❛✉

(1,0,1) ❡♥q✉❛♥t♦ ❛ s❡❣✉♥❞❛ t❡♠ ♠✉❧t✐❣r❛✉ (0,2,1)✳

❊①❡♠♣❧♦ ✶✳✹✳✸✳ ❖ ♣♦❧✐♥ô♠✐♦ st❛♥❞❛r❞ ❞❡ ❣r❛✉n✱ é ♠✉❧t✐❧✐♥❡❛r ❞❡ ♠✉❧t✐❣r❛✉(1,1, . . . ,1)

| {z }

n

✱

t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ♠✉❧t✐❧✐♥❡❛r ❞❡ ❣r❛✉ n✳

❉❡✜♥✐çã♦ ✶✳✹✳✹✳ ❉♦✐s ❝♦♥❥✉♥t♦s ❞❡ ♣♦❧✐♥ô♠✐♦s sã♦ P I✲❡q✉✐✈❛❧❡♥t❡s ♦✉ ❛♣❡♥❛s ❡q✉✐✲ ✈❛❧❡♥t❡s s❡ ❡❧❡s ❣❡r❛♠ ♦ ♠❡s♠♦ T✲✐❞❡❛❧✳

❚❡♦r❡♠❛ ✶✳✹✳✺✳ ❙❡❥❛f(x1, ..., xn) = n

X

i=0

fi(x1, ..., xn)✱ ♦♥❞❡fi é ❛ ❝♦♠♣♦♥❡♥t❡ ❤♦♠♦❣ê♥❡❛

❞❡ f ❞❡ ❣r❛✉ i ❡♠ x1✳

(i) ❙❡ F ❝♦♥té♠ ♠❛✐s q✉❡ n ❡❧❡♠❡♥t♦s ❡♥tã♦

hf0, f1, ..., fniT =hfiT.

(ii) ❙❡ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ F é ✐❣✉❛❧ ❛ ③❡r♦✱ ❡♥tã♦ f é P I✲❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐❧✐♥❡❛r❡s✳

(31)

✶✳✹✳ P♦❧✐♥ô♠✐♦s ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s

❉❡♠♦♥str❛çã♦✿ ❉❡♠♦♥str❛çã♦ ❞♦ ✐t❡♠ (i)✿

❆ ✐♥❝❧✉sã♦⊇ é ❞✐r❡t❛✳

P❛r❛ ♣r♦✈❛r ❛ ♦✉tr❛ ✐♥❝❧✉sã♦ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡✿

f0, f1, ..., fn ∈ hfiT.

❙❡❥❛♠ α0, α1, ..., αn∈F ❞✐st✐♥t♦s✳ ❈♦♠♦hfiT é T✲✐❞❡❛❧ t❡♠♦s q✉❡

f(αjx1, x2, ..., xn) = n

X

i=0

fi(αjx1, x2, ..., xn) = n

X

i=0

αijfi(x1, x2, ..., xn),

❡stá ❡♠hfiT _{♣❛r❛ t♦❞♦} _j _{= 0}_,₁_{, ..., n}_{✳ ❊♠ ♥♦t❛çã♦ ♠❛tr✐❝✐❛❧ ✜❝❛✿}

       

1 α0 α20 · · · α0n

1 α1 α21 · · · α1n

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

1 αn α2n · · · αnn

       

| {z }

A         f0 f1 ✳✳✳ fn         =        

f(α0x1, x2, ..., xn)

f(α1x1, x2, ..., xn)

✳✳✳

f(αnx1, x2, ..., xn)

       

❆ ♠❛tr✐③A✱ é ❛ ♠❛tr✐③ ❞❡ ❱❛♥❞❡r♠♦♥❞❡ q✉❡ t❡♠ ❞❡t❡r♠✐♥❛♥t❡

det(A) = Y

1≤i<j≤n

(αj −αi)6= 0.

▲♦❣♦✱ A é ✐♥✈❡rtí✈❡❧✱ ❛ss✐♠ t❡♠♦s q✉❡✿

        f0 f1 ✳✳✳ fn         =        

b11 b12 · · · b1n+1

b21 b22 · · · b2n+1

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ bn+1 1 bn+1 2 · · · bn+1n+1

       

| {z }

A−1

       

f(α0x1, x2, ..., xn)

f(α1x1, x2, ..., xn)

✳✳✳

f(αnx1, x2, ..., xn)

        ❡ ♣♦rt❛♥t♦✱

f0, f1, ..., fn∈Span{f(α0x1, x2, ..., xn), ..., f(αnx1, x2, ..., xn)} ⊆ hfiT.

❉❡♠♦♥str❛çã♦ ❞♦ ✐t❡♠ (ii)✿

❊s❝r❡✈❛

f(x1, ..., xm) =

X

n1,...,nm≥0

f(n1,...,nm)₍_x

1, ..., xm),

(32)

✶✳✹✳ P♦❧✐♥ô♠✐♦s ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s

♦♥❞❡ f(n1,...,nm) _∈ _F_h_x

1, ..., xmi é ❛ ❝♦♠♣♦♥❡♥t❡ ♠✉❧t✐✲❤♦♠♦❣ê♥❡❛ ❞❡ f ❝♦♠ ♠✉❧t✐❣r❛✉

(n1, ..., nm)✳

❆♣❧✐❝❛♥❞♦ ❛❧❣✉♠❛s ✈❡③❡s ♦ ✐t❡♠(i) t❡r❡♠♦s✿

hfiT ₌_h_f(n1,...,nm)_;_n

1, ..., nm ≥0iT.

❙❡ ♣r♦✈❛r♠♦s q✉❡ f(n1,...,nm) é_{P I}✲❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ♠✉❧t✐❧✐♥❡❛r❡s s(n1,...,nm) ❡♥tã♦

hfiT =h [

n1,...,nm≥0

s(n1,...,nm)_iT_.

❙❡{xij :i= 1, ...m ❡ j = 1, ..., ni} é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ❞✐st✐♥t❛s ❡♠ X ❡♥tã♦✿

f(n1,...,nm)₍_x

11+· · ·+x1n1, ..., xm1+· · ·+xmnm) =g, ❡stá ❡♠ hf(n1,...,nm)_iT✳

❉❡♥♦t❛♥❞♦ ♣♦rh(n1,...,nm)₍_x

11, ..., x1n1, ..., xm1, ..., xmnm)❛ ❝♦♠♣♦♥❡♥t❡ ♠✉❧t✐❧✐♥❡❛r ❞❡g ♥❛s ✈❛r✐á✈❡✐s xij✬s✱ t❡♠♦s q✉❡✱ ♣❡❧❛ ♣❛rt❡ (i)

h(n1,...,nm) _{∈ h}_f(n1,...,nm)_iT_.

◆♦t❡ t❛♠❜é♠ q✉❡

h(n1,...,nm)₍_x

1, ..., x1

| {z }

n1❢❛t♦r❡s

, ..., xm, ..., xm)

| {z }

nm❢❛t♦r❡s

= (n1!)· · ·(nm!)

| {z }

6

=0

f(n1,...,nm)_.

▲♦❣♦✱

hf(n1,...,nm)_iT ₌_h_h(n1,...,nm)_iT_.

❊st❡ ♣r♦❝❡ss♦ é ❝❤❛♠❛❞♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ❞❡ f(n1,...,nm)

❊①❡♠♣❧♦ ✶✳✹✳✻✳ ❙❡❥❛ F ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ③❡r♦✱ ❡♥❝♦♥tr❛r❡♠♦s ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✐❞❡♥✲ t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ♠✉❧t✐❧✐♥❡❛r❡s q✉❡ é P I✲❡q✉✐✈❛❧❡♥t❡ ❛ ✐❞❡♥t✐❞❛❞❡ f(x1) =x31✳

❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ❞✐st✐♥t❛s {y1, y2, y3} ❡ ❢❛③❡♥❞♦✿

f(y1+y2+y3) = (y1 +y2+y3)3 = (y1+y2+y3)(y1+y2+y3)(y1+y2+y3)

= (y1 +y2+y3)(y1y1+y1y2+y1y3+y2y1+y2y2+y2y3+y3y1+y3y2+y3y3)

= y1y2y3+y1y3y2+y2y1y3+y2y3y1+y3y1y2+y3y2y1

| {z }

h(y1,y2,y3)

+g(y1, y2, y3)

(33)

✶✳✹✳ P♦❧✐♥ô♠✐♦s ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦s ❡ ♠✉❧t✐❧✐♥❡❛r❡s

♦♥❞❡h(y1, y2, y3)é ❛ ❝♦♠♣♦♥❡♥t❡ ♠✉❧t✐❧✐♥❡❛r ❞❡f(y1+y2+y3)❡g(y1, y2, y3)é ❛ ❝♦♠♣♦♥❡t❡

♠✉❧t✐✲❤♦♠♦❣ê♥❡❛ ❞❡f(y1+y2+y3)✳ ▲♦❣♦✱ f é P I✲❡q✉✐✈❛❧❡♥t❡ ❛ {h}✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✼✳ ❚♦❞❛ P■✲á❧❣❡❜r❛ ✭s♦❜r❡ q✉❛❧q✉❡r ❝♦r♣♦✮ s❛t✐s❢❛③ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐✲ ♥♦♠✐❛❧ ♠✉❧t✐❧✐♥❡❛r✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛R ✉♠❛ P■✲á❧❣❡❜r❛ ✭s♦❜r❡ q✉❛❧q✉❡r ❝♦r♣♦✮ ❡ s✉♣♦♥❤❛ q✉❡

f(x1, . . . , xn)∈FhXi é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ R✳ ❙✉♣♦♥❤❛ q✉❡ ♦ ❣r❛✉ ❞❡ f ♥❛

✈❛r✐á✈❡❧x1 é n✳ ❆♣❧✐❝❛r❡♠♦s ♦ ♣r♦❝❡ss♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ❡♠ f✳ Pr✐♠❡✐r♦✱ s✉❜st✐t✉✐♥❞♦ x1

♣♦rx11+x12✱ t❡♠♦s

0 = f(x11+x12, x2, . . . , xn) =

(f(x11+x12, x2, . . . , xn)−f(x11, x2, . . . , xn)−f(x12, x2, . . . , xn))+

+f(x11, x2, . . . , xn) +f(x12, x2, . . . , xn).

▼❛s ❝♦♠♦f é ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ R t❡♠♦s q✉❡ ♦ ♣♦❧✐♥ô♠✐♦

f1(x11, x12, x2, . . . , xn) =f(x11+x12, . . . , xn)−f(x11, x2, . . . , xn)−f(x12, x2, . . . , xn),

é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥ô♠✐❛❧ ♣❛r❛R ❡ ♦ ❣r❛✉ ❞❡ f1 ❡♠ x11 ❡ ❡♠ x12 é ♥♦ ♠á①✐♠♦ n−1✳

❆❣♦r❛✱ ❛♣❧✐❝❛r❡♠♦s ♦ ♣r♦❝❡ss♦✱ ❝❛s♦ s❡❥❛ ♥❡❝❡ssár✐♦✱ ♣❛r❛f1 ♥❛s ✈❛r✐á✈❡✐sx11 ❡x12✳ ❙✉❜s✲

t✐t✉✐♥❞♦ x11 ♣♦r x11+x12 ❡ x12 ♣♦r x13+x14 t❡♠♦s

0 =f1(x11+x12, x13+x14, x2, . . . , xn) =





f1(x11+x12, x13+x14, x2, . . . , xn)−

X

i=1,2, j=3,4

f1(x1i, x1j, x2, . . . , xn)





+

+ X

i=1,2, j=3,4

f1(x1i, x1j, x2, . . . , xn).

▼❛s ❝♦♠♦f1 é ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛R t❡♠♦s q✉❡

f2(x11, x12, x13, x14, x2, . . . , xn) = f1(x11+x12, x13+x14, x2, . . . , xn)−

X

i=1,2, j=3,4

f1(x1i, x1j, x2, . . . , xn),

é ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛R ❡ ♦ ❣r❛✉ ❞❡ f2 ❡♠ x11, x12, x13 ❡x14 é ♥♦ ♠á①✐♠♦n−2✳

(34)

✶✳✺✳ ❖ ❚❡♦r❡♠❛ ❞❡ ❆♠✐ts✉r✲▲❡✈✐t③❦✐

❘❡♣❡t✐♥❞♦ ❡ss❡ ♣r♦❝❡ss♦✱ ❛♣ós ❛❧❣✉♥s ♣❛ss♦s✱ ❝❛s♦ s❡❥❛♠ ♥❡❝❡ssár✐♦s✱ ❡♥❝♦♥tr❛r❡♠♦s ✉♠❛ ✐❞❡♥t✐❞❛❞❡

fn−1(x11, x12, . . . , x1,2n−1, x₂. . . , x_n)

♣❛r❛ R t❛❧ q✉❡ ♦ ❣r❛✉ ❞❡ fn−1 ♥❛s ✈❛r✐á✈❡✐sx11, x12, . . . , x1,2n−1 é ♥♦ ♠á①✐♠♦ ✶✳

❆❣♦r❛✱ r❡♣❡t✐♥❞♦ ♦ ❛r❣✉♠❡♥t♦ ♣❛r❛ ❛s ✈❛r✐á✈❡✐sx2, . . . , xn❡♥❝♦♥tr❛r❡♠♦s ✉♠❛ ✐❞❡♥t✐❞❛❞❡

♣♦❧✐♥♦♠✐❛❧ ♠✉❧t✐❧✐♥❡❛r ♣❛r❛ R✳

✶✳✺ ❖ ❚❡♦r❡♠❛ ❞❡ ❆♠✐ts✉r✲▲❡✈✐t③❦✐

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛ ❞❡ ❆♠✐ts✉r✲▲❡✈✐t③❦✐ ❢❡✐t❛ ♣♦r ❘♦ss❡t ❬✶✺❪ ❡♠ ✶✾✼✻✱ ❛❧é♠ ❞❡st❛ ❞❡♠♦♥str❛çã♦ ❡①✐st❡♠ ♦✉tr❛s✱ ❛ ❞❡♠♦♥str❛çã♦ ♦r✐❣✐♥❛❧ ❞❡ ❆♠✐ts✉r ❡ ▲❡✈✐t③❦✐ ❬✶❪✱ ❇✳ ❑♦st❛♥t ❬✶✷❪✱ ❘✳ ❙✇❛♥ ❬✶✼❪ ❡ ❨✉✳ ❘❛③♠②s❧♦✈ ❬✶✹❪✱ t♦❞❛s ✉s❛♥❞♦ ❞✐❢❡r❡♥t❡s ❢❡rr❛♠❡♥t❛s ♣❛r❛ s✉❛ ❞❡♠♦♥str❛çã♦✳

■♥✐❝✐❛♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡ ❧❡♠❛✿

▲❡♠❛ ✶✳✺✳✶✳ ❆ á❧❣❡❜r❛ Mn(F) ♥ã♦ s❛t✐s❢❛③ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ❣r❛✉ ♠❡♥♦r q✉❡

2n✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ ♦ ❝♦♥trár✐♦✱ ♦✉ s❡❥❛✱ ❡①✐st❡f(x1, x2, ..., xm)∈FhXi✉♠❛ ✐❞❡♥✲

t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♥ã♦ ♥✉❧❛ ♣❛r❛ Mn(F)✱ ❝♦♠ m < 2n✳ P❡❧❛ ♣r♦♣♦s✐çã♦ ✶✳✹✳✼✱ ♣♦❞❡♠♦s

s✉♣♦rf(x1, x2, ..., xm) ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♠✉❧t✐❧✐♥❡❛r✳

❆❣♦r❛ ❢❛③❡♥❞♦✿

f(x1, x2, ..., xm)xm+1· · ·x2n−1 =g(x1, ..., x2n−1)

❡ ❡s❝r❡✈❡♥❞♦

g(x1, ..., x2n−1) =

X

σ∈S2n−1

ασxσ(1)· · ·xσ(2n−1),

♣♦❞❡♠♦s s✉♣♦r αid6= 0 ❡ ❛ss✐♠ t❡♠♦s

0 =g(E1,1, E1,2, E2,2, E2,3, ..., En−1,n−1, En−1,n, En,n) = αidE1,n ⇒αid= 0.

❆❜s✉r❞♦✳

(35)

✶✳✺✳ ❖ ❚❡♦r❡♠❛ ❞❡ ❆♠✐ts✉r✲▲❡✈✐t③❦✐

▲❡♠❛ ✶✳✺✳✷✳ ❙❡ ♦ ♣♦❧✐♥ô♠✐♦ st❛♥❞❛r❞ ❞❡ ❣r❛✉ 2n é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ Mn(Q)✱ ❡♥tã♦ é

t❛♠❜é♠ ♣❛r❛Mn(F)✳

❉❡♠♦♥str❛çã♦✿ ❉❡✜♥✐r❡♠♦s✿

ϕ:Mn(Z)−→Mn(F)✱ ❝♦♠ϕ((aij)ij) = (aij·1F)ij✱ ♦♥❞❡✱1F é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞♦ ❝♦r♣♦

F ❝♦♠ r❡❧❛çã♦ ❛♦ ♣r♦❞✉t♦✳ ◆♦t❡ q✉❡ ϕ é ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s✳

❙❡ E1, ..., E2n sã♦ ♠❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s ❡♠ Mn(Z) ❡♥tã♦ ϕ(E1), ..., ϕ(E2n) sã♦ ♠❛tr✐③❡s

❡❧❡♠❡♥t❛r❡s ❡♠Mn(F)✳ ❆ss✐♠

St2n(ϕ(E1), ..., ϕ(E2n)) = ϕ(St2n(E1, ..., E2n)) =ϕ(0) = 0.

▲❡♠❛ ✶✳✺✳✸✳ ❙❡❥❛ C ✉♠❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛ s♦❜r❡ Q✳ ❙❡ a∈Mn(C) ❡

tr❛ç♦(a) =tr❛ç♦(a2_{) =}_{· · ·}₌ _tr❛ç♦₍_an_{) = 0}_.

❡♥tã♦ an_{= 0}_✳

❆ ♣r♦✈❛ ❞❡st❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ✈✐st❛ ❡♠ ❬✺❪ ♣á❣✐♥❛ ✽✷✳

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ é ♦ ❢❛♠♦s♦ ❚❡♦r❡♠❛ ❞❡ ❆♠✐ts✉r ❡ ▲❡✈✐t③❦✐✳

❚❡♦r❡♠❛ ✶✳✺✳✹✳ ❖ ♣♦❧✐♥ô♠✐♦ st❛♥❞❛r❞ ❞❡ ❣r❛✉ 2n é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ ❛ á❧❣❡❜r❛ Mn(F)✳

❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❧❡♠❛ ✶✳✺✳✷ é s✉✜❝✐❡♥t❡ ❝♦♥s✐❞❡r❛r ♦ ❝❛s♦F =Q✳

❉❡❝♦♠♣♦♥❤❛ ❛ Q✲á❧❣❡❜r❛ ❞❡ ●r❛ss♠❛♥♥E ❡♠ E(0)_⊕_E(1) _{❝♦♠♦ ❡♠ ✶✳✷✳✺✳}

❉❛❞♦sa1, ..., a2n ∈Mn(Q) ❞❡✜♥✐r❡♠♦sb ∈Mn(E) ♣♦r✿

b =a1e1+a2e2+· · ·a2ne2n.

❚❡♠♦s

a=b2 = X

1≤i<j≤2n

(aiaj −ajai)eiej.

◆♦t❡ q✉❡ a∈Mn(E(0)) ♦♥❞❡ E(0) éQ✲á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛✳ ❆❧é♠ ❞✐ss♦✱

tr❛ç♦(a) = tr❛ç♦(a2_{) =} _{· · ·}₌_tr❛ç♦₍_an_{) = 0}_.

(36)

✶✳✻✳ ▼❛tr✐③❡s ●❡♥ér✐❝❛s

❉❡ ❢❛t♦✱

tr❛ç♦(a) =tr❛ç♦(b2j) =tr❛ç♦(b2j−1 ·b) =

−tr❛ç♦(b·b2j−1_{) =} ₋_tr❛ç♦₍_b2j_{) =}₋_tr❛ç♦₍_aj₎_⇒_tr❛ç♦₍_aj_{) = 0}_.

❙❡❣✉❡ ❞♦ ❧❡♠❛ ✶✳✺ q✉❡ an _{= 0}_{✳ ❊♥tã♦✿}

0 =an =b2n = X

σ∈S2n

(aσ(1)eσ(1))· · ·(aσ(2n)eσ(2n))

= X

σ∈S2n

(aσ(1)· · ·aσ(2n))eσ(1)· · ·eσ(2n)

= St2n(a1, ..., a2n)(e1· · ·e2n)

P♦rt❛♥t♦✱ St2n(a1, ..., an) = 0

❖❜s❡r✈❛çã♦ ✶✳✺✳✺✳ ▼❡s♠♦ ❝♦♥❤❡❝❡♥❞♦✲s❡ ♦ ❣r❛✉ ♠í♥✐♠♦ ♣❛r❛ ❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❛ á❧❣❡❜r❛ Mn(F) ♥ã♦ s❡ ❝♦♥❤❡❝❡ ✉♠❛ ❜❛s❡ ♥♦ ❝❛s♦ ❣❡r❛❧✳

✶✳✻ ▼❛tr✐③❡s ●❡♥ér✐❝❛s

❖✉tr♦ ❝♦♥❝❡✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦ tr❛❜❛❧❤♦ é ♦ ❞❡ ♠❛tr✐③❡s ❣❡♥ér✐❝❛s✱ ❡st❛ t❡♦r✐❛ é ✉♠❛ ✐♠♣♦rt❛♥t❡ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ✐❞❡♥t✐❞❛❞❡s Zn✲❣r❛❞✉❛❞❛s ❞❡ Mn(F)✱

q✉❡ ✈❡r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ✸✳ ❆q✉✐ ❞❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ s♦❜r❡ ❡st❡ ❛ss✉♥t♦✱ ♣❛r❛ ✐ss♦ ❞❡♥♦t❡ ♣♦r Ωn ❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛ ❧✐✈r❡

Ωn=F[ypq(i) :p, q = 1, ..., n ❡i∈N],

❧✐✈r❡♠❡♥t❡ ❣❡r❛❞❛ ♣❡❧❛s ✈❛r✐á✈❡✐s y(pqi).

❉❡✜♥✐çã♦ ✶✳✻✳✶✳ ❆s ♠❛tr✐③❡s yi = n

X

p,q=1

y(i)

pqEpq ♦♥❞❡ i ∈ N sã♦ ❝❤❛♠❛❞❛s ❞❡ ♠❛tr✐③❡s

❣❡♥ér✐❝❛s✳ ❆ s✉❜á❧❣❡❜r❛ Rn ❞❡ Mn(Ωn) ❣❡r❛❞❛ ♣♦ry1, y2, ..., yi, ... é ❝❤❛♠❛❞❛ ❞❡ á❧❣❡❜r❛

❞❛s ♠❛tr✐③❡s ❣❡♥ér✐❝❛s n×n✳

❊①❡♠♣❧♦ ✶✳✻✳✷✳ ❚♦♠❛♥❞♦ R2 ❛ s✉❜á❧❣❡❜r❛ ❞❡ M2(Ω2) t❡♠♦s q✉❡ ❡❧❛ é ❣❡r❛❞❛ ♣♦r✿

y1 =



 y

(1) 11 y

(1) 12

y(1)₂₁ y(1)₂₂



✱ y₂ = 

 y

(2) 11 y

(2) 12

y₂₁(2) y₂₂(2)



✱✳✳✳