NÚMEROS PRIMOS GAUSSIANOS PARA O ENSINO MÉDIO

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲

P❘❖❋▼❆❚

◆ú♠❡r♦s Pr✐♠♦s ●❛✉ss✐❛♥♦s

♣♦r

❈②❜❡❧❡ ❱❡r❞❡ ❆r❛❣ã♦ ❞❡ ❆❧♠❡✐❞❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞♦ ❈✉rs♦ ❛♣r❡✲ s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣♦st♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇

◆ú♠❡r♦s Pr✐♠♦s ●❛✉ss✐❛♥♦s

♣♦r

❈②❜❡❧❡ ❱❡r❞❡ ❆r❛❣ã♦ ❞❡ ❆❧♠❡✐❞❛

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞♦ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

ár❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛

❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦ ✲ ❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ✲ ❯❋P❇

Pr♦❢✳ ❉r✳ ❋r❛♥❝✐s❝♦ ❙✐❜ér✐♦ ❇❡③❡rr❛ ❆❧❜✉q✉❡rq✉❡ ✲ ❯❊P❇

❆❣♦st♦✴✷✵✶✹

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❊❯❙✱ q✉❡ s❡♠♣r❡ ❡stá ♣r❡s❡♥t❡ ♥♦ ♠❡✉ ❝❛♠✐♥❤♦ ♠❡ ❛❝♦♠♣❛✲ ♥❤❛♥❞♦✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♠❡✉s ♣❛✐s✱ ♣♦r s❡r❡♠ ♠❡✉s ♠❡❧❤♦r❡s ❛♠✐❣♦s✳ P♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r❡♠ ❡♠ ♠❡✉ ♣♦t❡♥❝✐❛❧ ❡ ✐♥✈❡st✐r❡♠ ♥♦ ♠❡✉ ❝r❡s❝✐♠❡♥t♦ ♣❡ss♦❛❧✳ P♦r ♠❡ ❝♦❧♦❝❛r ♥♦ ❝❛♠✐♥❤♦ ❞♦ ❜❡♠ ❡ s❡♠♣r❡ ♠❡ ❡♥s✐♥❛r ♦ q✉❡ é ❝❡rt♦ ❡ ♦ q✉❡ é ❥✉st♦✳ P♦r ❡st❛r❡♠ ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡ ♠✐♥❤❛ ✈✐❞❛✳ ❖❜r✐❣❛❞♦ ♣❛✐✱ ♦❜r✐❣❛❞♦ ♠ã❡ ♣♦r s❡r❡♠ ♦s ♠❡❧❤♦r❡s ♣❛✐s q✉❡ ❛❧❣✉é♠ ♣♦❞❡ t❡r✱ ♥ã♦ s❡✐ s❡ ✉♠ ❞✐❛ ✈♦✉ ♣♦❞❡r ❛❣r❛❞❡❝❡r ♦ s✉✜❝✐❡♥t❡✱ ❛♠♦ ✈♦❝ês✳ ❆♦s ♠❡✉s ✐r♠ã♦s✱ ❆❧❡①❛♥❞r❡✱ ❇r✉♥♦ ❡ ❏ú♥✐♦r✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ❛♠✐③❛❞❡ q✉❡ t❡♠ ♣♦r ♠✐♠✱ ♣❡❧♦ r❡s♣❡✐t♦ ❡ ❛♣♦✐♦ ❛s ♠✐♥❤❛s ❞❡❝✐sõ❡s✱ ♦❜r✐❣❛❞♦ ♣♦r ❝♦♥tr✐❜✉ír❡♠ ♣❛r❛ ✉♥✐ã♦ ❞❡ ♥♦ss❛ ❢❛♠í❧✐❛✱ s❡✐ q✉❡ s❡♠♣r❡ ♣♦ss♦ ❝♦♥t❛r ❝♦♠ ✈♦❝✄❡s✳ ❆s ♠❡✉s q✉❡r✐❞♦s s♦❜r✐♥❤♦s✱ ●❛❜r✐❡❧ ❡ ❈❛❡❧✱ ♠✐♥❤❛s ❝✉♥❤❛❞❛s ❊❧✐♥❡ ❡ ❋❛❜✐❛♥❡✱ ♣❡❧♦ ❝❛r✐♥❤♦✱ ♣❡❧❛ ❢♦rç❛ ❡ ❛♣♦✐♦ ♥❛s ❤♦r❛s ♠❛✐s ❞✐❢í❝❡✐s✳ ❙♦✉ ❣r❛t❛ ♣♦r ✈♦❝ês ❡st❛r❡♠ s❡♠♣r❡ ❝♦♠✐❣♦✳

◆❛t❤á❧✐❛ ❘❛♣❤❛❡❧❧❛ ♠✐♥❤❛ ❛♠✐❣❛ ❡ ❝♦♠♣❛♥❤❡✐r❛ ❡♠ t♦❞❛s ❛s ❤♦r❛s✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦ q✉❡ ✐♥❞✐❝♦✉ ♦ t❡♠❛ ❞❡st❡ tr❛❜❛❧❤♦✱ ❛❝♦♠♣❛♥❤♦✉ ♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ❛❥✉❞❛♥❞♦ s❡♠♣r❡ q✉❡ ❢♦✐ s♦❧✐❝✐t❛❞♦✳ ❙ó t❡♥❤♦ ❛❣r❛❞❡❝❡r ❡ss❛ ♦♣♦rt✉♥✐❞❛❞❡✳

❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ♠✐♥✐str❛r❛♠ ❛s ❛✉❧❛s ♥♦ P❘❖❋▼❆❚ ♥❛ ❯❋P❇✱ ❛ t♦❞♦s ♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ t✉r♠❛ ✷✵✶✷✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❆❧❡ss❛♥❞r♦ ▼✐❣♥❛❝ ✱ ▼❛❣♥✉♥ ❈❡s❛r✱ ❆♥❞ré ❙♦❛r❡s ❡ ❲❛s❤✐♥❣t♦♥ ●♦♥ç❛❧✈❡s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❡ ❛❝r❡❞✐t❛r❛♠ ❡♠ ♠❡✉ ♣♦t❡♥❝✐❛❧ ♣❛r❛ q✉❡ ❝❤❡❣❛ss❡ ♥♦ ✜♥❛❧ ❞❡ss❛ ❡t❛♣❛✳ ◗✉❡ ❞✉r❛♥t❡ ♦s ❞♦✐s ú❧t✐♠♦s ❛♥♦s✱ ❡♥❢r❡♥t❛r❛♠ ❛ ❡str❛❞❛ ❘❡❝✐❢❡ ✲ ❏♦ã♦ P❡ss♦❛ t♦❞♦s ♦s sá❜❛❞♦s✳

❆♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❡ ♠❡✉s ❛♠❛❞♦s ❛❧✉♥♦s ❞❛ ❊s❝♦❧❛ ❆❞❡❧❛✐❞❡ P❡ss♦❛ ❈â♠❛r❛ q✉❡ s♦✉❜❡r❛♠ ❝♦♠♣r❡❡♥❞❡r ♠✐♥❤❛s ❛✉sê♥❝✐❛s✱ ♠❡✉s ❡str❡ss❡s ❡ ♠✐♥❤❛s ❛♥❣✉st✐❛s ♥❡ss❡ ♠♦♠❡♥t♦ tã♦ ✐♠♣♦rt❛♥t❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳

❆ t♦❞♦s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞❛ ❢❛♠í❧✐❛ ❊❳❆❚❆❈❖❘✱ ♣♦r ❝❡❞❡r ❡ss❡ ❡s♣❛ç♦ ❡ ♣❡❧❛ ❝♦♠♣r❡❡♥sâ♦ ❞❛s ♠✐♥❤❛s ❢❛❧t❛s✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞❡ ❏♦ã♦ P❡ss♦❛✱ ❆❧✐ss♦♥ ❚❤♦♠❛s✱ ❋❡❧✐♣❡✱ ❙❛♠❛r❛ ❡ ❇r✉♥♦ ❋❛r✐❛s✱ ♣❡❧❛ ❤♦s♣❡❞❛❣❡♠ ❞✉r❛♥t❡ ❞❡ss❡s ❞♦✐s ❛♥♦s✳

❆ ❙❇▼ ✲ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ♦❢❡r❡❝✐❞❛ ❛♦s ♣r♦❢❡ss♦r❡s ❞❛ r❡❞❡ ♣ú❜❧✐❝❛✳

❆ ❈❆P❊❙ ✲ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r ♣❡❧❛ ❜♦❧s❛ ❝♦♥❝❡❞✐❞❛✳

❉❡❞✐❝❛tór✐❛

❆ t♦❞♦s ♦s q✉❡ s❡ ❛❧❡❣r❛♠ ❝♦♠ ♦ ♥♦ss♦ s✉❝❡ss♦✳

❊♣í❣r❛❢❡

❆ ▼❛t❡♠át✐❝❛ é ❛ r❛✐♥❤❛ ❞❛s ❝✐ê♥❝✐❛s ❡ ❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s é ❛ r❛✐♥❤❛ ❞❛s ♠❛t❡♠át✐❝❛s✳ ✭●❛✉ss✮

❘❡s✉♠♦

◆♦ss♦ ♣r♦♣ós✐t♦ ♥❡st❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ✉♠❛ ❝❛t❡❣♦r✐❛ ❡s♣❡❝✐❛❧ ❞❡ ♥ú♠❡r♦s✿ ♦s ♥ú♠❡r♦s ♣r✐♠♦s ❞❡ ●❛✉ss✳ ❋❛③❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ ❤✐stór✐❝❛ ❞❡ ❝♦♠♦ s✉r❣✐✉ ♦ ✐♥t❡r❡ss❡ ❞❡ ❈❛r❧ ❋r✐❡sr✐❝❤ ●❛✉ss ♥♦s ❡st✉❞♦s ♣❡❧♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ s✉r❣✐♥❞♦ ❡♠ s✉❛ ❤♦♠❡♥❛❣❡♠✱ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss✳ ❈✐t❛♠♦s ♦s t❡♦r❡♠❛s ♠❛✐s ✐♠♣♦r✲ t❛♥t❡s ❡✴♦✉ ✐♥t❡r❡ss❛♥t❡s ❝♦♠ s✉❛s ♣r♦♣♦s✐çõ❡s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❞❡♠♦♥str❛çõ❡s✱ ❡ ❞❡s❡♥✈♦❧✈❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ❝♦♠♣r❡❡♥sã♦✳ ❚❛♠❜é♠ s❡rã♦ ❛♣r❡✲ s❡♥t❛❞♦s ♦s ♥ú♠❡r♦s ♣r✐♠♦s ●❛✉ss✐❛♥♦s ❡ ✈❛♠♦s ❝♦♥❝❧✉✐r ❝♦♠ ❛t✐✈✐❞❛❞❡s ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡♥✈♦❧✈❡♥❞♦ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❝♦♠♣❛r❛çõ❡s ❡♥tr❡ ♦s ♣r✐♠♦s ✐♥t❡✐r♦s ❡ ♣r✐♠♦s ●❛✉ss✐❛♥♦s✱ ❡♥✈♦❧✈❡♥❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✳

❆❜str❛❝t

❙✉♠ár✐♦

✶ ◆ú♠❡r♦s ■♥t❡✐r♦s ❞❡ ●❛✉ss ✶

✶✳✶ ❈♦♥s✐❞❡r❛çõ❡s ❍✐stór✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❖s ■♥t❡✐r♦s ❞❡ ●❛✉ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷✳✶ ❆ ◆♦r♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳✷ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳✸ ❖ ❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✹ ❖ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✺ ❖ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✷✳✻ ❋❛t♦r❛çã♦ Ú♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷ ◆ú♠❡r♦s Pr✐♠♦s ●❛✉ss✐❛♥♦s ✷✻

✸ ❆t✐✈✐❞❛❞❡s ❝♦♠ ♦s ♥ú♠❡r♦s Pr✐♠♦s ●❛✉ss✐❛♥♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ✸✹ ✸✳✶ ❆t✐✈✐❞❛❞❡ ✶ ✲ ❆♣r❡s❡♥t❛çã♦ ❞♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✷ ❆t✐✈✐❞❛❞❡ ✷ ✲ ■❞❡♥t✐✜❝❛çã♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡♠Z[i] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✸ ❆t✐✈✐❞❛❞❡ ✸ ✲ ❘❡❧❛❝✐♦♥❛r ♦s ♣r✐♠♦s ✐♥t❡✐r♦s ❡ ♣r✐♠♦s ●❛✉ss✐❛♥♦s ✳ ✳ ✳ ✸✾

✹ ❙♦❧✉çõ❡s ❞❛s ❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s ✹✷ ✹✳✶ ❙♦❧✉çã♦ ❞❛ ❆t✐✈✐❞❛❞❡ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✷ ❙♦❧✉çã♦ ❞❛ ❆t✐✈✐❞❛❞❡ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

❙❯▼➪❘■❖

✹✳✸ ❙♦❧✉çã♦ ❞❛ ❆t✐✈✐❞❛❞❡ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

◆♦t❛çõ❡s ❡ ❙í♠❜♦❧♦s ✹✼

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✵

■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦ ✐r❡♠♦s ❛❜♦r❞❛r ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss q✉❡ ♣♦ss✐❜✐❧✐t❛r❛♠ ❡st✉❞♦s ♥❛s ár❡❛s ❞❡ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✱ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❆❧❣é❜r✐❝♦s✱ ❡st✉❞♦s ❞♦s ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s✳ ❊st❡s ♥ú♠❡r♦s ❛♣❛r❡❝❡♠ ♥❛t✉r❛❧♠❡♥t❡ ❡♠ q✉❡stõ❡s r❡❧❛✲ ❝✐♦♥❛❞❛s ❛♦ ú❧t✐♠♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ às r❡❝✐♣r♦❝✐❞❛❞❡s ❝ú❜✐❝❛ ❡ ❜✐q✉❛❞rát✐❝❛ ❡ à ❢❛t♦r❛çã♦ ú♥✐❝❛ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s✳

◆♦ ❈❛♣ít✉❧♦ ✶✱ ❢❛r❡♠♦s ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ s♦❜r❡ ❛ ♦❜r❛ ❞♦ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ ❈❛r❧ ❋✳ ●❛✉ss✱ q✉❡ ♣r♦❞✉③✐✉ ❝♦♠ ❞❡s❡♥✈♦❧t✉r❛ ❡♠ t♦❞♦s ♦s r❛♠♦s ❞❛ ♠❛t❡♠át✐❝❛✳ ➱ ♥♦tór✐♦ ♦ ♣r❛③❡r q✉❡ s❡♥t✐❛ ♣❡❧❛ ✐♥✈❡st✐❣❛çã♦ ❡♠ ❆r✐t♠ét✐❝❛✳ ❆ s✉❛ ♦❜r❛ ♠♦♥✉♠❡♥✲ t❛❧ ✧❉✐sq✉✐s✐t✐♦♥❡s ❆r✐t❤♠❡t✐❝❛❡✧ ❧❛♥ç♦✉ ♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ♠♦❞❡r♥♦✳ ❊♠ ✶✽✷✺✱ ♣✉❜❧✐❝♦✉ ✉♠ tr❛❜❛❧❤♦ ❡♠ q✉❡ ✐♥tr♦❞✉③✐❛ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❞❛ ❢♦r♠❛ a+bi, ❝♦♠ a ❡b _∈ Z ❡i2 _{= −1. ❊ss❡ ❝♦♥❥✉♥t♦ é ✐♥❞✐❝❛❞♦ ♣♦r Z[i] ❡ é}

❞❡♥♦♠✐♥❛❞♦ ✐♥t❡✐r♦s ❞❡ ●❛✉ss ♦✉ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s ❡♠ ❤♦♠❡♥❛❣❡♠ ❛♦ s❡✉ ❝r✐❛❞♦r✳ ❊♠ s❡❣✉✐❞❛✱ ✐♥tr♦❞✉③✐♠♦s ❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ■♥t❡✐r♦s ●❛✉ss✐❛♥♦s✿ ♥♦r♠❛✱ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✱ ❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦✱ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉✲ ❝❧✐❞❡s✱ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t ❡ ❋❛t♦r❛çã♦ ú♥✐❝❛✳ ❈❛❞❛ ✉♠❛ ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s t❡♠ t❡♦r❡♠❛s ❡ ❝♦r♦❧ár✐♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♣♦❞❡r♠♦s ❝♦♠♣r❡❡♥❞❡r ♦s ♥ú♠❡r♦s ♣r✐♠♦s ●❛✉ss✐❛♥♦s✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ✈❛♠♦s ❞❡✜♥✐r q✉❡♠ sã♦ ♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡♠ Z[i] ❡ ❛♣❧✐❝❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s ♣❛r❛ ✐❞❡♥t✐✜❝á✲❧♦s✳ ▼♦str❛r❡♠♦s q✉❡ ♥❡♠ t♦❞♦s ♣r✐♠♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s s❡rã♦ ♣r✐♠♦s ❞❡ ●❛✉ss✱ ♣♦✐s ✈❡r❡♠♦s q✉❡ ♦s ♣r✐♠♦s p ❡♠ Z ♥❛ ❢♦r♠❛ 4n+ 1 ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ♣❡❧♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ♣r♦♣♦st❛s ❞❡ ❛t✐✈✐❞❛❞❡s ♣❛r❛ ♦ ♣r♦❢❡ss♦r ❛♣❧✐❝❛r ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❞❡s❡♥✈♦❧✈✐❞❛s ♣❛r❛ t✉r♠❛s ❞♦ ✸♦ _{❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ t❡♥❞♦ ❝♦♠♦} ♦❜❥❡t✐✈♦ ❞❡ ❞❡s♣❡rt❛r ❝✉r✐♦s✐❞❛❞❡s ♥♦s ❛❧✉♥♦s ❡♠ r❡❧❛çã♦ ❛♦s ♣r✐♠♦s ●❛✉ss✐❛♥♦s✳

P❛r❛ ✜♥❛❧✐③❛r✱ ♥♦ ❈❛♣ít✉❧♦ ✹✱ ✐r❡♠♦s s♦❧✉❝✐♦♥❛r ❞❡t❛❧❤❛❞❛♠❡♥t❡ ❛s ❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛ ♥♦ ❈❛♣ít✉❧♦ ✸✳

❈❛♣ít✉❧♦ ✶

◆ú♠❡r♦s ■♥t❡✐r♦s ❞❡ ●❛✉ss

◆♦ í♥✐❝✐♦ ❞❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❢❛③❡r ✉♠❛ ❛❜♦r❞❛❣❡♠ ❤✐stór✐❝❛ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐✲ r♦s ❞❡ ●❛✉ss✱ ♠♦str❛♥❞♦ s✉❛ ✐♠♣♦rtâ♥❝✐❛✱ ❢♦r♥❡❝❡♥❞♦ ✐❞❡✐❛s ♣❛r❛ q✉❡ ♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ♣♦ss❛ tr❛❜❛❧❤❛r ❡st❡ ❛ss✉♥t♦ ♥❛ sér✐❡ ✜♥❛❧ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ◆♦ ❞❡❝♦rr❡r ❞♦ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❡st❛❜❡❧❡❝❡r ✈ár✐❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ t❡♦r❡♠❛s r❡❧❛❝✐♦♥❛✲ ❞♦s ❛♦s ♥ú♠❡r♦s ●❛✉ss✐❛♥♦s✳ ❖s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss ♥ã♦ ❡stá ♣r❡s❡♥t❡ ♥♦ ❝✉rrí❝✉❧♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♠❛s ♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ♣♦❞❡ ❢❛③❡r ❛❧❣✉♠❛s ❛❞❛♣✲ t❛çõ❡s ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ❛❧✉♥♦s ❡♠ r❡❧❛çã♦ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ❆s r❡❢❡rê♥❝✐❛s ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ ❝❛♣ít✉❧♦ ❢♦r❛♠ ❇❖❨❊❘ ❬✶❪✱ ❇❯✲ ❚▲❊❘ ❬✷❪✱ ❈❖◆❘❆❉ ❬✸❪✱ ❊❱❊❙ ❬✹❪✱ ❋❯❏■❲❆❘❆ ❬✻❪✱ ❍❊❋❊❩ ❬✼❪ ❡ ❙■❉❑■ ❬✽❪✳

✶✳✶ ❈♦♥s✐❞❡r❛çõ❡s ❍✐stór✐❝❛s

❈♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ ❞♦s ♠❛✐♦r❡s ♠❛t❡♠át✐❝♦s q✉❡ ❥á ❡①✐st✐✉✱ ❈❛r❧ ❋r✐❡sr✐❝❤ ●❛✉ss ♥❛s❝❡✉ ❡♠ ❇r✉♥s✇✐❝❤ ♥❛ ❆❧❡♠❛♥❤❛ ❡♠ ✸✵ ❞❡ ❆❜r✐❧ ❞❡ ✶✼✼✼ ❡ ❢❛❧❡❝❡✉ ♥❛ ❝✐❞❛❞❡ ❞❡ ●♦tt✐♥❣❡♥✱ ✷✸ ❞❡ ❋❡✈❡r❡✐r♦ ❞❡ ✶✽✺✺✳ ●❛✉ss ❞❡♠♦♥str♦✉ ❞❡s❞❡ ♠✉✐t♦ ❝❡❞♦ ♦s s❡✉s ❞♦t❡s ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛✳ ❆s s✉❛s ❝♦♥tr✐❜✉✐çõ❡s ♣❛r❛ ❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❞❛ ❣❡♦♠❡tr✐❛ ❡ ❞❛ á❧❣❡❜r❛ sã♦ ✐♥ú♠❡r❛s✳ P♦r ❡①❡♠♣❧♦✱ ❛ s✉❛ t❡s❡ ❞❡ ❞♦✉t♦r❛♠❡♥t♦ ❢♦✐ ❛ ♣r✐♠❡✐r❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ á❧❣❡❜r❛✳ ●❛✉ss t❡✈❡ t❛♠❜é♠ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝♦♥tr✐❜✉✐çã♦ ♣❛r❛ ❛ ❛str♦♥♦♠✐❛✱ t❡♥❞♦ s❡ ✐♥t❡r❡ss❛❞♦ ♣❡❧♦ ❡st✉❞♦ ❞❛s ór❜✐t❛s ♣❧❛♥❡tár✐❛s ❡ ♣❡❧❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ❢♦r♠❛ ❞❛ ❚❡rr❛✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ss❛ ❝♦♥tr✐❜✉✐çã♦ ❢♦✐ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ ♠ét♦❞♦ ♣❛r❛ ❝❛❧❝✉❧❛r✱ ❝♦♠ ❣r❛♥❞❡ ♣r❡❝✐sã♦✱ ♦s ♣❛râ♠❡tr♦s ❞❡ ✉♠❛ ór❜✐t❛ ♣❧❛♥❡tár✐❛ ❛ ♣❛rt✐r ❞❡ ❛♣❡♥❛s três ♦❜s❡r✈❛çõ❡s ❞❛ ♣♦s✐çã♦ ❞♦ ♣❧❛♥❡t❛✳

❊♥tr❡ ✶✽✵✽ ❡ ✶✽✷✺ ❈❛r❧ ❋✳ ●❛✉ss ✐♥✈❡st✐❣❛✈❛ q✉❡stõ❡s r❡❧❛❝✐♦♥❛❞❛s à r❡❝✐♣r♦❝✐✲

❖s ■♥t❡✐r♦s ❞❡ ●❛✉ss ❈❛♣ít✉❧♦ ✶

❞❛❞❡ ❝ú❜✐❝❛ ❡ à ❜✐q✉❛❞rát✐❝❛✱ q✉❛♥❞♦ ♣❡r❝❡❜❡✉ q✉❡ ❡ss❛ ✐♥✈❡st✐❣❛çã♦ s❡ t♦r♥❛✈❛ ♠❛✐s s✐♠♣❧❡s tr❛❜❛❧❤❛♥❞♦ s♦❜r❡ Z[i]✱ ❝❤❛♠❛❞♦ ♣♦st❡r✐♦r♠❡♥t❡ ❞❡ ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s✱ ❞♦

q✉❡ ❡♠Z✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❖ ❝♦♥❥✉♥t♦Z[i]é ❢♦r♠❛❞♦ ♣❡❧♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❞❛ ❢♦r♠❛ a+bi✱ ♦♥❞❡ a ❡b sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ i2 _{=−1✳ ❖s ✐♥t❡✐r♦s}

❞❡ ●❛✉ss ❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ ✉♠❛ sér✐❡ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ♣❛r❡❝✐❞❛s ❝♦♠ ❛ ❞♦s ✐♥t❡✐r♦s r❡❛✐s✱ ♣♦ré♠ ♠❛✐s ❣❡r❛✐s✳ ❆ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ t♦r♥❛✲s❡ ♠❛✐s ❝♦♠♣❧❡①❛✿ t♦♠❡✱ ♣♦r ❡①❡♠♣❧♦✱ ✺ ♦ q✉❛❧ é ♣r✐♠♦ ❡♠Z✱ ♠❛s q✉❡ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s ❞❡ Z[i]✱ ❛ s❛❜❡r5 = (1 + 2i)(1₋2i)✳ ❊♠ ✈❡r❞❛❞❡✱ ♥❡♥❤✉♠ ♣r✐♠♦ r❡❛❧ ❞❛

❢♦r♠❛ (4n+ 1)é ✉♠ ✧♣r✐♠♦ ❞❡ ●❛✉ss✧✱ ❛♦ ♣❛ss♦ q✉❡ ♣r✐♠♦s r❡❛✐s ❞❛ ❢♦r♠❛(4n₋1) ♣❡r♠❛♥❡❝❡♠ ♣r✐♠♦s ♥♦ s❡♥t✐❞♦ ❣❡♥❡r❛❧✐③❛❞♦✳ ◆♦ ❧✐✈r♦ ❉✐sq✉✐s✐t✐♦♥❡s ❆r✐t❤♠❡t✐❝❛❡✱ ●❛✉ss ✐♥❝❧✉✐ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✳ ❚♦❞♦ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡ ❡♠ q✉❡ ❛ ❢❛t♦r❛çã♦ é ú♥✐❝❛ é ❝❤❛♠❛❞♦ ❤♦❥❡ ❞❡ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡ ❞❡ ●❛✉ss✳

●❛✉ss ❡st❡♥❞❡✉ ❛ ✐❞❡✐❛ ❞❡ ♥ú♠❡r♦ ✐♥t❡✐r♦ q✉❛♥❞♦ ❞❡✜♥✐✉ ♦ ❝♦♥❥✉♥t♦ Z[i]✱ ♣♦✐s ❞❡s❝♦❜r✐✉ q✉❡ ♠✉✐t♦ ❞❛ t❡♦r✐❛ ❞❡ ❊✉❝❧✐❞❡s s♦❜r❡ ❢❛t♦r❛çã♦ ❞❡ ✐♥t❡✐r♦s ♣♦❞❡r✐❛ s❡r tr❛♥s♣♦rt❛❞❛ ♣❛r❛ Z[i] ❝♦♠ ❝♦♥s❡q✉ê♥❝✐❛s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳ ❊❧❡ ❞❡s❡♥✈♦❧✈❡✉ ✉♠❛ t❡♦r✐❛ ❞❡ ❢❛t♦r❛çã♦ ❡♠ ♣r✐♠♦s ♣❛r❛ ❡ss❡s ♥ú♠❡r♦s ❝♦♠♣❧❡✲ ①♦s ❡ ❞❡♠♦♥str♦✉ q✉❡ ❡ss❛ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ♣r✐♠♦s é ú♥✐❝❛✱ ❝♦♠♦ ❛❝♦♥t❡❝❡ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❆ ❢❡rr❛♠❡♥t❛ q✉❡ ●❛✉ss ❢❡③ ❞❡ss❡ ♥♦✈♦ t✐♣♦ ❞❡ ♥ú♠❡r♦ ❢♦✐ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ Ú❧t✐♠♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✳ ❖s ✐♥t❡✐r♦s ❞❡ ●❛✉ss sã♦ ❡①❡♠♣❧♦s ❞❡ ✉♠ t✐♣♦ ♣❛rt✐❝✉❧❛r ❞❡ ♥ú♠❡r♦ ❝♦♠✲ ♣❧❡①♦✱ ❛ s❛❜❡r✱ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s q✉❡ sã♦ s♦❧✉çõ❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧✿

anxn_+an

−1xn−1+· · ·+a1x+a0 = 0✱ ♦♥❞❡ t♦❞♦s ♦s ❝♦❡✜❝✐❡♥t❡san, an−1, . . . a1 ❡ a0 sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❊ss❡s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s q✉❡ sã♦ r❛í③❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ♣♦✲ ❧✐♥♦♠✐❛❧ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ✐♥t❡✐r♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❛❧❣é❜r✐❝♦s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛ i, é ✉♠ ✐♥t❡✐r♦ ❛❧❣é❜r✐❝♦✱ ♣♦✐s s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ x2_{+ 1 = 0✳}

✶✳✷ ❖s ■♥t❡✐r♦s ❞❡ ●❛✉ss

❉❡✜♥✐çã♦ ✶ ❖s ✐♥t❡✐r♦s ❞❡ ●❛✉ss sã♦ ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦✿

Z[i] =_{a+bi:a, b_∈Z_}.

❱❛♠♦s ♠♦str❛r ❛❧❣✉♠❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❛r✐t♠ét✐❝❛s ❞♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss✳ Pr✐✲ ♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❛♠♦s q✉❡ Z[i] é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ C✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦ Z[i] ♠✉♥✐❞♦ ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❤❡r❞❛❞❛s ❞❡ C✳ ■st♦ é✱ s❡z1 =a+bi ❡ z2 =c+di, ❡♥tã♦✿

z1+z2 = (a+c) + (b+d)i ❡

❆ ◆♦r♠❛ ❈❛♣ít✉❧♦ ✶

z1·z2 = (ac−bd) + (ad+bc)i.

❊❧❡♠❡♥t♦s ✐♥✈❡rsí✈❡✐s ❝♦♠ r❡s♣❡✐t♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ sã♦ ❝❤❛♠❛❞♦s ❞❡ ✉♥✐❞❛❞❡s✳ ❆s ✉♥✐❞❛❞❡s ❡♠Z[i]✱ ❛♥❛❧♦❣❛♠❡♥t❡ ❛Z✱ sã♦ t♦❞♦s ♦s ❡❧❡♠❡♥t♦sz _∈Z[i]q✉❡ ♣♦ss✉❡♠ ✐♥✈❡rs♦✱ ♦✉ s❡❥❛✱ é ✉♠ ❡❧❡♠❡♥t♦ z′ _{∈ Z[i]t❛❧ q✉❡ z ·z}′ _{= 1✳ ❆s ✉♥✐❞❛❞❡s ❞❡ Z sã♦} 1 ❡ ₋1✱ ❞❡s❞❡ q✉❡1_·1 = 1 ❡ (₋1)_·(₋1) = 1✳ ❊♠Z[i]sã♦ q✉❛tr♦ ✉♥✐❞❛❞❡s✳ ▼❛✐s ✉♠❛ ✈❡③✱ t❡♠♦s1 ❡ ₋1, ♠❛s t❛♠❜é♠ t❡♠♦s i ❡ ₋i✱ ♦❜s❡r✈❛♥❞♦ q✉❡i_·(₋i) = 1 ❡ (₋i)_·

i = 1✳ ❈♦♠♦ ♣♦❞❡♠♦s t❡r ❝❡rt❡③❛ ❞❡ q✉❡ ❡①✐st❡♠ ❛♣❡♥❛s ♦s q✉❛tr♦ ❡❧❡♠❡♥t♦s ♠✉❧t✐♣❧✐❝❛t✐✈♦❄

❆ ♣r♦✈❛ q✉❡ ❡①✐t❡♠ ❛♣❡♥❛s q✉❛tr♦ ✉♥✐❞❛❞❡s ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss é ❢❡✐t❛ ♣♦r ❝♦♥tr❛❞✐çã♦✳ ❖ ♣r✐♠❡✐r♦ ♣❛ss♦ ❝♦♥s✐st❡ ❡♠ ❞❡✜♥✐r ❛ ♥♦r♠❛ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ●❛✉ss✐❛♥♦✱ q✉❡ ❢❛r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡✱ ❡♠❜♦r❛ s❡❥❛ ♦ ♠❡s♠♦ q✉❡ ♣❛r❛ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳

✶✳✷✳✶ ❆ ◆♦r♠❛

❊♠Z✱ t❛♠❛♥❤♦ é ♠❡❞✐❞♦ ♣❡❧♦ ✈❛❧♦r ❛❜s♦❧✉t♦✳ ❊♠ Z[i]✱ ✉s❛♠♦s ❛ ♥♦r♠❛✳

❉❡✜♥✐çã♦ ✷ ❆ ♥♦r♠❛ ❞❡ a+bi_∈Z[i] é N(a+bi) = a2_+b2_✳

❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ N(57₋11i) = 572_{+ 11}2 _{= 3370✳ ❆❧❣✉♥s ♠❛t❡♠át✐❝♦s ♣r❡❢❡r❡♠}

❞❡✜♥✐r ❛ ♥♦r♠❛ ❝♦♠♦ ♦ ✈❛❧♦r √a2_+b2✱ q✉❡ é ❛ r❛í③ q✉❛❞r❛❞❛ ❞❛ ♥♦r♠❛✱ ❛q✉✐✱ s❡rá

♠❛✐s ❝♦♥✈❡♥✐❡♥t❡ ♥ã♦ ❡①tr❛ír♠♦s ❡stá r❛í③✱ ♣♦✐s ❛ ❛r✐t♠ét✐❝❛ t♦r♥❛✲s❡✲ á ♠❛✐s s✐♠♣❧❡s✳ ❆ s❡❣✉✐r✱ ♦ ▲❡♠❛ q✉❡ ❛❥✉❞❛rá ♥❛s ❞❡♠♦s♥tr❛çõ❡s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss✳

▲❡♠❛ ✶✳✶ P❛r❛ t♦❞♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss s ❡ t, t❡♠♦s N(s) _·N(t) =

N(st)✳

Pr♦✈❛✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛rs=a+bi_⇒N(s) = a2_+b2 _{❡t =c+di⇒N(t) =c}2_+d2_✳

◆♦t❛✲s❡ q✉❡✿ st= (a+bi)_·(c+di) = (ac₋bd) + (ad+bc)i✳ ❯t✐❧✐③❛♥❞♦ ✉♠❛ á❧❣❡❜r❛✿ N(st) = (ac₋bd)2+ (ad+bc)2

= a2c2₋2abcd+b2d2+a2d2 + 2abcd+b2c2

= a2c2+b2d2+a2d2+b2c2

= a2(c2+d2) +b2(c2+d2) = (a2+b2)(c2+d2)

= N(s)N(t).

❆ ◆♦r♠❛ ❈❛♣ít✉❧♦ ✶

❆❣♦r❛ ♣♦❞❡♠♦s ❡s❝❧❛r❡❝❡r ❛ q✉❡stã♦ ❞❛s ✉♥✐❞❛❞❡s ❞❡Z[i]✳ ❈♦♠♦ ✉♠❛ ♣r✐♠❡✐r❛

❛♣❧✐❝❛çã♦ ❞♦ ▲❡♠❛ ✶ ✳✶✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss✱ q✉❡ tê♠ ✐♥✈❡rs♦s ♠✉❧t✐♣❧✐❝❛t✐✈♦s ❡♠Z[i]✳ ❆ ✐❞❡✐❛ é ❛♣❧✐❝❛r ♥♦r♠❛s ♣❛r❛ r❡❞✉③✐r ❛ q✉❡stã♦ ❛ ✐♥✈❡rs✐❜✐✲ ❧✐❞❛❞❡ ❡♠ Z✳

❈♦r♦❧ár✐♦ ✶✳✶ ❖s ú♥✐❝♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss q✉❡ sã♦ ✐♥✈❡rsí✈❡✐s ❡♠ Z[i] sã♦ _±1 ❡ _±

i✳

Pr♦✈❛✳ P♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ±1 ❡ _±i tê♠ ✐♥✈❡rs❛s ❡♠ Z[i]✳ ❖♥❞❡ 1 ❡ ₋1 sã♦ ❛ s✉❛ ♣ró♣r✐❛ ✐♥✈❡rs❛ ❡ i ❡₋i sã♦ ✐♥✈❡rs♦s ✉♠ ❞♦ ♦✉tr♦ ❝♦♠♦ ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✳

❙✉♣♦♥❤❛ q✉❡ α _∈ Z[i] s❡❥❛ ✐♥✈❡rsí✈❡❧✱ ❞✐❣❛♠♦s α_· β = 1✱ ♣❛r❛ ❛❧❣✉♠ β _∈ Z[i]✳

◗✉❡r❡♠♦s ♠♦str❛r q✉❡ α _{∈ {±}1,_±i_}. ❚♦♠❛♥❞♦ ❛ ♥♦r♠❛ ❞❡ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛

❡q✉❛çã♦ α_·β = 1✱ ❡♥❝♦♥tr❛♠♦s N(α)N(β) = 1✳ ❊st❛ é ✉♠❛ ❡q✉❛çã♦ ❡♠ Z✱ ❡♥tã♦ s❛❜❡♠♦s q✉❡ N(α) =_±1✳ ❯♠❛ ✈❡③ q✉❡ ❛ ♥♦r♠❛ ♥ã♦ t❡♠ ✈❛❧♦r❡s ♥❡❣❛t✐✈♦s✱N(α) = 1✳ ❈♦♥s✐❞❡r❡α =a+bi✱ t❡♠♦s a2_+b2 _{= 1✱ ❡ ❛s s♦❧✉çõ❡s ✐♥t❡✐r❛s ♥♦s ♠♦str❛ q✉❛tr♦}

✈❛❧♦r❡s ♣❛r❛ α =_±1,_±i✳

❆ ♥♦r♠❛ ❞❡ ❝❛❞❛ ✐♥t❡✐r♦ ❞❡ ●❛✉ss é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♥ã♦✲♥❡❣❛t✐✈♦✱ ♠❛s ♥ã♦ é ✈❡r❞❛❞❡ q✉❡ t♦❞♦ ✐♥t❡✐r♦ ♥ã♦✲♥❡❣❛t✐✈♦ é ✉♠❛ ♥♦r♠❛✳ ❉❡ ❢❛t♦✱ ❛s ♥♦r♠❛s sã♦ ♦s ✐♥t❡✐r♦s ❞❛ ❢♦r♠❛ a2_+b2_{✱ ❡ ❛ r❡❝í♣r♦❝❛ ♥ã♦ é ✈❡r❞❛❞❡✐r❛ ♣♦✐s ❡①✐st❡ ♥ú♠❡r♦ ✐♥t❡✐r♦}

♣♦s✐t✐✈♦ q✉❡ ♥ã♦ é ❛ s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳

▲❡♠❛ ✶✳✷ P❛r❛ t♦❞♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss s ❡ t, ❝♦♠ t ₆= 0✱ t❡♠♦s q✉❡

Ns

t

= N(s)

N(t).

Pr♦✈❛✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r q✉❡ s =a+bi _⇒ N(s) =a2 _+b2 _{❡t =c+di ⇒N(t) =} c2_+d2_{✳ ◆♦t❛✲s❡ q✉❡✿} s

t =

(a+bi) (c+di) =

(a+bi) (c+di)·

(c₋di) (c₋di) =

ac+db

c2_+d2

+

bc₋ad

c2_+d2

i✳

❉✐✈✐s✐❜✐❧✐❞❛❞❡ ❈❛♣ít✉❧♦ ✶

❯t✐❧✐③❛♥❞♦ ❛ á❧❣❡❜r❛✿

Ns

t

=

ac+db

c2_+d2

2

+

bc₋ad

c2_+d2

2

=

a2_c2_{+ 2abcd+d}2_b2_+b2_c2_−2abcd+a2_d2

(c2_+d2₎2

=

a2_c2_+d2_b2_+b2_c2_+a2_d2

(c2_+d2₎2

=

a2_(c2_+d2_{) +b}2_(c2_+d2₎

(c2_+d2₎2

=

a2_+b2 c2_+d2

= N(s)

N(t).

✶✳✷✳✷ ❉✐✈✐s✐❜✐❧✐❞❛❞❡

❖ ▲❡♠❛ ❛❝✐♠❛ ♣♦❞❡ ♣❛r❡❝❡r s❡♠ ✐♠♣♦rtâ♥❝✐❛✱ ♠❛s N(s)

N(t) é ✉♠ ✐♠♣♦rt❛♥t❡ ♣r♦✲ ❜❧❡♠❛✳ ❉❛❞♦s ✐♥t❡✐r♦s ❞❡ ●❛✉ss s ❡ t✱ s❡rá q✉❡ s

t t❛♠❜é♠ é ✉♠ ✐♥t❡✐r♦ ●❛✉ss✐❛♥♦❄

❆ r❡s♣♦st❛ é s✐♠♣❧❡s✿ às ✈❡③❡s✳ ◆❡♠ t♦❞❛s ❡ss❛s ❞✐✈✐sõ❡s ♣r♦❞✉③❡♠ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❡ ●❛✉ss✱ ❡ ❝♦♠♦ s❡♠♣r❡ ❛ ❞✐✈✐sã♦ ♣♦r ③❡r♦ é ♣r♦✐❜✐❞❛✳

❚♦♠❛♥❞♦ α = (57₋ 11i) ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ♥ú♠❡r♦ β = (a +bi) ♣❛r❛

β _∈ Z[i]✱ ♦♥❞❡ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦ ❡♥tr❡ α ❡ β✱ ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú✲

♠❡r♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s✳ P♦r ❡①❡♠♣❧♦✱ (57−11i)

(14 + 3i) = 3,731−1,585i✱ ❝♦♥s✐❞❡r❛♥❞♦ três ❝❛s❛s ❞❡❝✐♠❛✐s✱ ❝❧❛r❛♠❡♥t❡ q✉❡ ♦ q✉♦❝✐❡♥t❡ ♥ã♦ ♣❡rt❡♥❝❡ Z[i]✳ ❱❛♠♦s ❛❣♦r❛

❡♥❝♦♥tr❛r ✉♠ ✐♥t❡✐r♦ ❞❡ ●❛✉ss β = (a+bi) q✉❡ ❞✐✈✐❞❡ α = (57₋11i) ❝♦♠ q✉♦❝✐✲ ❡♥t❡ s❡♥❞♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❡ ●❛✉ss✳ P❡r❝❡❜❡♠♦s q✉❡ ❛ ❞✐✈✐sã♦ t❡♠ ❛ ♠❡s♠❛ ❞❡✜♥✐çã♦ ♥♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✿

❉❡✜♥✐çã♦ ✸ ❉✐③❡♠♦s q✉❡ ♦ ✐♥t❡✐r♦ ❞❡ ●❛✉ss (a+ bi) ❞✐✈✐❞❡ ♦ ✐♥t❡✐r♦ ❞❡ ●❛✉ss (c +di) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ✐♥t❡✐r♦ ❞❡ ●❛✉ss (e + f i) t❛❧ q✉❡ (c+di) = (a+bi)(e+f i)✳ ❙❡ (a+bi) ❞✐✈✐❞❡ (c+di)✱ ✉t✐❧✐③❛♠♦s ❛ ♥♦t❛çã♦

(a+bi)_|(c+di)✳

❉✐✈✐s✐❜✐❧✐❞❛❞❡ ❈❛♣ít✉❧♦ ✶

❊①❡♠♣❧♦ ✶ ❱❡❥❛♠♦s (7₋25i)✳ ❚❡♠♦s✿

(7₋25i)_·(a+bi) = (57₋11i) 7a+ 7bi₋25ai₋25bi2 = (57₋11i) (7a+ 25b) + (7b₋25a)i = (57₋11i)

❱❛♠♦s ❢♦r♠❛r ✉♠ s✐st❡♠❛✿

7a+ 25b= 57

−25a+ 7b=₋11 ⇒a= 1 ❡ b= 2✳ (1 + 2i)∈Z[i]✳ P♦rt❛♥t♦✱ ✈✐♠♦s q✉❡ (57−11i)

(7₋25i) = 1 + 2i✳

▲♦❣♦✱ (7₋25i)_|(57₋11i) ❡ (1 + 2i)_|(57₋11i)✳ _⋄

❊①❡♠♣❧♦ ✷ ❊♥tã♦ (14 + 3i)∤(57₋11i)✱ ♣♦✐s

(14 + 3i)(a+bi) = (57₋11i) 14a+ 14bi+ 3ai+ 3bi2 = (57₋11i) (14a₋3b) + (14b+ 3a)i = (57₋11i)

❱❛♠♦s ❢♦r♠❛r ✉♠ s✐st❡♠❛✿

14a₋3b= 57

3a+ 14b=₋11 ⇒a= 153

41 ❡ b =− 65 41✳ 153 41 − 65 41i

6∈Z[i]✳ _⋄

Pr♦♣♦s✐çã♦ ✶ ❖ ✐♥t❡✐r♦ ❞❡ ●❛✉ss α =a+bi é ❞✐✈✐sí✈❡❧ ♣♦r ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ c

s❡✱ ❡ s♦♠❡♥t❡ s❡✱ c_|a ❡ c_|b ❡♠ Z✳

Pr♦✈❛✳ ❚♦♠❡c_|(a+bi) ❡♠ Z[i] é ✐❣✉❛❧a+bi=c_·(m+ni)♣❛r❛ q✉❛❧q✉❡r m, n_∈Z

q✉❡ é ❡q✉✐✈❛❧❡♥t❡ a=cm❡ b =cn, ♦✉c_|a ❡ c_|b✳

❚♦♠❛♥❞♦b= 0♥♦ Pr♦♣♦s✐çã♦ ✶ ♥♦s ❞✐③ q✉❡ ❛ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ✐♥t❡✐r♦s ❝♦♠✉♥s ♥ã♦ ♠✉❞❛ q✉❛♥❞♦ s❡ tr❛❜❛❧❤❛ ❡♠ Z[i] : ♣❛r❛ a, c _∈ Z, ❡♥tã♦ c _| a ❡♠ Z[i] s❡✱ ❡ s♦♠❡♥t❡ s❡✱ c _| a ❡♠ Z. ◆♦ ❡♥t❛♥t♦✱ ✐ss♦ ♥ã♦ s✐❣♥✐✜❝❛ q✉❡ ♦✉tr♦s ❛s♣❡❝t♦s ❞❡

Z ♣❡r♠❛♥❡❝❡♠ ♦ ♠❡s♠♦ ♣❛r❛ Z[i]✳ P♦r ❡①❡♠♣❧♦✱ ✈❡r❡♠♦s q✉❡ ❛❧❣✉♥s ♣r✐♠♦s ❡♠

Z ♥ã♦ sã♦ ♣r✐♠♦ ❡♠ Z[i]✳ ❆ ♠✉❧t✐♣❧✐❝❛t✐✈✐❞❛❞❡ ❞❛ ♥♦r♠❛ tr❛♥s❢♦r♠❛ ❛s r❡❧❛çõ❡s ❞❡

❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♠ Z[i] ❡♠ r❡❧❛çõ❡s ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♠ Z✳

Pr♦♣♦s✐çã♦ ✷ P❛r❛ α, β ❡♠ Z[i], s❡ β _|α ❡♠ Z[i], ❡♥tã♦ N(β)_|N(α) ❡♠ Z✳

Pr♦✈❛✳ ❈♦♥s✐❞❡r❡ α =β_·γ ♣❛r❛ Z[i]✳ ❚♦♠❛♥❞♦ ❛ ♥♦r♠❛ ❞❡ ❛♠❜♦s ♦s ❧❛❞♦s✱ t❡♠♦s

q✉❡✿ N(α) =N(β)_·N(γ)_⇒N(β)_|N(α)✳

❉✐✈✐s✐❜✐❧✐❞❛❞❡ ❈❛♣ít✉❧♦ ✶

❈♦r♦❧ár✐♦ ✶✳✷ ❯♠ ✐♥t❡✐r♦ ❞❡ ●❛✉ss t❡♠ ♥♦r♠❛ ♣❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❡ é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ (1 +i)✳

Pr♦✈❛✳ ❈♦♠♦ N(1 +i) = 2, q✉❛❧q✉❡r ♠ú❧t✐♣❧♦ ❞❡(1 +i) t❡♠ ♥♦r♠❛ ♣❛r✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s✉♣♦♥❤❛ q✉❡ (a+bi) t❡♠ ♥♦r♠❛ ♣❛r✳ ❊♥tã♦ a2_+b2 _{≡0 ✭♠♦❞ ✷✮✳ ❆♥❛❧✐s❛♥❞♦}

♦s ❝❛s♦s✿

✐ ✲ P❛r❛ a s❡♥❞♦ ✉♠ ♥ú♠❡r♦ ♣❛r ❡ b s❡♥❞♦ ✉♠ ♥ú♠❡r♦ í♠♣❛r. ❚❡♠♦s q✉❡✿ a= 2m ❡ b= 2n+ 1, ♦♥❞❡ m, n_∈Z,❡♥tã♦✿

(2m)2 + (2n+ 1)2 _≡1 ✭♠♦❞ ✷✮✳

✐✐ ✲ P❛r❛ a ❡ b ❛♠❜♦s ♥ú♠❡r♦s ♣❛r❡s. ❚❡♠♦s✿ a = 2m ❡ b = 2n, ♦♥❞❡ m, n _∈ Z,

❡♥tã♦✿

(2m)2+ (2n)2 _≡0 ✭♠♦❞ ✷✮✳

✐✐✐ ✲ P❛r❛ a ❡ b ❛♠❜♦s ♥ú♠❡r♦s í♠♣❛r❡s. ❚❡♠♦s q✉❡ ✿ a= 2m+ 1 ❡ b = 2n₋1, ♦♥❞❡ m, n_∈Z, ❡♥tã♦✿

(2m+ 1)2+ (2n₋1)2 _≡0✭♠♦❞ ✷✮✳

m₋n _≡0✭♠♦❞ ✷✮✳

m _≡n ✭♠♦❞ ✷✮✳

P♦rt❛♥t♦ a_≡b ✭♠♦❞ ✷✮✳

❈♦♥s✐❞❡r❡ a+bi = (1 +i)_·(u+vi), ♣❛r❛ ❛❧❣✉♥s u, v _∈Z✳ ❚❡♠♦s q✉❡✿

a+bi = u+vi+ui+vi2

= (u₋v) + (u+v)i

❱❛♠♦s ❢♦r♠❛r ✉♠ s✐st❡♠❛ ✿

u₋v =a

u+v =b ⇒u=

b+a

2 ❡ v =

b₋a

2 ✳

❚❡♠♦s q✉❡u❡vsã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❞❡s❞❡ q✉❡a_≡b ✭♠♦❞ ✷✮✳ ❆ss✐♠✱(1+i)_|(a+bi) s❡♥❞♦ a ❡ b ❛♠❜♦s ♣❛r❡s ♦✉ ❛♠❜♦s í♠♣❛r❡s✳

❊①❡♠♣❧♦ ✸ N(1 + 3i) = 12_{+ 3}2 _{= 10≡0 ✭♠♦❞ ✷✮✱ t❡♠♦s q✉❡ (1 +i)|(1 + 3i)✱ ♦✉}

s❡❥❛✱ (1 + 3i) = (1 +i)(2 +i)✳_⋄

❖ ❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦ ❈❛♣ít✉❧♦ ✶

❊①❡♠♣❧♦ ✹ N(1 + 2i) = 12_{+ 2}2 _{= 5≡1 ✭♠♦❞ ✷✮✱ t❡♠♦s q✉❡ (1 +i) ∤(1 + 2i)✱ ♦✉}

s❡❥❛✱ (1 + 2i) ♥ã♦ é ♠ú❧t✐♣❧♦ ❞❡(1 +i)✳ _⋄

❊①❡♠♣❧♦ ✺ N(2 + 6i) = 22_{+ 6}2 _{= 40≡0 ✭♠♦❞ ✷✮✱ t❡♠♦s q✉❡ (1 +i)|(2 + 6i)✱ ♦✉}

s❡❥❛✱ (2 + 6i) = (1 +i)(4 + 2i)✳_⋄

❆ Pr♦♣♦s✐çã♦ ✷ é ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♠♦str❛r ❞❡ ✉♠❛ ❢♦r♠❛ rá♣✐❞❛ ❡ ♣rát✐❝❛ ❝♦♠♦ ✈❡r✐✜❝❛r s❡ ♦ ✐♥t❡✐r♦ ❞❡ ●❛✉ss ♥ã♦ é ❞✐✈✐sí✈❡❧ ♣♦r ♦✉tr♦ ✐♥t❡✐r♦ ❞❡ ●❛✉ss✳ ❚r❛♥s❢♦r♠❛r ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♠ Z[i] ❡♠ Z✱ t❡♠ ✉♠ ❛♣❡❧♦ ó❜✈✐♦✱ ✉♠❛ ✈❡③ q✉❡ é ♠❛✐s ❝♦♥❢♦rtá✈❡❧ tr❛❜❛❧❤❛r ❝♦♠ ❛ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♠ Z✳

❆ r❡❝í♣r♦❝❛ ♥❡♠ s❡♠♣r❡ é ✈❡r❞❛❞❡✐r❛✳ ❱❡❥❛♠♦s✱ ♣♦r ❡①❡♠♣❧♦ ♣❛r❛ α = (14 + 3i) ❡β = (4 + 5i)✳ ❚❡♠♦s q✉❡ N(β) = 42 _{+ 5}2 _{= 41 ❡N(α) = 14}2 _{+ 3}2 _{= 205 =}

41_·5, ❡♥tã♦ N(β)_|N(α), ❡♠ Z✱ ♣ór❡♠ (4 + 5i)∤(14 + 3i)✳

❊♠ Z, s❡_|m_| = _|n_|, ❡♥tã♦ m = _±n✱ ♦✉ s❡❥❛✱ m ❡ n sã♦ ❛ss♦❝✐❛❞♦s✳ ❊♠ Z[i] é ❢❛❧s♦✿ ♣♦✐s s❡ N(α) = N(β) ♥ã♦ s✐❣♥✐✜❝❛ q✉❡ s❡❥❛ ✈❡r❞❛❞❡ q✉❡α ❡β sã♦ ♠ú❧t✐♣❧♦s

✉♥✐tár✐♦s ❡♥tr❡ s✐✳ ❈♦♥s✐❞❡r❡ (4 + 5i)❡ (4₋5i).❚❡♠♦s N(4₋5i) = N(4 + 5i) = 41✱ ♠❛s ♦s ♠ú❧t✐♣❧♦s ✉♥✐tár✐♦s ❞❡ (4 + 5i) sã♦✿

✭❛✮ (4 + 5i)_·1 = 4 + 5i

✭❜✮ (4 + 5i)_·(₋1) =₋4₋5i

✭❝✮ (4 + 5i)_·i= 4i₋5 = ₋5 + 4i

✭❞✮ (4 + 5i)_·(₋i) =₋4i+ 5 = 5₋4i

❖ ♥ú♠❡r♦(4₋5i)♥ã♦ ❡stá ♥❡st❛ ❧✐st❛✱ ❡♥tã♦(4 + 5i) ❡ (4₋5i)♥ã♦ sã♦ ♠ú❧t✐♣❧♦s ✉♥✐tár✐♦s✳ ■r❡♠♦s ✈❡r ♠❛✐s t❛r❞❡ q✉❡ (4 + 5i)❡ (4₋5i) sã♦ ❛✐♥❞❛ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ❡♠ Z[i].❊♠ s✉♠❛✱ t♦♠❛r ❛ ♥♦r♠❛ ❡♠ Z[i]é ✉♠ ♣❛ss♦ ♠❛✐s ❞rást✐❝♦ ❞♦ q✉❡ ❛ r❡♠♦çã♦ ❞❡ ✉♠ s✐♥❛❧ ❡♠ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳

✶✳✷✳✸ ❖ ❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦

❖ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ❡♠Z[i] é ❛♥á❧♦❣♦ à ❞✐✈✐sã♦ ❝♦♠ r❡st♦ ❡♠ Z.

❚❡♦r❡♠❛ ✹ ✭❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦✮✳ P❛r❛ α, β _∈ Z[i] ❝♦♠ β ₆= 0, ❡①✐st❡♠ γ, δ _∈

Z[i] ❞❡ t❛❧ ❢♦r♠❛ q✉❡ α = βγ+δ ❡ δ > 0 ♦✉ N(δ) < N(β)✳ ◆❛ ✈❡r❞❛❞❡✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r δ ❞❡ t❛❧ s♦rt❡ q✉❡ N(δ)6 N(β)

2 ✳

❖ ❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦ ❈❛♣ít✉❧♦ ✶

❖s ♥ú♠❡r♦s ❞♦s γ ❡ δ sã♦ ♦ q✉♦❝✐❡♥t❡ ❡ r❡st♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ ♦ r❡st♦ é ❧✐♠✐✲

t❛❞♦ ✭❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ s✉❛ ♥♦r♠❛✮ ♣❡❧♦ t❛♠❛♥❤♦ ❞♦ β q✉❡ r❡♣r❡s❡♥t❛ ♦ ❞✐✈✐s♦r✳

❆♥t❡s ❞❡ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ✹✱ ♥♦t❛♠♦s q✉❡ ❤á ✉♠❛ s✉t✐❧❡③❛ ♥❛ t❡♥t❛t✐✈❛ ❞❡ ❝❛❧❝✉❧❛r γ ❡δ. ■st♦ ♣♦❞❡ s❡r ♠❡❧❤♦r ❝♦♠♣r❡❡♥❞✐❞♦ ♣♦r ♠❡✐♦ ❞♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✳

❊①❡♠♣❧♦ ✻ ❈♦♥s✐❞❡r❡ α= 27₋23i ❡ β = 8 +i_⇒N(β) = 82_{+ 1}2 _{= 65✳ ◗✉❡r❡♠♦s}

❡s❝r❡✈❡r α=βγ+δ ❝♦♠ N(δ)<65✳

(27₋23i) = (8 +i)_·(a+bi)_⇒a= 193

65 ❡ b=− 211

65

❚❡♠♦s q✉❡✿ 193

65 = 2,969. . . ❡ − 211

65 =−3,246. . .✳ ❈♦♠♦ a ❡ b6∈Z✱ ❡♥tã♦ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦ s❡✉ ✐♥t❡✐r♦ ❡ t❡♠♦s q✉❡ γ = 2₋3i. ◆♦ ❡♥t❛♥t♦✱ α=βγ+δ _⇒

δ = α₋βγ

= (27₋23i)₋(8 +i)(2₋3i) = (27₋23i)₋(16 + 2i₋24i+ 3) = 8₋i.

❈♦♠♦ δ = 8₋i_⇒N(8₋i) = 82_{+ 1}2 _{= 65 =N(β)✳ ⋄}

❖ ✐❞❡❛❧ é q✉❡ ❛ ♥♦r♠❛ ❞♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ s❡❥❛ ✐♥❢❡r✐♦r ❛ ♥♦r♠❛ ❞♦ ❞✐✈✐s♦r✳ ❆ss✐♠✱ ❛ ♥♦ss❛ ❡s❝♦❧❤❛ ♣❛r❛ γ ❡ δ ♥ã♦ é ❞❡s❡❥á✈❡❧✳ P❛r❛ ❝♦rr✐❣✐r ❛ ♥♦ss❛ ❛❜♦r❞❛❣❡♠✱ t❡♠♦s

q✉❡ ♣❡♥s❛r ❝✉✐❞❛❞♦s❛♠❡♥t❡ s♦❜r❡ ❛ ♥♦ss❛ ❢♦r♠❛ ❞❡ s✉❜st✐t✉✐r 193

65 = 2,969. . . ❡ − 211

65 =−3,246. . . ❝♦♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣ró①✐♠♦s✳ ◆♦t❡✲s❡ q✉❡

193 65 ❡ −

211 65 ❡stã♦ ♠❛✐s ♣ró①✐♠♦ à ❞✐r❡✐t❛ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ ❱❛♠♦s ✉s❛r ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♠❛✐s ♣ró①✐♠♦ ❡♠ ✈❡③ ❞❡ ♦ ♠❛✐♦r ✐♥t❡✐r♦✱ ♣♦rt❛♥t♦✱

193 65

é ♠❛✐s ♣ró①✐♠♦ ❞❡ ✸ ❡

−211₆₅

é ♠❛✐s ♣ró①✐♠♦ ❞❡ −3✱ ❧♦❣♦ γ = 3₋3i✳ ❚❡♠♦s q✉❡ ✿

δ = α₋βγ

= (27₋23i)₋(8 +i)(3₋3i) = (27₋23i)₋(24 + 3i₋24i+ 3) = ₋2i

❈♦♠♦ δ =₋2i_⇒ N(δ)<65. ❊♥tã♦✱ ♥ós ✉s❛♠♦s γ = 3₋3i ❡ δ =₋2i. ❊s❝♦❧❤❡♥❞♦

♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♠❛✐s ♣ró①✐♠♦✱ ❡♠ ✈❡③ ❞♦ q✉❡ ♦ ♠❛✐♦r ✐♥t❡✐r♦✱ t❛♠❜é♠ ♣♦❞❡r✐❛ s❡r ❢❡✐t♦ ❡♠ Z.

❖ ❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦ ❈❛♣ít✉❧♦ ✶

❊♠ ✈❡r❞❛❞❡✱ ♦ q✉❡ ❡st❛♠♦s ❢❛③❡♥❞♦ é t♦♠❛♥❞♦ ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♠❛✐♦r ❞♦ q✉❡ ❛❧❣✉é♠✳ P♦r ❡①❡♠♣❧♦✱ 34

9 = 3,77. . . ❡stá ♠❛✐s ♣ró①✐♠♦ ❞❡ ✹ ❞♦ q✉❡ ❞❡ ✸✳ ❊♠

r❡❧❛çã♦ ❛ ❞✐✈✐sã♦ ❝♦♠ r❡st♦✱ ✐st♦ ❝♦rr❡s♣♦♥❞❡ ❛✿

34 = 9_·4₋2

♦✉

34 = 9_·3 + 7.

❖ r❡st♦ ❞❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ é ♥❡❣❛t✐✈♦✱ ♠❛s é ♠❡♥♦r ❡♠ ✈❛❧♦r ❛❜s♦❧✉t♦✳ ❖ q✉❡ ❡♥❝♦♥tr❛♠♦s ❛q✉✐ é ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ♠♦❞✐✜❝❛❞❛ ❡♠ Z. ◆♦ ❚❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦

♥♦r♠❛❧♠❡♥t❡ ♦ r❡st♦ ♥ã♦ é ♥❡❣❛t✐✈♦ ❡ é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡ ♣♦r |β_|✳

➚s ✈❡③❡s✱ ♦ ♥ú♠❡r♦ ♣♦❞❡ ❡st❛r ♥♦ ♠❡✐♦ ❡♥tr❡ ❞♦✐s ♠ú❧t✐♣❧♦s ❞❡b, ❝❛s♦ ❡♠ q✉❡ ♦

q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ♥ã♦ sã♦ ú♥✐❝♦s✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ a= 27 ❡ b= 6✱ ❡♥tã♦ ❡①✐st❡ ✉♠

♥ú♠❡r♦ ❡q✉✐❞✐st❛♥t❡ ❡♥tr❡ 4b ❡ 5b✿

27 = 6.4 + 3 ❡27 = 6.5₋3.

❆ss✐♠✱ t❡♠♦s ❞✉❛s ❡s❝♦❧❤❛s ❞❡ r = 3♦✉ r = ₋3✳ ❖ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ✉s✉❛❧ ❡♠ Z t❡♠ ✉♠ q✉♦❝✐❡♥t❡ ❡ ✉♠ r❡st♦ ú♥✐❝♦✱ ♠❛s ❛ ✈❡rsã♦ ♠♦❞✐✜❝❛❞❛ ♥ã♦ ❛♣r❡s❡♥t❛ ❡st❛ s✐♥❣✉❧❛r✐❞❛❞❡✳ ■st♦ ♣♦❞❡ ♣❛r❡❝❡r ✉♠❛ ❝❛❧❛♠✐❞❛❞❡✱ ♠❛s é ❡①❛t❛♠❡♥t❡ ♦ q✉❡ ♣r❡❝✐s❛♠♦s ♣❛r❛ ♣r♦✈❛r ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ❡♠ Z[i] ✭❚❡♦r❡♠❛ ✻✮✳ ❆♣ós ❛ ♣r♦✈❛✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ♠❛✐s ❡①❡♠♣❧♦s✳

Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✹✿ ❈♦♥s✐❞❡r❡ α, β _∈ Z[i], ❝♦♠ β ₆= 0✳ ◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r

γ, δ _∈Z[i]t❛✐s q✉❡ α=βγ+δ ❡ N(δ)_≤ 1

2N(β)✳ ❈♦♥s✐❞❡r❛♥❞♦ q✉❡β¯é ♦ ❝♦♥❥✉❣❛❞♦ ❞❡ β✳ ❚❡♠♦s q✉❡

α

β =

αβ¯ ββ¯ =

αβ¯ N(β) =

m+ni

N(β) =

m N(β)

+

n N(β)

i. ✭✶✳✶✮

❯s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ♠♦❞✐✜❝❛❞♦ ❡♠ Z, t❡♠♦s

m =N(β)_·q1+r1 ❡ n=N(β)·q2+r2,

❡♠ q✉❡ q1 ❡ q2 ❡stã♦ ❡♠ Z ❡0_≤| r1 _|,_|r2 _|≤ 1

2N(β). ❙✉❜st✐t✉✐♥❞♦ m ❡ n ❡♠ ✭✶✳✶✮✱ t❡♠♦s q✉❡

α

β =

N(β)q1+r1+ (N(β)q2+r2)i N(β)

= q1+q2i+ r1+r2i

N(β)

❖ ❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦ ❈❛♣ít✉❧♦ ✶

❞♦♥❞❡

α₋βγ = r1+_¯r2i

β , ✭✶✳✷✮

❡♠ q✉❡ γ =q1+q2i✳ ❈♦♥s✐❞❡r❡ δ=α₋βγ✳

❆♣❧✐❝❛♥❞♦ ♥♦r♠❛ ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡ ✭✶✳✷✮ ❡ s❛❜❡♥❞♦ q✉❡N(β) = N( ¯β) ♦❜t❡♠♦s

N(α₋βγ) = N(r1+r2i)

N( ¯β) =

r2 1 +r22 N(β) .

❙❛❜❡♠♦s q✉❡ 0_≤|r1 _| ❡ _|r2 _|≤ 1

2N(β)✱ ❞♦♥❞❡

N(α₋βγ)_≤ (1/4)N(β)

2 _{+ (1/4)N(β)}2

N(β) =

1 2N(β).

❊①❡♠♣❧♦ ✼ ❈♦♥s✐❞❡r❡ α = 11 + 10i ❡ β = 4 +i_⇒N(β) = 42 _{+ 1}2 _{= 17✳ ❆ss✐♠✱}

α

β =

αβ¯ N(β)

= (11 + 10i)(4−i) 17

= 44−11i+ 40i+ 10 17

= 54 + 29i 17 .

❈♦♠♦ 54

17 = 3,1764. . . ❡ 29

17 = 1,7058. . .✱ t❡♠♦s γ = 3 + 2i✳ ❊♥tã♦✿

δ=α₋βγ = (11 + 10i)₋(4 +i)(3 + 2i) = (11 + 10i)₋(12 + 8i+ 3i₋2) = 1₋i.

P♦r ✜♠ q✉❡ N(δ) = 12_{+ (−1)}2 _{= 2≤} 1

2N(β)✳ ⋄

❊①❡♠♣❧♦ ✽ ❈♦♥s✐❞❡r❡ α = 41 + 24i ❡ β = 11₋2i _⇒N(β) = 112 _{+ (−2)}2 _{= 125✳}

❖ ❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦ ❈❛♣ít✉❧♦ ✶

❆ss✐♠✱

α

β =

αβ¯ N(β)

= (41 + 24i)(11 + 2i) 125

= 451 + 82i+ 264i−48 125

= 403 + 346i 125 .

❈♦♠♦ 403

125 = 3,224. . . ❡ 346

125 = 2,768. . .✱ t❡♠♦s γ = 3 + 3i✳ ❊♥tã♦✿

δ =α₋βγ = (41 + 24i)₋(11₋2i)(3 + 3i) = (41 + 24i)₋(33 + 33i₋6i+ 6) = 2₋3i.

P♦r ✜♠ q✉❡ N(δ) = 22_{+ (−3)}2 _{= 13≤} 1

2N(β)✳ ⋄

❍á ✉♠❛ ❞✐❢❡r❡♥ç❛ ✐♥t❡r❡ss❛♥t❡ ❡♥tr❡ ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ❡♠ Z[i] ❡ ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ❡♠ Z✱ ❛ s❛❜❡r✱ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ♥ã♦ sã♦ ú♥✐❝♦s ❡♠ Z[i]✳

❊①❡♠♣❧♦ ✾ ❈♦♥s✐❞❡r❡ α = 37 + 2i ❡ β = 11 + 2i _⇒ N(β) = 112 _{+ 2}2 _{= 125.}

❈❛❧❝✉❧❛♠♦s✿

α

β =

αβ¯ N(β)

= (37 + 2i)(11−2i) 125

= 407−74i+ 22i+ 4 125

= 411−52i 125 .

❈♦♠♦ 411

125 = 3,288. . . ❡ − 52

125 =−0,416. . .✱ t❡♠♦s✿ γ1 = 3✱ ❝♦♥s✐❞❡r❛♥❞♦ b = 0 ❡

γ2 = 3₋i✱ ❝♦♥s✐❞❡r❛♥❞♦ b=₋1✳ ❊♥tã♦ ♣❛r❛ γ1 = 3✱ t❡♠♦s

δ=α₋βγ = (37 + 2i)₋(11 + 2i)(3) = (37 + 2i)₋33₋6i

= 4₋4i

❖ ❚❡♦r❡♠❛ ❞❛ ❉✐✈✐sã♦ ❈❛♣ít✉❧♦ ✶

◆♦t❡ q✉❡ N(δ) = 42_{+ (−4)}2 _{= 32≤} 1

2N(β)✳ ❏á ♣❛r❛ γ2 = 3₋i✳ t❡♠♦s

δ=α₋βγ = (37 + 2i)₋(11 + 2i)(3₋i) = (37 + 2i)₋(33 + 6i₋11i+ 2) = 2 + 7i

◆♦t❡ q✉❡ N(δ) = 22 _{+ 7}2 _{= 53 ≤} 1

2N(β)✳ ❈♦♥❝❧✉✐♠♦s q✉❡ é ✈❡r❞❛❞❡ ♣❛r❛ α =

β_·3 + (4₋4i) ♦✉ α=β_·(3₋i) + (2 + 7i)✳_⋄

◆♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ♦ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦ ♥ã♦ s❡r✐❛ ❞♦ ♥♦ss♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❡♠ Z[i].

❱❡❥❛♠♦s ❛❜❛✐①♦ ✉♠ ❡①❡♠♣❧♦ ♦♥❞❡ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ♣❡r♠✐t❡ ❞♦✐s r❡s✉❧t❛❞♦s ❞✐❢❡r❡♥t❡s✳

❊①❡♠♣❧♦ ✶✵ ❈♦♥s✐❞❡r❡α = 1+8i ❡ β = 2₋4i_⇒N(β) = 22₊₍₋₄₎2 _{= 20✳ ❆ss✐♠✱}

α

β =

αβ¯ N(β)

= (1 + 8i)(2 + 4i) 20

= 2 + 4i+ 16i−32 20

= −30 + 20i 20

=

−3₂

+i.

❈♦♠♦−3₂ =₋1,5♣♦❞❡♠♦s ✉s❛rγ1 =₋2+i ♦✉ γ2 =₋1+i✳ ❊♥tã♦ ♣❛r❛γ1 =₋2+i,

t❡♠♦s

δ =α₋βγ = (1 + 8i)₋(2₋4i)(₋2 +i) = (1 + 8i)₋(₋4 + 2i+ 8i+ 4) = 1₋2i.

◆♦t❡ q✉❡ N(δ) = 12_{+ (−2)}2 _{= 5≤} 1

2N(β)✳

❖ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ❈❛♣ít✉❧♦ ✶

❏á ♣❛r❛ γ2 =−1 +i, t❡♠♦s

δ =α₋βγ = (1 + 8i)₋(2₋4i)(₋1 +i) = (1 + 8i)₋(₋2 + 2i+ 4i+ 4) = ₋1 + 2i.

◆♦t❡ q✉❡ N(δ) = (₋1)2 _{+ 2}2 _{= 5 ≤} 1

2N(β). ❙❡ ❡s❝♦❧❤❡r♠♦s ♦ γ1 t❡r❡♠♦s α =

β_·(₋2 +i) + (1₋2i), ❡ s❡ ❡s❝♦❧❤❡r♠♦s ♦ γ2 t❡r❡♠♦s α =β_·(₋1 +i) + (₋1 + 2i)✳_⋄

✶✳✷✳✹

❖ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s

❉❡✜♥✐çã♦ ✺ P❛r❛ α ₆= 0 ❡ β _∈ Z[i], ✉♠ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ α ❡ β é ❞✐✈✐s♦r

❝♦♠✉♠ ❝♦♠ ♥♦r♠❛ ♠á①✐♠❛✳

■st♦ é ❛♥á❧♦❣♦ ❛ ❞❡✜♥✐çã♦ ✉s✉❛❧ ❞❡ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ✭♠❞❝✮ ❡♠ Z, ❡①❝❡t♦ ♦

❝♦♥❝❡✐t♦ ♥ã♦ ❡stá ♣r❡s♦ ❝♦♠ ✉♠ ♥ú♠❡r♦ ❡s♣❡❝í✜❝♦✳ ❙❡ δ é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠

❞❡ α ❡ β, ♣♦r ✐ss♦ sã♦ ✭♣❡❧♦ ♠❡♥♦s✮ s❡✉s ♠ú❧t✐♣❧♦s ✉♥✐tár✐♦s ₋δ, iδ ❡ ₋iδ ✳ ❚❛❧✈❡③

❡①✐st❛♠ ♦✉tr♦s ❣r❛♥❞❡s ❞✐✈✐s♦r❡s ❝♦♠✉♥s✱ ♣♦ré♠ s✐♠♣❧❡s♠❡♥t❡ ♥ã♦ s❛❜❡♠♦s ❛✐♥❞❛✳ ✭❱❛♠♦s ❞❡s❝♦❜r✐r ♥♦ ❈♦r♦❧ár✐♦ ✹✳✼✮✳ P♦❞❡♠♦s ❢❛❧❛r ❞❡ ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✱ ♠❛s ♥ã♦ é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠✳ ❆ s❡♠❡❧❤❛♥t❡ t❡❝♥✐❝✐❞❛❞❡ ♦❝♦rr❡r✐❛ ❡♠ Z s❡ ❞❡✜♥íss❡♠♦s ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❝♦♠♦ ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❝♦♠ ♦ ♠❛✐♦r ✈❛❧♦r ❛❜s♦❧✉t♦✱ ❡♠ ✈❡③ ❞❡ ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ♣♦s✐t✐✈♦✳

❚❡♦r❡♠❛ ✻ ✭❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✮✳ ❚♦♠❡ α, β _∈Z[i] ♦♥❞❡ sã♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❆♣❧✐❝❛♥❞♦ r❡♣❡t✐❞❛♠❡♥t❡ ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ♦♥❞❡ ♦ r❡st♦ é ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ t❡✲ r❡♠♦s

α = βγ1 +δ1, com N(δ1)< N(β)

β = δ1γ2 +δ2, com N(δ2)< N(δ1)

δ1 = δ2γ3 +δ3, com N(δ3)< N(δ2) ✳✳✳

❖ ú❧t✐♠♦ r❡st♦ s❡♥❞♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ é ❞✐✈✐sí✈❡❧ ♣♦r t♦❞♦s ♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡

α ❡ β, ❡ s❡rá ♦ ❞✐✈✐s♦r ❝♦♠✉♠✱ ♣♦r ✐ss♦ é ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ α ❡ β.

❖ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ❈❛♣ít✉❧♦ ✶

Pr♦✈❛✳ ❆ ♣r♦✈❛ é ❛♥á❧♦❣❛ ❛♦ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ❡♠ Z. ❖ r❛❝✐♦❝í♥✐♦ é ❛ ♣❛rt✐r

❞❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ♦♥❞❡ ❝❛❞❛ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ α ❡ β ❞✐✈✐❞❡ ♣❡❧♦ ú❧t✐♠♦ r❡st♦

✭❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✮✳ P♦rt❛♥t♦✱ ❡st❡ ú❧t✐♠♦ r❡st♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠✱ q✉❡ é ❞✐✈✐sí✈❡❧ ♣♦r t♦❞♦s ♦s ♦✉tr♦s✳ P♦r ✐ss♦ é ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳

❉❡✜♥✐çã♦ ✼ ◗✉❛♥❞♦ α ❡ β _∈ Z[i] tê♠ ❛s ✉♥✐❞❛❞❡s ❝♦♠♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✱ ❝❤❛♠❛♠♦s ❞❡ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✱ ♦✉ s❡❥❛✱ ♣r✐♠♦s ❡♥tr❡ s✐✳

❊①❡♠♣❧♦ ✶✶ ❈❛❧❝✉❧❛♠♦s ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡α = 32 + 9i ❡ β = 4 + 11i✳

Pr✐♠❡✐r♦ r❡❛❧✐③❛♠♦s ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦ ❡✱ t❡r❡♠♦s γ = 2 ₋ 2i ❡ δ = 2 ₋ 5i✳

❆♣❧✐❝❛♠♦s ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✱ ❡♥❝♦♥tr❛♠♦s

32 + 9i = (4 + 11i)(2₋2i) + (2₋5i) 4 + 11i = (2₋5i)(₋2 +i) + (3₋i)

2₋5i = (3₋i)(1₋i)₋i

3₋i = (₋i)(1 + 3i) + 0.

❖ ú❧t✐♠♦ r❡st♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ é (₋i), ❡♥tã♦ mdc(32 + 9i,4 + 11i) = ₋i, sã♦

r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✳ ⋄

❊①❡♠♣❧♦ ✶✷ ▼♦str❛r q✉❡ ♦s ❝♦♥❥✉❣❛❞♦s(4+5i) ❡ (4₋5i),sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s

❡♠ Z[i]✳ ❈♦♥s✐❞❡r❡ α = 4 + 5i ❡ β = 4 ₋ 5i✱ ✉t✐❧✐③❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❛ ❞✐✈✐sã♦

❡♥❝♦♥tr❛r❡♠♦s γ =i ❡ δ =₋(1₋i), ♣♦rt❛♥t♦ t❡♠♦s q✉❡

4 + 5i = (4₋5i)i₋(1₋i) 4₋5i = ₋(1₋i)(₋4)₋i

−(1₋i) = ₋i(1 +i) + 0.

❖ ú❧t✐♠♦ r❡st♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ é ✉♠❛ ✉♥✐❞❛❞❡✱ ♦✉ s❡❥❛✱ mdc(4 + 5i,4₋5i) = ₋i✱

t❡♠♦s q✉❡ (4 + 5i) ❡ (4₋5i) sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✳_⋄

❊①❡♠♣❧♦ ✶✸ ❱❛♠♦s ✈❡r✐✜❝❛r ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ♣❛r❛ α = 11 + 3i ❡ β = 1 + 8i✳ ❊♥❝♦♥tr❛♠♦s γ = 1₋i ❡ δ= 2₋4i, ❡♥tã♦

11 + 3i = (1 + 8i)(1₋i) + (2₋4i) 1 + 8i = (2₋4i)(₋1 +i) + (₋1 + 2i) 2₋4i = (₋1 + 2i)(₋2) + 0.

❖ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t ❈❛♣ít✉❧♦ ✶

❖ mdc(11 + 3i,1 + 8i) =₋1 + 2i✳ P♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❢♦r♠❛ ❞✐❢❡r❡♥t❡✱ ❡ ♦❜t❡r

♥♦ ú❧t✐♠♦ r❡st♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ❛ss✐♠

11 + 3i = (1 + 8i)(1₋i) + (2₋4i) 1 + 8i = (2₋4i)(₋2 +i) + (1₋2i) 2₋4i = (1₋2i)2 + 0.

▲♦❣♦✱ mdc(11 + 3i,1 + 8i) = 1₋2i✱ ❛ss✐♠✱ s❛❜❡♠♦s q✉❡ (1₋2i) = (₋1)(₋1 + 2i)✳ _⋄

❙❡ ξ é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ α ❡ β✱ ❡♥tã♦ N(ξ) _| N(α) ❡N(ξ) _| N(β)✱ ❞❡ ♠♦❞♦ q✉❡ N(ξ)_|(N(α), N(β))✳ ◆♦ ❡♥t❛♥t♦✱ ♣♦❞❡ ❛❝♦♥t❡❝❡r N(ξ)<(N(α), N(β)).

◆♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ♦♥❞❡ α = 4 + 5i ❡ β = 4₋5i sã♦ ♣r✐♠♦s ❡♥tr❡ s✐ ❡✱ ♣♦rt❛♥t♦✱ N(ξ) = 1✱ ♠❛s N(α) = N(β) = 41. ◆♦ ❡①❡♠♣❧♦ ♦♥❞❡ α = 11 + 3i ❡β = 1 + 8i _⇒ N(α) = 112 _{+ 3}2 _{= 130 ❡ N(β) = 8}2_{+ 1}2 _{= 65, t❡♥❞♦ mdc(130,65) = 65, ❡} mdc(α, β) = ₋1 + 2i, ♦♥❞❡ s✉❛ ♥♦r♠❛N(₋1 + 2i) = (₋1)2_{+ 2}2 _{= 5✳}

❙✉♣♦♥❤❛(N(α), N(β)) = 1✳ ❊♥tã♦✱ q✉❛❧q✉❡r ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡α ❡ β t❡♠ ♥♦r♠❛

❞✐✈✐❞✐♥❞♦ ♣♦r ✶✱ ❧♦❣♦✱ s✉❛ ♥♦r♠❛ ❞❡✈❡ s❡r ✶ ❡✱ ♣♦rt❛♥t♦✱ ♦ ❞✐✈✐s♦r ❝♦♠✉♠ é ✉♠❛ ✉♥✐❞❛❞❡✳ ❱✐♠♦s q✉❡ ♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s ❝♦♠ ♥♦r♠❛ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s tê♠ ❞❡ s❡r ♣r✐♠♦s ❡♥tr❡ s✐✳ ❆ r❡❝í♣r♦❝❛ ♥ã♦ é ✈❡r❞❛❞❡✐r❛✱ ❝♦♠♦ ♠♦str❛ ♥♦ ❡①❡♠♣❧♦ (4 + 5i) ❡ (4₋5i)✳

▼❛s✱ ❡♠ ❣❡r❛❧ é ♥❡❝❡ssár✐♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ❡♠ Z[i]✱ ❛ ✜♠ ❞❡ ❝❛❧❝✉❧❛r ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❡♠ Z[i]✳

❈♦r♦❧ár✐♦ ✶✳✸ ✳ P❛r❛ α ₆= 0 ❡ β _∈ Z[i]✱ ✈❛♠♦s t❡r δ ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠

♣r♦❞✉③✐❞♦ ♣❡❧♦ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✳ ◗✉❛❧q✉❡r ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ α ❡ β é

✉♠ ♠ú❧t✐♣❧♦ ✉♥✐tár✐♦ ❞❡ δ✳

Pr♦✈❛✳ ❱❛♠♦s t❡r δ ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ α ❡β✳ ❆ ♣❛rt✐r ❞❛ ♣r♦✈❛ ❞♦

❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✱ δ′

| δ, ❝♦♥s✐❞❡r❛♥❞♦ δ′ _{é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠✳ ❋❛ç❛ δ = δ}′_γ, ❛ss✐♠ t❡♠♦s q✉❡

N(δ) =N(δ′₎

·N(γ)_≥N(δ′_).

❈♦♠♦ δ′ _{é ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✱ s✉❛ ♥♦r♠❛ é ♠á①✐♠❛ ❡♥tr❡ ❛s ♥♦r♠❛s ❞♦s} ❞✐✈✐s♦r❡s ❝♦♠✉♥s✱ ❛ss✐♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ N(δ) _≥ N(δ′_{) t❡♠ ❞❡ s❡r ✉♠❛ ✐❣✉❛❧❞❛❞❡✳} ■ss♦ ✐♠♣❧✐❝❛ N(γ) = 1✱ ❡♥tã♦ γ =_±1♦✉ _±i✳ ❆ss✐♠✱ δ ❡δ′ _{sã♦ ♠ú❧t✐♣❧♦s ✉♥✐tár✐♦s} ✉♠ ❞♦ ♦✉tr♦✳

❖ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t ❈❛♣ít✉❧♦ ✶

✶✳✷✳✺

❖ ❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t

❉❛❞♦s ✐♥t❡✐r♦s a ❡ b, ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ ❡①✐t❡♠ ✐♥t❡✐r♦s x ❡ y t❛✐s q✉❡ ax+by =

mdc(a, b).❆ ♠❡s♠❛ ✐❞❡✐❛ ❢✉♥❝✐♦♥❛ ❡♠ Z[i].

❚❡♦r❡♠❛ ✽ ✭❚❡♦r❡♠❛ ❞❡ ❇é③♦✉t✮✳ ❙❡❥❛α ❡ β, ❛♠❜♦s ♥ã♦ ♥✉❧♦s✱ ❡♥tã♦ mdc(α, β) =

δ. ❊♥tã♦ δ=αx+βy, ♣❛r❛ ❛❧❣✉♠ x, y _∈Z[i].

Pr♦✈❛✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦C s❡♥❞♦ t♦❞❛s ❛s ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ❞❡α ❡ β ❡ n=

αx0 +βy0, ♦♥❞❡ n é ♦ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡ C✳ ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ n ∤ α.

❊♥tã♦ α=nq+r, ❝♦♠ 0< r < n✳

r = α₋nq=α₋(αx0+βy0)q

= α₋αx0q₋βy0q

= α(1₋x0q) +β(₋y0q).

❊♥tã♦ r _∈ C, ♣♦✐s r > 0 ❡ r < n, ♦♥❞❡ n é ♦ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡ C, ♣♦rt❛♥t♦✱ n_|α. ❆♥❛❧♦❣❛♠❡♥t❡✱ n_|β. ❆ss✐♠✱ n é ♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡α ❡ β.❱❛♠♦s ♠♦str❛r q✉❡ n =d. ❉❡ ❢❛t♦✱ ♣♦✐s s❡ d=mdc(α, β), ❡♥tã♦

d_|α _⇒α =dq1 ❡ d_|β _⇒β =dq2.

❈♦♠♦ n=αx0+βy0 _⇒n=d(q1x0 +q2y0)_⇒d _|n_⇒d_≤n_⇒d=n✳

❈♦r♦❧ár✐♦ ✶✳✹ ❙❡♥❞♦ α ❡ β ✐♥t❡✐r♦s ❞❡ ●❛✉ss✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ ❡♥tã♦ sã♦ r❡❧❛t✐✲

✈❛♠❡♥t❡ ♣r✐♠♦s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

1 = αx+βy

♣❛r❛ ❛❧❣✉♠ x, y _∈Z[i]✳

Pr♦✈❛✳ (_⇒)✿ ❙❡ α ❡β sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✱ ❡♥tã♦ ✶ é ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r

❝♦♠✉♠ ❞❡ α ❡β. P♦rt❛♥t♦✱ ♣♦❞❡♠♦s t♦♠❛r x, y _∈ Z[i] t❛✐s q✉❡ 1 = αx+βy ♣❡❧♦

t❡♦r❡♠❛ ❞❡ ❇é③♦✉t✳

(_⇐)✿ ❙❡ 1 = αx+βy ♣❛r❛ ❛❧❣✉♥s x, y _∈ Z[i], ❡♥tã♦ q✉❛❧q✉❡r ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ α ❡ β é ✉♠ ❞✐✈✐s♦r ❞❡ ✶✱ ❡ ♣♦rt❛♥t♦✱ é ✉♠ ♠ú❧t✐♣❧♦ ✉♥✐tár✐♦✱ ❧♦❣♦✱ t❡♠♦s q✉❡ α ❡β

sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ✳

◆♦ ❊①❡♠♣❧♦ ✶✶ t❡♠♦s q✉❡ α = 32 + 9i ❡β = 4 + 11i, q✉❡ sã♦ r❡❧❛t✐✈❛♠❡♥t❡

♣r✐♠♦s✱ ✉♠❛ ✈❡③ q✉❡ ♦ ú❧t✐♠♦ r❡st♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ é (₋i). P♦❞❡♠♦s ✐♥✈❡rt❡r ♦s