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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

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

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

❖s ■♥t❡✐r♦s ●❛✉ss✐❛♥♦s ✈✐❛

▼❛tr✐③❡s

♣♦r

❋❛❜rí❝✐♦ ❞❡ P❛✉❧❛ ❋❛r✐❛s ❇❛r❜♦s❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛

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

❆❣♦st♦ ✴ ✷✵✶✺ ❏♦ã♦ P❡ss♦❛ ✲ P❇

❖s ■♥t❡✐r♦s ●❛✉ss✐❛♥♦s ✈✐❛

▼❛tr✐③❡s

♣♦r

❋❛❜rí❝✐♦ ❞❡ P❛✉❧❛ ❋❛r✐❛s ❇❛r❜♦s❛

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

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❆♣r♦✈❛❞❛ ♣♦r✿

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

Pr♦❢✳ ❉r✳ ❏♦ã♦ ❇♦s❝♦ ❇❛t✐st❛ ▲❛❝❡r❞❛ ✲ ❯❋P❇

Pr♦❢✳ ❉r✳ ❏❛♠✐❧s♦♥ ❘❛♠♦s ❈❛♠♣♦s ✲ ❯❋P❇

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r ♠❡ ❛❥✉❞❛r ♥❛s ❤♦r❛s ❡♠ q✉❡✱ ♥♦ ♠❡✉ ✐s♦❧❛♠❡♥t♦✱ ♣❡♥s❛✈❛ ❡♠ ❞❡s✐st✐r ❞❛♥❞♦✲♠❡ ❢♦rç❛s ♣❛r❛ ❝♦♥t✐♥✉❛r✳

❆♦s ♠❡✉s ♣❛✐s q✉❡ ❛❝r❡❞✐t❛r❛♠ ♥♦ ♠❡✉ ♣♦t❡♥❝✐❛❧ ❡ ✐♥✈❡st✐r❛♠ ♥♦ ♠❡✉ ❝r❡s❝✐✲ ♠❡♥t♦ ♣❡ss♦❛❧✱ ❢❛③❡♥❞♦ q✉❡ ❛ ❝❛❞❛ ❞✐❛ ♠❡ t♦r♥❛ss❡ ✉♠❛ ♣❡ss♦❛ ♠❡❧❤♦r✳

❆ ♠✐♥❤❛ ✐r♠ã ❡ ❛♠✐❣❛ q✉❡✱ à s✉❛ ♠❛♥❡✐r❛✱ ❢❛③✐❛ ❝♦♠ q✉❡ ♥ã♦ ❞❡s✐st✐ss❡✳

❊♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ❛♠✐❣♦s ❍❡❧❞❡r ❡ ▼♦r❛✐s q✉❡ ♣♦r ♠❡✐♦ ❞❡ ✧❜r✐❣❛s✧✱ r✐s❛❞❛s ❡ ♣❛❧❛✈r❛s ❞❡ ❝♦♥❢♦rt♦✱ ✜③❡♠♦s ❛ ♥♦ss❛ ❥♦r♥❛❞❛✳

❆ ♠✐♥❤❛ ✜❧❤❛ ❡ ❡s♣♦s❛✱ ▼❛r✐❛ ▲✉ís❛ ❡ ❆❧✉s❦❛✱ q✉❡ ✈❡✐♦ ♥♦ ✜♥❛❧ ❞❛ ❝❛♠✐♥❤❛❞❛✱ ❡ ❢❡③ ♦ s❡✉ ♣❛♣❡❧ ❞❡ ♦①✐❣❡♥❛r ♠❡✉s ♣✉❧♠õ❡s ♥♦s ú❧t✐♠♦s ♠❡tr♦s✳

❆ t♦❞♦ ❝♦r♣♦ ❞♦❝❡♥t❡ q✉❡ ❡♠ ♠✉✐t❛s ✈❡③❡s ❢♦r❛♠ ♥ã♦ ❡❞✉❝❛❞♦r❡s ❡ s✐♠ ❛♠✐❣♦s q✉❡ ❛♣♦♥t❛♠ ♦ ❝❛♠✐♥❤♦ ❝❡rt♦✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ q✉❡ ✐♥❞✐❝♦✉ ♦ t❡♠❛ ❞❡ss❡ tr❛❜❛❧❤♦✱ ❛❝♦♠♣❛♥❤♦✉ ♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛❥✉❞❛♥❞♦ s❡♠♣r❡ q✉❡ ❢♦✐ s♦❧✐❝✐✲ t❛❞♦✳

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

❉❡❞✐❝❛tór✐❛

❉❡❞✐❝♦ ❡ss❡ tr❛❜❛❧❤♦ ❛ t♦❞♦s q✉❡ ❝♦♥✲ tr✐❜✉ír❛♠ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ❡st❛ ♥♦✈❛ ❝❛♠✐♥❤❛❞❛✳

❘❡s✉♠♦

◆♦ss♦ ❡st✉❞♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ✉♠❛ ❝❛t❡❣♦r✐❛ ❡s♣❡❝✐❛❧ ❞❡ ♥ú♠❡r♦s✱ ♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s✱ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ♦♣❡r❛çõ❡s✱ t❡r ✉♠❛ ✈✐sã♦ ❣❡r❛❧ s♦❜r❡ ❡ss❡s ♥ú♠❡r♦s✱ s✉❛ ❤✐stór✐❛ ❡ s✉r❣✐♠❡♥t♦✳ ❚❛♠❜é♠ ❡st✉❞❛r❡♠♦s ♥ú♠❡r♦s ♣r✐♠♦s ●❛✉ss✐❛♥♦s✱ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❛♣❧✐❝❛çã♦ ❝♦♠ r❡♣r❡s❡♥t❛çã♦ ❡♠ ❧✐♥❣✉❛❣❡♠ ♠❛tr✐✲ ❝✐❛❧ ❞♦ t✐♣♦ 2×2✳

P❛❧❛✈r❛s ❈❤❛✈❡s✿ ■♥t❡✐r♦s ❞❡ ●❛✉ss✱ ◆ú♠❡r♦s Pr✐♠♦s ●❛✉ss✐❛♥♦s✱ ❋❛t♦r❛çã♦ Ú♥✐❝❛ ❡ ▼❛tr✐③❡s✳

❆❜str❛❝t

❖✉r st✉❞② ❛✐♠s t♦ ♣r❡s❡♥t ❛ s♣❡❝✐❛❧ ❝❛t❡❣♦r② ♦❢ ♥✉♠❜❡rs✱ t❤❡ ●❛✉ss✐❛♥ ✐♥t❡❣❡rs✱ t❤❡✐r ♣r♦♣❡rt✐❡s ❛♥❞ ♦♣❡r❛t✐♦♥s✱ ❤❛✈❡ ❛♥ ♦✈❡r✈✐❡✇ ❛❜♦✉t t❤❡s❡ ♥✉♠❜❡rs✱ t❤❡✐r ❤✐st♦r② ❛♥❞ ❡♠❡r❣❡♥❝❡✳ ❲❡ ✇✐❧❧ ❛❧s♦ st✉❞② ●❛✉ss✐❛♥ ♣r✐♠❡ ♥✉♠❜❡rs✱ t❤❡✐r ♣r♦♣❡rt✐❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ♠❛tr✐① ❧❛♥❣✉❛❣❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ 2×2t②♣❡✳

❑❡②✇♦r❞s✿ ●❛✉ss✐❛♥ ■♥t❡❣❡rs✱ ●❛✉ss✐❛♥ Pr✐♠❡s ♥✉♠❜❡rs✱ ✉♥✐q✉❡ ❢❛❝t♦r✐③❛t✐♦♥ ❛♥❞ ♠❛tr✐①✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ①

✶ ❘❡s✉❧t❛❞♦s ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s s♦❜r❡ ▼❛tr✐③ ❡ ❉❡t❡r♠✐♥❛♥t❡ ✶ ✶✳✶ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❉❡t❡r♠✐♥❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✷ ❖s ■♥t❡✐r♦s ●❛✉ss✐❛♥♦s ✽

✷✳✶ ❖s ■♥t❡✐r♦s ●❛✉ss✐❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✷ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✸ ❖s Pr✐♠♦s ●❛✉ss✐❛♥♦s ✷✶

✸✳✶ ❋❛t♦r❛çã♦ Ú♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✷ ❈r✐tér✐♦s ❞❡ Pr✐♠❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✸✺

◆♦t❛çõ❡s

◆♦t❛çõ❡s ●❡r❛✐s

• N é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s

• Z é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s

• Z+ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s

• Z∗ _{=Z− {0} é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♠❡♥♦s ♦ 0} • Q é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s

• R é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s

• C é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s

• Z[i]é ♦ ❛♥❡❧ ❞♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s

• U(Z[i]) é ♦ ❣r✉♣♦ ❞❛s ✉♥✐❞❛❞❡s ❞❡ Z[i]

• R é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✐❞❡♥t✐❞❛❞❡

• M2(R) é ♦ ❛♥❡❧ ❞❛s ♠❛tr✐③❡s 2×2s♦❜r❡ R

• A r❡♣r❡s❡♥t❛ ✉♠❛ ♠❛tr✐③

• ◆(A) é ❛ ♥♦r♠❛ ❞❡ A

• At r❡♣r❡s❡♥t❛ ❛ ♠❛tr✐③ tr❛♥s♣♦st❛

• det(A) é ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡A

• ❚r(A) é ♦ tr❛ç♦ ❞❡A

• J r❡♣r❡s❡♥t❛ ❛ ♠❛tr✐③

0 −1 1 0

• | é ❛ ♦♣❡r❛çã♦ ❞✐✈✐❞❡

• ∤ é ❛ ♦♣❡r❛çã♦ ♥ã♦ ❞✐✈✐❞❡

• ≡ r❡❧❛çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛

• ♠❞❝ é ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠

■♥tr♦❞✉çã♦

❉❡ ❛❝♦r❞♦ ❝♦♠ ♦s ❛①✐♦♠❛s ❞❡ ❈❛♥t♦r✲❉❡❞❡❦✐♥❞ ♣❛r❛ r❡♣r❡s❡♥t❛r ♦s ♥ú♠❡r♦s r❡❛✐s R✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ♣♦❞❡♠ s❡r ✐❞❡♥t✐✜❝❛❞♦s ❝♦♠ ♦s ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛ ❞♦

s❡❣✉✐♥t❡ ♠♦❞♦✿ ✜①❡♠♦s s♦❜r❡ ❛ r❡t❛ ✉♠ ♣♦♥t♦O ✭❛ ♦r✐❣❡♠✮ ❡ ❡s❝♦❧❤❛♠♦s ✉♠ ♦✉tr♦

♣♦♥t♦ U s♦❜r❡ ❛ ♠❡s♠❛ r❡t❛ ❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ 1✱ ❞❡ ♠♦❞♦ q✉❡ 1

s❡❥❛ ✐❣✉❛❧ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ OU✳ ◆ã♦ ♦❜st❛♥t❡✱ ♥❡❝❡ss✐t❛♠♦s

❞❡ ✉♠ ♣❧❛♥♦ ♣❛r❛ r❡♣r❡s❡♥t❛r ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✱ ♦✉ s❡❥❛✱ ❝❤❛♠❛♠♦s ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❞❡ ❡✐①♦ r❡❛❧ ❡ ♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s ❞❡ ❡✐①♦ ✐♠❛❣✐♥ár✐♦✳ P♦rt❛♥t♦✱ ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z = x +yi é r❡♣r❡s❡♥t❛❞♦ s♦❜r❡ ♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❝♦♠♦ ♦ ♣♦♥t♦ ❞❡

❝♦♦r❞❡♥❛❞❛s (x, y)✳ ❊st❛ ✐♥t❡r♣r❡t❛çã♦ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ❡ ✉s❛❞❛ ❡♠ 1796 ♣♦r ●❛✉ss✳

❖ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❝❤❛♠❛✲s❡ ❞❡ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ ✭●❛✉ss✐❛♥♦✮✳ ❆✐♥❞❛ ♣♦❞❡♠♦s ✈❡r ❝❛❞❛ ♣♦♥t♦(x, y)❞♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ ❝♦♠♦ ✉♠ ✏✈❡t♦r✑ ✭s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♦r✐❡♥t❛❞♦✮

❞❡ ♦r✐❣❡♠ (0,0) ❡ ❡①tr❡♠✐❞❛❞❡ (x, y)✳ ◆❡st❡ ❝❛s♦✱ s❡ x 6= 0✱ ❡♥tã♦ m = yx−1 _{é ❛}

✐♥❝❧✐♥❛çã♦ ❞♦ ✈❡t♦r✳ ❆ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❝♦♠♦ ♣♦♥t♦s ♥♦ ♣❧❛♥♦ é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣rát✐❝❛✳

❆ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s t❛✐s ❝♦♠♦

x3 ≡q (mod p) ❡ x4 ≡q (mod p),

❝♦♠p❡q♥ú♠❡r♦s ♣r✐♠♦s✱ ♠♦t✐✈❛r❛♠ ●❛✉ss ❡♠1825❛ ✐♥tr♦❞✉③✐r ♦s ♥ú♠❡r♦ ✐♥t❡✐r♦s

❝♦♠♣❧❡①♦s ❞❛ ❢♦r♠❛

α =a+bi, a, b ∈Z,

❝❤❛♠❛❞♦s ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s✱ ❝♦♠♦ ✉♠ s✉❜❛♥❡❧ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s

C ❡ ❞❡♥♦t❛❞♦ ♣♦rZ[i]✱ ♦✉ s❡❥❛✱

Z[i] ={a+bi:a, b∈Z}

◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ ✈❛♠♦s ❡st✉❞❛r ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s r❡❧❛❝✐✲ ♦♥❛❞♦s ❛♦s ✐♥t❡✐r♦s ❡ ♣r✐♠♦s ●❛✉ss✐❛♥♦s ✈✐❛ ♠❛tr✐③❡s2×2 ❡s♣❡❝✐❛✐s✱ ❞♦ t✐♣♦✿

A=

a −b

b a

,

❞❡ ♠♦❞♦ q✉❡ ❡ss❡ ❡st✉❞♦ ♣♦ss❛ s❡r tr❛t❛❞♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳

◆♦ ❈❛♣ít✉❧♦ ✶ ✈❡r❡♠♦s ❛s ♣r✐♥❝✐♣❛✐s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s s♦❜r❡ ♠❛tr✐③❡s2×2

❡ ❞❡t❡r♠✐♥❛♥t❡s ✭s♦❜r❡ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✐❞❡♥t✐❞❛❞❡✮ q✉❡ s❡rã♦ ♥❡❝❡ssár✐❛s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ◆ã♦ ♦❜st❛♥t❡✱ t♦❞❛s ❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❝♦♥t✐♥✉❛♠ ✈á❧✐❞♦s ♣❛r❛ ♠❛tr✐③❡sn×n✳

❏á ♥♦ ❈❛♣✐t✉❧♦ ✷ ❛♣r❡s❡♥t❛r❡♠♦s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ t❡♦r❡♠❛s r❡❧❛❝✐♦✲ ♥❛❞♦s ❛♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s✱ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦

Z[i]✱ ♦s q✉❛✐s ❛♣r❡s❡♥t❛♠ s❡♠❡❧❤❛♥ç❛s ❝♦♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❡

❛♣r❡s❡♥t❛r❡♠♦s ♦ ❆❧❣♦rít✐♠♦ ❞❛ ❉✐✈✐sã♦ ♣❛r❛ ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ ✷✳

P♦r ✜♠✱ ♥♦ ❈❛♣✐t✉❧♦ ✸ ❛❜♦r❞❛r❡♠♦s ♦s ♣r✐♠♦s ●❛✉ss✐❛♥♦s ❞❡t❛❧❤❛♥❞♦ s✉❛s ♣r♦✲ ♣r✐❡❞❛❞❡s ❡ t❡♦r❡♠❛s✱ ❝♦♠♦ ♦ ❞❛ s✉❛ ❢❛t♦r❛çã♦ ú♥✐❝❛ ❡ ♦ ❞♦ ❝r✐tér✐♦ ❞❡ s✉❛ ♣r✐♠❛❧✐✲ ❞❛❞❡✳

❈❛♣ít✉❧♦ ✶

❘❡s✉❧t❛❞♦s ❡ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s s♦❜r❡

▼❛tr✐③ ❡ ❉❡t❡r♠✐♥❛♥t❡

◆❡st❡ ❝❛♣✐t✉❧♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ s♦❜r❡ ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ 2✱ s❡✉s ❝♦♥✲

❝❡✐t♦s ❡ ♦♣❡r❛çõ❡s✳ ❆s ▼❛tr✐③❡s sã♦ ❢❡rr❛♠❡♥t❛s ❞❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r ♠✉✐t♦ út❡✐s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✳ ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ❡♠ ♠❛✐s ❞❡t❛❧❤❡s ♣♦❞❡ ❝♦♥s✉❧t❛r ❛s r❡❢❡rê♥❝✐❛s ❇♦❧❞r✐♥❡❬✷❪✱ ❍♦✛♠❛♥✲❑✉♥③❡ ❬✼❪ ❡ ▲✐♣s❝❤✉t③❬✾❪✳

✶✳✶ ▼❛tr✐③❡s

❊♠ t✉❞♦ q✉❡ s❡❣✉❡✱ s❛❧✈♦ ♠❡♥çã♦ ❡①♣❧í❝✐t❛ ❡♠ ❝♦♥trár✐♦✱ R r❡♣r❡s❡♥t❛ ✉♠ ❛♥❡❧

❝♦♠✉t❛t✐✈♦ ❝♦♠ ✐❞❡♥t✐❞❛❞❡✳

❯♠ ❛rr❛♥❥♦ ❞❡ q✉❛tr♦ ❡❧❡♠❡♥t♦s a, b, c, d∈R ❡♠ q✉❡

A=

a b c d

♦✉ A=

a11 a12

a21 a22

❝❤❛♠❛✲s❡ ✉♠❛ ♠❛tr✐③ 2×2s♦❜r❡R✭❧ê✲s❡ ✏♠❛tr✐③ ❞♦✐s ♣♦r ❞♦✐s✑✮ ♦✉ s✐♠♣❧❡s♠❡♥t❡

✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ 2 s♦❜r❡ R✳ ❊♠ ❛❧❣✉♠ s❡♥t✐❞♦✱ ✉♠❛ ♠❛tr✐③ 2×2

♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦s ♥♦ss♦s ✏♣❛r❡s✑ ♦r❞❡♥❛❞♦s ✭①✱ ②✮✱ ❡ ♣♦❞❡ s❡r ✈✐st❛ ❞❡ ❞♦✐s ♠♦❞♦s✿ ✈❡♥❞♦ ❝♦♠♦ ❞✉❛s ❧✐♥❤❛s

(a, b) ❡ (c, d),

❛s q✉❛✐s ❝❤❛♠❛♠✲s❡ ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ❧✐♥❤❛ ❞❡A✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦✉ ✈❡♥❞♦ ❝♦♠♦

❞✉❛s ❝♦❧✉♥❛s

a c

❡

b d

,

❛s q✉❛✐s ❝❤❛♠❛♠✲s❡ ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ❝♦❧✉♥❛ ❞❡ A✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ➱ ✉s✉❛❧

❞❡♥♦t❛r ✉♠❛ ♠❛tr✐③ ♣♦r

A= (aij).

✶✳✶✳ ▼❆❚❘■❩❊❙

❖s ❡❧❡♠❡♥t♦s aij ∈ R ❝❤❛♠❛♠✲s❡ ❞❡ ❡♥tr❛❞❛s ❞❛ ♠❛tr✐③✳ ◆❡st❡ ❝❛s♦✱ ❢♦r♠❛❧♠❡♥t❡✱

✉♠❛ ♠❛tr✐③ 2×2 é ✉♠❛ ❢✉♥çã♦

f :{1,2} × {1,2} →R

❞❡✜♥✐❞❛ ❝♦♠♦ f(i, j) =aij✳

❉✉❛s ♠❛tr✐③❡s

A=

a11 a12

a21 a22

❡ B=

b11 b12

b21 b22

sã♦ ✐❣✉❛✐s✱ ❡♠ sí♠❜♦❧♦s✱ A=B s❡✱ ❡ s♦♠❡♥t❡ s❡✱

a11 =b11, a12=b12, a21=b21 ❡ a22=b22.

❱❛♠♦s ❞❡♥♦t❛r ♣♦r M2(R)♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♠❛tr✐③❡s ❞❡ ♦r❞❡♠2 s♦❜r❡R✳

❆ss✐♠✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡M2(R)✱ ♠✉♥✐❞♦ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦

A+B = (aij +bij)

❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r

cA= (caij), ∀ c∈R,

s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

✶✳ A+ (B+C) = (A+B) +C✱ ♣❛r❛ q✉❛✐sq✉❡rA,B,C∈M2(R)✳

✷✳ ❊①✐st❡ O∈M2(R) t❛❧ q✉❡ A+O=O+A=A✱ ♣❛r❛ q✉❛❧q✉❡r A∈M2(R)✳

✸✳ P❛r❛ ❝❛❞❛ A ∈M2(R)✱ ❡①✐st❡ −A∈M2(R) t❛❧ q✉❡

A+ (−A) = −A+A =O.

✹✳ A+B=B+A✱ ♣❛r❛ q✉❛✐sq✉❡r A,B∈M2(R)✳

✺✳ c(A+B) = cA+cB✱ ♣❛r❛ q✉❛✐sq✉❡r A,B∈M2(R)❡ c∈R✳

✻✳ (c+d)A=cA+dA✱ ♣❛r❛ q✉❛✐sq✉❡r c, d∈R ❡ A∈M2(R)✳

✼✳ c(dA) = (cd)A✱ ♣❛r❛ q✉❛✐sq✉❡rc, d∈R ❡ A∈M2(R)✳

✽✳ 1·A=A✱ ♣❛r❛ q✉❛❧q✉❡r A∈M2(R)✳

✶✳✶✳ ▼❆❚❘■❩❊❙

◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ ♦ t❡r♥♦ (M2(R),+,·) é ✉♠ ♠ó❞✉❧♦ s♦❜r❡ R✳

❏á ❡q✉✐♣❛♠♦s ♦ ❝♦♥❥✉♥t♦ M2(R) ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ♠ó❞✉❧♦✳ ❆❣♦r❛✱ ❡q✉✐✲

♣❛r❡♠♦s M2(R) ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ✏á❧❣❡❜r❛✑✳ P❛r❛ ✐st♦✱ s❡❥❛♠ A = (aij),B =

(bij)∈M2(R)✳ ❖ ♣r♦❞✉t♦ ❞❡ A ♣♦r B é ❞❡✜♥✐❞♦ ❝♦♠♦

AB=

a11b11+a12b21 a11b12+a12b22

a21b11+a22b21 a21b12+a12b22

.

❖ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ ✶✳ A(BC) = (AB)C✱ ♣❛r❛ q✉❛✐sq✉❡r A,B,C∈M2(R)✳

✷✳ A(B+C) = AB+AC✱ ♣❛r❛ q✉❛✐sq✉❡r A,B,C∈M2(R)✳

✸✳ (A+B)C=AC+BC✱ ♣❛r❛ q✉❛✐sq✉❡rA,B,C∈M2(R)✳

✹✳ ❊①✐st❡ I ∈M2(R) t❛❧ q✉❡ A·I =I·A=A✱ ♣❛r❛ q✉❛❧q✉❡r A∈M2(R)✳

✺✳ c(AB) = (cA)B=A(cB)✱ ♣❛r❛ q✉❛✐sq✉❡r A,B ∈M2(R) ❡c∈R✳

◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ ♦ t❡r♥♦(M2(R),+,·)é ✉♠ ❛♥❡❧ ❝♦♠ ✐❞❡♥t✐❞❛❞❡ ❝♦♠ ❛ ♠❛tr✐③

✐❞❡♥t✐❞❛❞❡I ♥ã♦ ❝♦♠✉t❛t✐✈♦ s♦❜r❡R✳ ❆❧é♠ ❞✐ss♦✱(M2(R),+,·)é ✉♠❛ ❛❧❣é❜r❛ s♦❜r❡

R✳

❆s ♠❛tr✐③❡s

E₁₁=

1 0 0 0

, E₁₂ =

0 1 0 0

, E₂₁=

0 0 1 0

❡ E₂₂=

0 0 0 1

.

❝❤❛♠❛♠✲s❡ ♠❛tr✐③❡s ✉♥✐tár✐❛s✳ ❙❡❥❛ A= (aij)∈M2(R)✳ ❊♥tã♦✿

✶✳ A=a11E11+a12E12+a21E21+a22E22.

✷✳ E2_ii=Eii ❡E2ij =O✱ s❡ i6=j✳

✸✳ E11+E22 =I✱ ❝♦♠ I ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ♦✉ ❛ ♠❛tr✐③ ✉♥✐❞❛❞❡✳

✹✳ EkmA=am1Ek1+am2Ek2✳

✺✳ AEkm =a1kE1m+a2kE2m✳

✻✳ EijAEkm =ajkEim✳

✶✳✷✳ ❉❊❚❊❘▼■◆❆◆❚❊❙

❆ tr❛♥s♣♦st❛ ❞❛ ♠❛tr✐③

A=

a11 a12

a21 a22

∈M2(R)

é ❛ ♠❛tr✐③

At =

a11 a21

a12 a22

,

✐st♦ é✱At_{é ❛ ♠❛tr✐③ ♦❜t✐❞❛ ❡s❝r❡✈❡♥❞♦✲s❡ ❛s ❧✐♥❤❛s ❞❡A❝♦♠♦ ❝♦❧✉♥❛s✳ ❖s ❡❧❡♠❡♥t♦s}

a11❡a22❢♦r♠❛♠ ❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ ❞❛ ♠❛tr✐③A✳ ❉✐r❡♠♦s q✉❡A= (aij)∈M2(R)

é ✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ s❡ aij = 0 q✉❛♥❞♦ i6=j✳

❆ ♠❛tr✐③ tr❛♥s♣♦st❛ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✶✳ (A+B)t_{= (A)}t_{+ (B)}t✱ ♣❛r❛ q✉❛✐sq✉❡r _A,B∈M

2(R)✳

✷✳ (cA)t _=cAt_{✱ ♣❛r❛ q✉❛❧q✉❡r A∈M}

2(R) ❡c∈R✳

✸✳ (At₎t_=A✱ ♣❛r❛ q✉❛❧q✉❡r _A∈M

2(R)✳

✹✳ (AB)t_=Bt_At_{✱ ♣❛r❛ q✉❛✐sq✉❡r A,B ∈M}

2(R)✳

✶✳✷ ❉❡t❡r♠✐♥❛♥t❡s

A=

a b c d

✉♠❛ ♠❛tr✐③ ❡♠ M2(R)✳ ❉❡✜♥✐♠♦s ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ A ❝♦♠♦ ♦ ❡❧❡♠❡♥t♦

ad−bc∈R

q✉❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r

|A|=

a b c d

♦✉

det(A).

➱ út✐❧ ✉s❛r ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦ ♣❛r❛ ♦ ❞❡t❡r♠✐♥❛♥t❡✿ s❡ C1 ❡ C2 sã♦ ❛s ❝♦❧✉♥❛s ❞❡

A✱ ❡♥tã♦

det(A) = det(C₁,C₂).

❙❡❥❛ A∈M2(R)✳ ❊♥tã♦ ♦ ❞❡t❡r♠✐♥❛♥t❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✳ det(A) = det(At_)✳

✷✳ det(C1+C′1,C2) = det(C1,C2) + det(C′1,C2)✳

✶✳✷✳ ❉❊❚❊❘▼■◆❆◆❚❊❙

✸✳ det(cC1,C2) =cdet(C1,C2)✱ ♣❛r❛ t♦❞♦ c∈R✳

✹✳ det(C₁,C₁) = 0✳

✺✳ det(C1+cC2,C2) = det(C1,C2)✱ ♣❛r❛ t♦❞♦ c∈R✳

✻✳ det(C₁,C₂) = −det(C₂,C₁)✳

➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ✈❛❧❡♠ ❞❡ ♠♦❞♦ ♥❛t✉r❛❧ ♣❛r❛ ❛ s❡❣✉♥❞❛ ❝♦❧✉♥❛✱ ♣♦r ❡①❡♠♣❧♦ det(C1, cC2) = cdet(C1,C2)✳ ❈♦♠♦ ✉♠❛ ✐❧✉str❛çã♦ ♣r♦✈❛r❡✲

♠♦s à ❝♦♥❞✐çã♦ ✭2✮✳ ❙❡❥❛♠

C1 =

a c

, C′1 =

x y

❡ C2 =

b d , ❊♥tã♦

det(C1+C′1,C2) =

a+x b c+y d

= (a+x)d−(c+y)b

= (ad−bc) + (dx−by) = det(C1,C2) + det(C′1,C2),

❆ ❝♦♥❞✐çã♦ 2 ❡3 ♣r♦✈❛ q✉❡ ❛ ❢✉♥çã♦ ❞❡t❡r♠✐♥❛♥t❡ det :M2(R)→R ❞❡✜♥✐❞❛ ❝♦♠♦

det(A) = det(C1,C2)

é ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r s♦❜r❡R✳

❚❡♦r❡♠❛ ✶✳✶ ✭❇✐♥❡t✲❈❛✉❝❤②✮ ❙❡❥❛♠ A,B ∈M2(R)✳ ❊♥tã♦

det(AB) = det(A) det(B).

Pr♦✈❛✳ ❙❡❥❛♠ A= a b c d

❡ B=

p q r s . ❊♥tã♦

det(A) det(B) = (ad−bc)(ps−qr)

= adps−adqr−bcps+bcqr

= acpq+adps+bcqr+bdrs−acpq−adqr−bcps−bdrs

= (ap+br)(cq+ds)−(aq+bs)(cp+dr) = det(AB),

q✉❡ é ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳

✶✳✷✳ ❉❊❚❊❘▼■◆❆◆❚❊❙

❆ ❢✉♥çã♦ ❚r:M2(R)→R ❞❡✜♥✐❞❛ ❝♦♠♦

❚r(A) =a+d, ∀ A=

a b c d

∈M2(R)

❝❤❛♠❛✲s❡ ❢✉♥çã♦ tr❛ç♦✳

❙❡❥❛♠ A,B∈M2(R)✳ ❊♥tã♦ ❛ ❢✉♥çã♦ tr❛ç♦ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

✶✳ ❚r(cA+B) = c❚r(A) +❚r(B)✱ ♣❛r❛ t♦❞♦ c∈R✳

✷✳ ❚r(AB) = ❚r(BA)✳

✸✳ ❚r(A) =❚r(At_)✳

❆ ❝♦♥❞✐çã♦ ✶ ♣r♦✈❛ q✉❡ ❛ ❢✉♥çã♦ tr❛ç♦ é ✉♠❛ ❢♦r♠❛ ❧✐♥❡❛r s♦❜r❡R✳

❖ ❝♦♥❥✉♥t♦

U(R) ={u∈R :uv =vu= 1, ♣❛r❛ ❛❧❣✉♠ v ∈R} ❝❤❛♠❛✲s❡ ❝♦♥❥✉♥t♦ ❞❛s ✉♥✐❞❛❞❡s ❞❡ R✳ P♦r ❡①❡♠♣❧♦✱

U(Z) = {−1,1}.

❙❡❥❛ A∈M2(R)✳ ❉✐r❡♠♦s q✉❡ A é ✉♠❛ ♠❛tr✐③ ✐♥✈❡rtí✈❡❧ s❡ ❡①✐st✐r B∈M2(R)

t❛❧ q✉❡

AB=BA=I.

➱ ❝♦♠✉♠ ❞❡♥♦t❛r B = A−1 _{♦♥❞❡ ❛ ♠❛tr✐③ B ❝❤❛♠á✲❧❛ ♠❛tr✐③ ✐♥✈❡rs❛ ❞❡ A✳ ❖}

❝♦♥❥✉♥t♦

U(M2(R)) =●▲2(R)

é ✉♠ ❣r✉♣♦ ❝❤❛♠❛❞♦ ❞❡ ❣r✉♣♦ ❧✐♥❡❛r ❣❡r❛❧✳ ◆♦t❡ q✉❡ s❡

A=

a b c d

∈M2(R),

❡♥tã♦ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞♦ ❞✐r❡t❛♠❡♥t❡ q✉❡

A2−❚r(A)A+ det(A)I =0,

♦✉ s❡❥❛✱ A ❛♥✉❧❛ ♦✉ A é ✉♠ r❛✐③ ❞♦ ♣♦❧✐♥ô♠✐♦

f =x2 −❚r(A)x+ det(A)∈R[x].

P♦rt❛♥t♦✱ s❡ A∈●▲2(R)✱ ❡♥tã♦ ∆ = det(A)∈ U(R)❡

A−1 = ∆−1(❚r(A)I−A).

✶✳✷✳ ❉❊❚❊❘▼■◆❆◆❚❊❙

P♦r ❡①❡♠♣❧♦✱ s❡

A=

2 0 0 1

∈M2(Z),

❡♥tã♦ A∈/●▲2(Z)✱ ♣♦✐sdet(A) = 2∈ U/ (Z) ={−1,1}✳

❈♦♥s✐❞❡r❡♠♦s ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ♥ã♦ ❤♦♠♦❣ê♥❡♦

ax+by = r cx+dy = s ♦✉

a b c d x y = r s .

❖ ♠ét♦❞♦ tr❛❞✐❝✐♦♥❛❧ ❞❡ r❡s♦❧✈❡r ❡ss❡ s✐st❡♠❛ é ♦ s❡❣✉✐♥t❡✿ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ♣♦rd✱ ❛ s❡❣✉♥❞❛ ♣♦r−b ❡ s♦♠❛♥❞♦✱ ♦❜t❡♠♦s

(ad−bc)x=dr−bs.

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ♦❜t❡♠♦s

(ad−bc)y=as−cr.

P♦rt❛♥t♦✱ s❡ ∆ = det(A)∈ U(R)✱ ❡♥tã♦ t❡r❡♠♦s ❛ ❘❡❣r❛ ❞❡ ❈r❛♠❡r

x= ∆−1

r b s d ❡

y= ∆−1

a r c s .

◆♦t❡ q✉❡ s❡

a b c d x y = r s

♦✉ AX=R

❡ det(A)∈ U(R)✱ ❡♥tã♦

x y = a b c d −1 r s

♦✉ X=A−1R.

❈❛♣ít✉❧♦ ✷

❖s ■♥t❡✐r♦s ●❛✉ss✐❛♥♦s

◆❡st❡ ❝❛♣✐t✉❧♦ ❡st✉❞❛r❡♠♦s ✈ár✐❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ t❡♦r❡♠❛s r❡❧❛❝✐♦✲ ♥❛❞♦s ❛♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s✱ ♦s q✉❛✐s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦ ❝♦♥❥✉♥t♦

Z[i]✳ ❖s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ✉♠ t✐♣♦ ♣❛rt✐❝✉❧❛r ❞❡ ♥ú♠❡r♦s

❝♦♠♣❧❡①♦s ❡ é ❢♦r♠❛❞♦ ♣❡❧❛s ♠❛tr✐③❡s ❞❛ ❢♦r♠❛

A=

a −b

b a

,

♦♥❞❡ a, b∈Z✱ ❝♦♠ ✉♠❛ sér✐❡ ❞❡ ♣r♦♣r✐❡❞❛❞❡s s❡♠❡❧❤❛♥t❡s ❛♦s ♥ú♠❡r♦s ✐♥t❡✐r♦sZ✳

●❛✉ss✱ ❛♦ ❞❡✜♥✐r ♥ú♠❡r♦ ✐♥t❡✐r♦ ❝♦♠♦ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ Z[i]✱ ♦❜s❡r✈♦✉ q✉❡

♠✉✐t♦ ❞❛ t❡♦r✐❛ ❞❡ ❊✉❝❧✐❞❡s s♦❜r❡ ❢❛t♦r❛çã♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦❞❡r✐❛ s❡r ❛♣❧✐❝❛❞❛ ♥❡ss❡ ♥♦✈♦ ❝♦♥❥✉♥t♦ ♣♦r ❡❧❡ ❞❡✜♥✐❞♦✳ ❆ss✐♠✱ ❞❡s❡♥✈♦❧✈❡✉ ✉♠❛ t❡♦r✐❛ ❞❡ ❢❛t♦r❛çã♦ ❡♠ ♣r✐♠♦s ♣❛r❛ A ∈ Z[i] ❡ ❡ss❛ ❞❡❝♦♠♣♦s✐çã♦ é ú♥✐❝❛✱ ✐❣✉❛❧ ❛♦s ❡❧❡♠❡♥t♦s ❞♦

❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s Z✱ ❞❛♥❞♦ ❝♦♠ ✐ss♦ ✉♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥tr✐❜✉✐çã♦

♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ú❧t✐♠♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ✭❍❡❢❡③ ❬✻❪✮✳

✷✳✶ ❖s ■♥t❡✐r♦s ●❛✉ss✐❛♥♦s

Z[i] =

a −b

b a

:a, b∈Z

♠✉♥✐❞♦ ❝♦♠ ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✐♥❞✉③✐❞❛s ♣❡❧♦ ❛♥❡❧M2(Z)é ✉♠ ❛♥❡❧

❝♦♠✉t❛t✐✈♦ ❝♦♠ ✐❞❡♥t✐❞❛❞❡✱ ❝❤❛♠❛❞♦ ❞❡ ❛♥❡❧ ❞♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s✳ P♦♥❞♦

J=

0 −1 1 0

∈Z[i],

♦❜t❡♠♦s

A=

a −b

b a

=aI+bJ, ∀ a, b∈Z,

✷✳✶✳ ❖❙ ■◆❚❊■❘❖❙ ●❆❯❙❙■❆◆❖❙

❛ q✉❛❧ ❝❤❛♠❛✲s❡ ❢♦r♠❛ ♥♦r♠❛❧ ❞❡ A✳ ❖❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ σ : Z → Z[i] ❞❡✜♥✐❞❛

❝♦♠♦

σ(a) =

a 0 0 a

=aI

❝❧❛r❛♠❡♥t❡ ♣r❡s❡r✈❛ ❛s ♦♣❡r❛çõ❡s ❞♦s ❛♥❡✐s ❡ é ✐♥❥❡t♦r❛✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r ♦ ❛♥❡❧ Z❝♦♠ ♦ s✉❜❛♥❡❧

R =

a 0 0 a

:a∈Z

❞❡ Z[i]✳ ◆❡st❡ ❝❛s♦✱ ❛ ♥♦ss❛ ♠✉❧t✐♣❧✐❝❛çã♦ s♦❜r❡ Z[i] ❡st❡♥❞❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ s♦❜r❡

Z✱ ♦✉ s❡❥❛✱

aIbI =abI

❡ I é ♦ ❡❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ s♦❜r❡ Z[i]✳

❖ ❝♦♥❥✉♥t♦

Q[i] =

a −b

b a

:a, b∈Q

♠✉♥✐❞♦ ❝♦♠ ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✐♥❞✉③✐❞❛s ♣❡❧♦ ❛♥❡❧ M2(Q) é ✉♠

❝♦r♣♦✳ ◆♦t❡ q✉❡ s❡ A∈Q[i]✱ ❝♦♠A 6=O✱ ❡♥tã♦ ∆ = det(A)∈ U(Q) =Q− {0}❡

A−1 = ∆−1(❚r(A)I−A).

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡

A=

a −b

b a

❡♥tã♦

A−1 = ∆−1_{(❚r(A)I−A) =}

a

a2_+b2

b a2_+b2 − b

a2_+b2

a a2_+b2

∈Q[i].

❆❧é♠ ❞✐ss♦✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ Z[i] é ✉♠ s✉❜❛♥❡❧ ✭❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✮ ❞❡ Q[i]

❡ q✉❡

Q[i] ={B−1A:A,B∈M2(Q), ❝♦♠ B∈ U(M2(Q))}.

◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡Q[i]é ♦ ❝♦r♣♦ q✉♦❝✐❡♥t❡ ✭✈❡r ❬✶✵❪✮ ❞❡ Z[i]✱ ❝♦♥✜r❛ ❞✐❛❣r❛♠❛

❛❜❛✐①♦✳

Q[i]

Q Z[i]

Z

✷✳✶✳ ❖❙ ■◆❚❊■❘❖❙ ●❆❯❙❙■❆◆❖❙

❙❡❥❛

A=

a −b

b a

∈Z[i].

❊♥tã♦ ❛ ♠❛tr✐③ tr❛♥s♣♦st❛

At=

a b

−b a

=aI−bJ ∈Z[i]

❝❤❛♠❛✲s❡ ❝♦♥❥✉❣❛❞❛ ❞❡ A✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ A é ✐♥✈❡rtí✈❡❧✱ ❡♥tã♦

A−1 = 1

a2_+b2A

t_.

❖❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ φ:Z[i]→Z[i] ❞❡✜♥✐❞❛ ❝♦♠♦

φ(A) =At

é ❜✐❥❡t♦r❛ ❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ ✶✳ φ(A+B) = φ(A) +φ(B).

✷✳ φ(AB) =φ(B)φ(A)✳

✸✳ φ2 _=I✳

❈♦♥s✐❞❡r❡♠♦s ❛ ✐❞❡♥t✐✜❝❛çã♦

Z+←→R+={aI :a ∈Z+} ❡ a≤b⇔aI≤bI.

❆ ❢✉♥çã♦ ◆:Z[i]→R+ ❞❡✜♥✐❞❛ ❝♦♠♦

◆(A) = det(A)I, ∀ A∈Z[i]

❝❤❛♠❛✲s❡ ♥♦r♠❛✳ ❖❜s❡r✈❡ q✉❡

◆(A) = (a2+b2)I=AAt, ∀ A=

a −b

b a

∈Z[i].

❆❧é♠ ❞✐ss♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✱

◆(A) = ◆(At) ❡ ◆(AB) = ◆(A)◆(B), ∀ A,B∈Z[i].

◆❡st❡ ❝❛s♦✱

◆(A)≤◆(A)◆(B), ∀ A,B∈Z[i],

♣♦✐s

◆(A) = 0 ⇔A=O.

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

Pr♦♣♦s✐çã♦ ✷✳✶ ❙❡❥❛

A=

a −b

b a

∈Z[i].

❊♥tã♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s:

✶✳ A∈ U(Z[i]);

✷✳ ◆(A) =I✱ ♦✉ s❡❥❛✱ a2_+b2 _{= 1;}

✸✳ A∈ {−I,I,−J,J}=

Jk:k = 0,1,2,3 ✳

Pr♦✈❛✳ (1⇒2)❙✉♣♦♥❤❛♠♦s q✉❡ A∈ U(Z[i])✳ ❊♥tã♦ ❡①✐st❡B∈Z[i] t❛❧ q✉❡

AB=I.

❆ss✐♠✱ ◆(A)◆(B) = I✳ ❈♦♠♦ a2 _+b2 _{∈ Z}

+ t❡♠♦s q✉❡ 0 < a2+b2 ≤1✳ P♦rt❛♥t♦✱

◆(A) = I✳

(2⇒3)❙✉♣♦♥❤❛♠♦s q✉❡ ◆(A) = I ❡ q✉❡ A=

a −b

b a

.

❊♥tã♦

a2+b2 = 1.

❆ss✐♠✱ |a| ≤1❡|b| ≤1✳ ▲♦❣♦✱a, b∈ {−1,0,1}✳ P♦rt❛♥t♦✱b = 0❡a =±1♦✉a= 0✱

b=±1✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

A∈ {−I,I,−J,J}=

Jk :k = 0,1,2,3 .

(3⇒1)➱ ✐♠❡❞✐❛t❛✳

✷✳✷ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦

◆❡st❛ s❡çã♦ ✈❛♠♦s ♣r♦✈❛r q✉❡ ♦ ❛♥❡❧ ❞♦s ✐♥t❡✐r♦s ●❛✉ss✐❛♥♦s Z[i] ♣♦ss✉✐ q✉❛s❡

t♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❛♥❡❧ ❞♦s ✐♥t❡✐r♦sZ✳ ❈♦♠♦ ✈❡r❡♠♦s✱ 2♣❡r❞❡ ❛ ♣r✐♠❛❧✐❞❛❞❡

❡♠ Z[i]✳

❙❡❥❛♠ A,B ∈Z[i]✱ ❝♦♠ A6=O✳ ❉✐r❡♠♦s q✉❡ A ❞✐✈✐❞❡ B s❡ ❡①✐st✐r Q∈Z[i]t❛❧

q✉❡

B =QA.

❈♦♠♦ ✈❡r❡♠♦s ❡st❛ ❞❡✜♥✐çã♦ ♥♦s ♣❡r♠✐t❡ ❡st✉❞❛r ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ✭♠❞❝✮✱ ♣r✐♠♦s ●❛✉ss✐❛♥♦s✱ ❡t❝✳ ❖❜s❡r✈❡ q✉❡ I ❞✐✈✐❞❡ A ❡ A ❞✐✈✐❞❡ O✱ ♣❛r❛ t♦❞♦ A∈Z[i]✳

❆ ♥♦t❛çã♦ ✉s❛❞❛ é A|B s❡ A ❞✐✈✐❞❡ B ❡ A∤B✱ ❝❛s♦ ❝♦♥trár✐♦✳

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

❊①❡♠♣❧♦✿ ❙❡❥❛♠

A =

7 −25 25 7

,B =

57 −11 11 57

∈Z[i].

▼♦str❡ q✉❡ A ❞✐✈✐❞❡ B✳ ⋄ ❙♦❧✉çã♦✳ ❉❡✈❡♠♦s ♣r♦✈❛r q✉❡ ❡①✐st❡

Q=

x −y

y x

∈Z[i]

t❛❧ q✉❡

B =QA.

▼❛s✱ ✐ss♦ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ s✐st❡♠❛

7x−25y = 57 25x+ 7y = 11

♣♦ss✉✐r s♦❧✉çã♦ ❡♠Z✳ P❡❧❛ ❘❡❣r❛ ❞❡ ❈r❛♠❡r✱ ♦ s✐st❡♠❛ ♣♦ss✉✐ s♦❧✉çã♦ s❡✱ ❡ s♦♠❡♥t❡

s❡✱

x= 7·57 + 25·11

7·7 + 25·25 ∈Z ❡ y=

7·11−25·57 7·7 + 25·25 ∈Z.

◆❡st❡ ❝❛s♦✱x= 1 ❡y=−2✳ P♦rt❛♥t♦✱ A ❞✐✈✐❞❡ B✳

❊♠ ❣❡r❛❧✱ ✉♠ ❝r✐tér✐♦ ♣❛r❛ ❡①❛♠✐♥❛r ❛s ❝♦♥❞✐çõ❡s ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♠ Z[i] é

❝♦♠♦ s❡❣✉❡✳ ❙❡❥❛♠

A=

a −b

b a

,B=

c −d

d c

∈Z[i].

❊♥tã♦A ❞✐✈✐❞❡ B s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡

Q=

x −y

y x

∈Z[i]

t❛❧ q✉❡

B =QA.

▼❛s✱ ✐ss♦ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ s✐st❡♠❛

ax−by = c bx+ay = d

♣♦ss✉✐r s♦❧✉çã♦ ❡♠Z✳ P❡❧❛ ❘❡❣r❛ ❞❡ ❈r❛♠❡r✱ ♦ s✐st❡♠❛ ♣♦ss✉✐ s♦❧✉çã♦ s❡✱ ❡ s♦♠❡♥t❡

s❡✱

x= ac+bd

a2_+b2 ∈Z ❡ y=

ad−bc a2_+b2 ∈Z.

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

❊①❡♠♣❧♦✿ ❙❡

A=

14 −3 3 14

,B=

57 −11 11 57

∈Z[i],

❡♥tã♦ A ♥ã♦ ❞✐✈✐❞❡ B✱ ♣♦✐s

ac+bd a2_+b2 =

831 205 ∈/Z.

⋄

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ A=aI✱ ♣❛r❛ t♦❞♦ a∈Z✱ ❡♥tã♦ A ❞✐✈✐❞❡ B s❡✱ ❡ s♦♠❡♥t❡ s❡✱

x= c

a ∈Z ❡ y= d a ∈Z

s❡✱ ❡ s♦♠❡♥t❡ s❡✱a ❞✐✈✐❞❡ c❡ d ❡♠ Z✳

❖❜s❡r✈❡ q✉❡✱ ❞❛❞♦s A,B ∈Z[i]✱ ❝♦♠ A6=O✱ s❡A ❞✐✈✐❞❡ B✱ ❡♥tã♦ ◆(A)❞✐✈✐❞❡

◆(B) ❡♠ Z✳ ▼❛s✱ ❛ r❡❝í♣r♦❝❛ é ❢❛❧s❛✱ ♣♦r ❡①❡♠♣❧♦✱ s❡

A=

2 −1 1 2

,B =

3 −1 1 3

∈Z[i],

❡♥tã♦ ◆(A) = 5I ❞✐✈✐❞❡ ◆(B) = 10I ❡♠ Z✳ ◆♦ ❡♥t❛♥t♦✱ A ♥ã♦ ❞✐✈✐❞❡B✱ ♣♦✐s

7 5 ∈/ Z.

❋✐♥❛❧♠❡♥t❡✱ s❡❥❛ A ∈ Z[i]✳ ❊♥tã♦ ◆(A) é ✐❣✉❛❧ ❛ ✉♠ ♥ú♠❡r♦ ♣❛r s❡✱ ❡ s♦♠❡♥t❡ s❡✱

I+J ❞✐✈✐❞❡ A✳

➱ ♣❡rt✐♥❡♥t❡ ♥♦t❛r q✉❡ ❡st❛s ♦❜s❡r✈❛çõ❡s ♥♦s ❧❡✈❛♠ ❞❡ ❢♦r♠❛ ♣rát✐❝❛ ❡ rá♣✐❞❛ ❞❡❝✐❞✐r s❡ ✉♠ ❞❛❞♦ ✐♥t❡✐r♦ ●❛✉ss✐❛♥♦ ❞✐✈✐❞❡ ♦✉ ♥ã♦ ✉♠ ♦✉tr♦ ✐♥t❡✐r♦ ●❛✉ss✐❛♥♦✳

❙❡❥❛♠ A,B ∈ Z[i]✳ ❉✐r❡♠♦s q✉❡ A é ❛ss♦❝✐❛❞♦ ❛ B s❡ ❡①✐st✐r U ∈ U(Z[i]) t❛❧

q✉❡

B=UA.

❱❛♠♦s r❡s✉♠✐r ❡ss❡s r❡s✉❧t❛❞♦s ♥❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿

Pr♦♣♦s✐çã♦ ✷✳✷ ❙❡❥❛♠ A,B∈Z[i]✳ ❊♥tã♦:

✶✳ ❙❡ A ❞✐✈✐❞❡ B✱ ❡♥tã♦ ◆(A) ❞✐✈✐❞❡ ◆(B) ❡♠ Z✳

✷✳ ❙❡ A ❞✐✈✐❞❡ B ❡ B ❞✐✈✐❞❡ A✱ ❡♥tã♦ A ❡ B sã♦ ❛ss♦❝✐❛❞♦s✳

✸✳ ❙❡ aI ❞✐✈✐❞❡ bI✱ ❡♥tã♦ a ❞✐✈✐❞❡ b ❡♠ Z✳

✹✳ ❙❡ A ❞✐✈✐❞❡ B✱ ❡♥tã♦ At ❞✐✈✐❞❡ Bt✳

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

✺✳ ❙❡ A ❞✐✈✐❞❡ B ❡ A ♥ã♦ é ✉♠❛ ✉♥✐❞❛❞❡ ❡ ♥❡♠ ❛ss♦❝✐❛❞♦ ❛ B✱ ❡♥tã♦

I <◆(A)<◆(B).

❆ ❢✉♥çã♦ f :R→Z❞❡✜♥✐❞❛ ❝♦♠♦

f(x) = ⌊x⌋= (x−1, x]∩Z

❝❤❛♠❛✲s❡ ❢✉♥çã♦ ♠❛✐♦r ✐♥t❡✐r♦ s♦❜r❡ R✳ ◆♦t❡ q✉❡

⌊x⌋= max{n ∈Z:n ≤x} ❡

0≤x− ⌊x⌋<1

❝❤❛♠❛✲s❡ ♣❛rt❡ ❢r❛❝✐♦♥ár✐❛ ❞❡ x✳

◆♦t❡ q✉❡ ♥ã♦ ♣♦❞❡♠♦s ❡st❡♥❞❡r ❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ s♦❜r❡ Z ♣❛r❛ Z[i]✳ P♦r

❡①❡♠♣❧♦✱ ♥ã♦ ✈❛❧❡ ❛ ▲❡✐ ❞❛ ❚r✐❝♦t♦♠✐❛ ❡♠ Z[i]✳ ❈❛s♦ ❝♦♥trár✐♦✱ s❡ J < O✱ ❡♥tã♦ O<−J✳ ❆ss✐♠✱

O<(−J)2 =−I,

♦ q✉❡ é ✐♠♣♦ssí✈❡❧✳ ❙❡ J>O✱ ❡♥tã♦

O<J2 =−I,

♦ q✉❡ t❛♠❜é♠ é ✐♠♣♦ssí✈❡❧✳ ▼❛s✱ ✉♠❛ ❝♦♠♣❛r❛çã♦ ❢r❛❝❛ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡Z[i]

♣♦❞❡ s❡r ❡❢❡t✉❛❞♦ ♣♦r ❝♦♠♣❛r❛çã♦ ❞❡ ♥♦r♠❛s✱ ❛ q✉❛❧ é ❞❛❞❛ ♣❡❧♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿

❚❡♦r❡♠❛ ✷✳✶ ✭❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦✮ ❙❡❥❛♠ A,B ∈ Z[i]✱ ❝♦♠ B 6= O✳ ❊♥tã♦

❡①✐st❡♠ Q,R∈Z[i] t❛✐s q✉❡

A =QB+R, ❝♦♠ R=O ♦✉ ◆(R)<◆(B).

Pr♦✈❛✳ ❙❡❥❛♠

A=

a −b

b a

,B=

c −d

d c

∈Z[i].

❊♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

B−1A = 1

c2_+d2B

t_A=

ac+bd c2_+d2

−(ad−bc)

c2_+d2

ad−bc c2_+d2

ac+bd c2_+d2

=

x −y

y x

,

♦♥❞❡ x, y ∈Q✱ ♣♦✐sB 6=O✳ P♦♥❞♦ r =⌊x⌋, s=⌊y⌋ ∈Z✱ ♦❜t❡♠♦s

|x−r| ≤ 1

2 ❡ |y−s| ≤ 1 2.

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

❆ss✐♠✱

◆

x −y

y x

−

r −s

s r

=◆

x−r −(y−s)

y−s x−r

≤ 1

4I <I.

▲♦❣♦✱ ❡s❝♦❧❤❡♥❞♦

Q=

r −s

s r

❡ R =A−QB,

t❡r❡♠♦sR=O ♦✉

◆(R) = ◆(B(B−1A−Q)) =◆(B)◆(B−1A−Q)<◆(B),

q✉❡ é ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳

❊①❡♠♣❧♦✿ ❉❡t❡r♠✐♥❡ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❡

A=

4 −5 5 4

∈Z[i] ❡ B =

1 −1 1 1

∈Z[i].

⋄

❙♦❧✉çã♦✳ ❈♦♠♦

B−1A=

₉ 2 − 1 2 1 2 9 2

t❡♠♦s q✉❡ ❡s❝♦❧❤❡r

Q=

r −s

s r t❛❧ q✉❡ 9 2 −r

≤ 1 2 ❡ 1 2 −s

≤ 1 2.

❆ss✐♠✱ r ∈ {4,5} ❡s ∈ {0,1}✳ P♦rt❛♥t♦✱ ❛s ♣♦ssí✈❡✐s s♦❧✉çõ❡s(Q,R)✱ ❝♦♠

Q=

r −s

s r

❡ R =A−QB,

sã♦

(4I,J),(4I+J,I),(5I,−I) ❡ (5I+J,−J).

▲❡♠❛ ✷✳✶ ❙❡❥❛♠ A,B ∈ Z[i]✱ ❝♦♠ B 6= O✳ ❊♥tã♦ ♦ q✉♦❝✐❡♥t❡ Q ❡ ♦ r❡st♦ R ❞♦

❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ sã♦ ú♥✐❝♦s s❡✱ ❡ s♦♠❡♥t❡ s❡✱

◆(A+B)≤max{◆(A),◆(B)}.

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

Pr♦✈❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡

◆(A+B)>max{◆(A),◆(B)}.

❊♥tã♦

A=O(A+B) +A ❡ A=I(A+B) + (−B),

❝♦♠

◆(A)<◆(A+B) ❡ ◆(−B) = ◆(B)<◆(A+B).

P♦rt❛♥t♦✱ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ♥ã♦ sã♦ ú♥✐❝♦s✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛♠ Q,Q₁,R,R₁ ∈Z[i] t❛✐s q✉❡

A=QB+R, ❝♦♠ R=O ♦✉ ◆(R)<◆(B).

❡

A=Q₁B+R₁, ❝♦♠ R₁ =O ♦✉ ◆(R₁)<◆(B).

❊♥tã♦

R−R₁ = (Q₁−Q)B.

❆ss✐♠✱

◆(B)≤◆((Q₁−Q)B) =◆(R−R₁)≤max◆(R),◆(R₁)<◆(B),

♦ q✉❡ é ✐♠♣♦ssí✈❡❧✱ ❛ ♠❡♥♦s q✉❡

R−R1 =O ♦✉ Q1−Q=O,

♦✉ s❡❥❛✱ R=R1 ❡Q=Q1✳

❊①❡♠♣❧♦✿ ❉❡t❡r♠✐♥❡ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❡

A =

7 −11 11 7

∈Z[i] ❡ B=

3 −5 5 3

∈Z[i].

⋄

❙♦❧✉çã♦✳ ❈♦♠♦

A+B=

10 −16 16 10

t❡♠♦s q✉❡

◆(A+B)>max{◆(A),◆(B)}.

❆ss✐♠✱

A=QB+R,

❝♦♠ Q= 2I ❡ R=I+J✱ é ✉♠❛ ❞❛s s♦❧✉çõ❡s ♣♦ssí✈❡✐s✳

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

▲❡♠❛ ✷✳✷ ❙❡❥❛ f : N −→ N ✉♠❛ ❢✉♥çã♦ (s❡q✉ê♥❝✐❛) ❞❡❝r❡s❝❡♥t❡✳ ❊♥tã♦ ❡①✐st❡

n0 ∈N t❛❧ q✉❡ f(n) =f(n0)✱ ♣❛r❛ t♦❞♦ n∈N✱ ❝♦♠ n ≥n0✳

Pr♦✈❛✳ ❙❡❥❛

S =f(N) ={f(n) :n∈N}.

❊♥tã♦S /∈ ∅✳ ❆ss✐♠✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✱ ❡①✐st❡s0 ∈S t❛❧ q✉❡s0 ≤s✱

♣❛r❛ t♦❞♦ s∈S✳ ❈♦♠♦s0 ∈S t❡♠♦s q✉❡ ❡①✐st❡ n0 ∈N t❛❧ q✉❡ f(n0) =s0✳ ❆ss✐♠✱

f(n)≤ f(n0) = s0✱ ♣❛r❛ t♦❞♦ n ∈ N✱ ❝♦♠ n ≥n0✱ ♣♦✐s f é ❞❡❝r❡s❝❡♥t❡✳ P♦r ♦✉tr♦

❧❛❞♦✱ s0 ≤ f(n)✱ ♣❛r❛ t♦❞♦ n ∈ N✱ ❝♦♠ n ≥ n0✳ P♦rt❛♥t♦✱ f(n) = f(n0)✱ ♣❛r❛ t♦❞♦

n∈N✱ ❝♦♠ n ≥n0✳

❈♦♠ ❜❛s❡ ♥♦ ❚❡♦r❡♠❛ ✷✳✶✱ ♦ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ♣♦❞❡ s❡r ❣❡♥❡r❛❧✐③❛❞♦ ♣❛r❛

Z[i]✱ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦

A = Q₁B+R₁, ❝♦♠ R₁ =O ♦✉ ◆(R₁)<◆(B)

B = Q2R1+R2, ❝♦♠ R2 =O ♦✉ ◆(R2)<◆(R1)

R1 = Q3R2+R3, ❝♦♠ R3 =O ♦✉ ◆(R3)<◆(R4)

✳✳✳

Rn−2 = QnRn−1+Rn, ❝♦♠ Rn =O ♦✉ ◆(Rn)<◆(Rn−1)

Rn−1 = Qn+1Rn, ❝♦♠ Rn+1 =O,

♣♦✐s✱ ♣❡❧♦ ▲❡♠❛ ✷✳✷✱ ❛ s❡q✉ê♥❝✐❛

{◆(B),◆(R1),◆(R2), . . .}

t❡r♠✐♥❛✳

❙❡❥❛♠ A,B∈Z[i]✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦✳ ❉✐r❡♠♦s q✉❡D∈Z[i]é ✉♠ ♠á①✐♠♦ ❞✐✈✐s♦r

❝♦♠✉♠ ❞❡ A ❡ B s❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿

✶✳ D|A ❡ D|B✳

✷✳ ❙❡ D₁ |A ❡D₁ |B✱ ❡♥tã♦ D₁ |D✳

❖❜s❡r✈❡ q✉❡ s❡ D,D₁ ∈Z[i] sã♦ ❞♦✐s ♠á①✐♠♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ A ❡ B✱ r❡s♣❡❝✲

t✐✈❛♠❡♥t❡✱ ❡♥tã♦✱ ♣❡❧❛s ❝♦♥❞✐çõ❡s (1) ❡ (2)✱ D1 |D ❡D |D1✳ ❆ss✐♠✱ D é ❛ss♦❝✐❛❞♦

❛ D1✳ P♦rt❛♥t♦✱ ❛ ♠❡♥♦s ❞❡ ❛ss♦❝✐❛❞♦s✱

D=♠❞❝(A,B)

é ú♥✐❝♦✳ P♦rt❛♥t♦✱ ♥ã♦ ❤á ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❡♠ ❡s❝♦❧❤❡r

D=

r −s

s r

, ❝♦♠ r >0 ❡ s ≥0.

❉✐r❡♠♦s q✉❡ A ❡B sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s q✉❛♥❞♦

♠❞❝(A,B) =I.

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

▲❡♠❛ ✷✳✸ ❙❡❥❛♠ A,B∈Z[i]− {O}✳ ❙❡

A =QB+R, ❝♦♠ R=O ♦✉ ◆(R)<◆(B),

❡♥tã♦

♠❞❝(A,B) = ♠❞❝(B,R) =♠❞❝(B,A−QB).

Pr♦✈❛✳ P❡❧❛ s✉❛ s✐♠♣❧✐❝✐❞❛❞❡ s✉❛ ❞❡♠♦♥str❛çã♦ ✜❝❛ ❝♦♠♦ s✉❣❡stã♦ ❞❡ ❡①❡r❝í❝✐♦✳

❚❡♦r❡♠❛ ✷✳✷ ✭❇é③♦✉t✮ ❙❡❥❛♠ A,B ∈ Z[i]✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦s✳ ❊♥tã♦ ❡①✐st❡♠

X,Y ∈Z[i] t❛✐s q✉❡

♠❞❝(A,B) =AX+BY.

Pr♦✈❛✳ ❇❛st❛ ✉s❛r ♦ ▲❡♠❛ ✷✳✸ ❡ ♦ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ❞❡ trás ♣❛r❛ ❢r❡♥t❡✳ ❊①❡♠♣❧♦✿ ❙❡❥❛♠

A=

7 −11 11 7

,B =

3 −5 5 3

∈Z[i].

❉❡t❡r♠✐♥❡ X,Y∈Z[i] t❛✐s q✉❡

♠❞❝(A,B) = AX+BY.

⋄

❙♦❧✉çã♦✳ ❏á ✈✐♠♦s✱ ♥♦ ❊①❡♠♣❧♦ ✷✳✷✱ q✉❡✿

A=Q1B+R1,

❝♦♠ Q₁ = 2I ❡ R₁ =I+J✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ B=Q2R1, ❡♠ q✉❡ Q2 =

4 −1 1 4

.

❆ss✐♠✱

R1 =A−Q1B=AX+BY,

❝♦♠ X=I ❡ Y =−Q1✳ P♦rt❛♥t♦✱ ♠❞❝(A,B) =I+J✳

❊①❡♠♣❧♦✿ ❙❡❥❛♠

A=

32 −9 9 32

,B=

4 −11 11 4

∈Z[i].

❉❡t❡r♠✐♥❡ X,Y∈Z[i] t❛✐s q✉❡

♠❞❝(A,B) = AX+BY.

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

⋄

❙♦❧✉çã♦✳ ❈♦♠♦

B−1A=

₂₂₇ 137 316 137 −316 137 227 137 =

2− 47 137 2 +

42 137

−2− 42 137 2−

47 137 = 2 2

−2 2

+ 1 137

−47 42

−42 −47

t❡♠♦s q✉❡

A=Q1B+R1,

❝♦♠

Q₁ =

2 2

−2 2

❡ R₁ = 1 137B

−47 42

−42 −47

=

2 5

−5 2

.

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱

B = Q2R1+R2, ❝♦♠ Q2 =

−2 −1 1 −2

❡ R2 =

3 1

−1 3

R1 = Q3R2+R3, ❝♦♠ Q3 =

1 1

−1 1

❡ R3 =

0 1

−1 0

R₂ = Q₄R₃, ❝♦♠ Q₄ =

1 −3 3 1

.

❆ss✐♠✱ ❝♦♠ ❛❧❣✉♥s ❝á❧❝✉❧♦s✱ ♦❜t❡♠♦s

♠❞❝(A,B) =I =AX+BY, ❝♦♠ X= 3I ❡ Y =

−5 −7 7 −5

.

P♦rt❛♥t♦✱ A ❡ B sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s✳

▲❡♠❛ ✷✳✹ ❙❡❥❛♠ A,B,C∈Z[i]✳ ❙❡ ♠❞❝(A,B) =I ❡ ♠❞❝(A,C) = I✳ ❊♥tã♦

♠❞❝(A,BC) =I.

Pr♦✈❛✳ ❚❡♠♦s q✉❡ ❡①✐st❡♠X,Y,V,W ∈Z[i] t❛✐s q✉❡

AX+BY=I ❡ AV+CW =I.

❆ss✐♠✱

I= (AX+BY)(AV+CW)

✐♠♣❧✐❝❛ q✉❡

I =A(AXV+BYV+CXW) +BC(YW).

P♦rt❛♥t♦✱ ♠❞❝(A,BC) = I✳

Pr♦♣♦s✐çã♦ ✷✳✸ ❙❡❥❛♠ A,B,C∈Z[i]✳ ❙❡ A|C✱ B|C ❡ ♠❞❝(A,B) = I✳ ❊♥tã♦ AB|C.

✷✳✷✳ ❆▲●❖❘■❚▼❖ ❉❆ ❉■❱■❙➹❖

Pr♦✈❛✳ ❚❡♠♦s q✉❡ ❡①✐st❡♠S,T,X,Y ∈Z[i] t❛✐s q✉❡

C=AS, C=BT ❡ AX+BY=I.

❆ss✐♠✱

C=ACX+BCY = (TX+SY)AB.

P♦rt❛♥t♦✱ AB|C✳

❈❛♣ít✉❧♦ ✸

❖s Pr✐♠♦s ●❛✉ss✐❛♥♦s

◆❡st❡ ❝❛♣✐t✉❧♦ ❛❜♦r❞❛r❡♠♦s ❞❡ ♠❛♥❡✐r❛ ❞❡t❛❧❤❛❞❛ ♦s ♣r✐♠♦s ●❛✉ss✐❛♥♦s✱ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ t❡♦r❡♠❛s✳

✸✳✶ ❋❛t♦r❛çã♦ Ú♥✐❝❛

❙❡❥❛ P ∈ Z[i]✳ ❉✐r❡♠♦s q✉❡ P é ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦ ❡♠ Z[i] s❡ ❛s s❡❣✉✐♥t❡s

❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿ ✶✳ P6=O ❡ P∈ U/ (Z[i])✳

✷✳ ❙❡ P|AB✱ ❡♥tã♦ P|A ♦✉ P|B ♦✉ ❛♠❜♦s✳

◆♦t❡ q✉❡ ❛ ❝♦♥❞✐çã♦ (2) é ❡q✉✐✈❛❧❡♥t❡ ❛✿

P=AB⇒A∈ U(Z[i]) ♦✉ B∈ U(Z[i]),

♦✉ s❡❥❛✱ s❡A |P✱ ❡♥tã♦

A∈ U(Z[i]) ♦✉ A é ❛ss♦❝✐❛❞♦ ❛ P.

P♦r ❡①❡♠♣❧♦✱P=I+J é ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦✱ ♣♦✐s P=AB⇒◆(A)◆(B) = 2I.

❆ss✐♠✱ ◆(A) = I ♦✉ ◆(B) = I✳ P♦rt❛♥t♦✱ A∈ U(Z[i])♦✉B ∈ U(Z[i])✳ ◆♦ ❡♥t❛♥t♦✱ 2 é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡♠Z✱ ♠❛s 2I ♥ã♦ é ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦✱ ♣♦✐s

2I = (I+J)(I−J) = J(I−J)2 =−J(I+J)2.

❖❜s❡r✈❡ q✉❡ P é ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Pt t❛♠❜é♠ ♦ é✱ ♣♦✐s

P=AB⇔Pt=AtBt.

✸✳✶✳ ❋❆❚❖❘❆➬➹❖ Ú◆■❈❆

➱ ♣❡rt✐♥❡♥t❡ ♦❜s❡r✈❛r q✉❡ P é ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ P ♣♦ss✉✐

❡①❛t❛♠❡♥t❡ ♦✐t♦ ❞✐✈✐s♦r❡s✱ ❛ s❛❜❡r✱

±I, ±J, ±P ❡ ±JP.

P♦rt❛♥t♦✱ s❡A∈Z[i]♥ã♦ é ❞✐✈✐sí✈❡❧ ♣♦r ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦P✱ ❡♥tã♦ ♠❞❝(A,P) =

I✳

Pr♦♣♦s✐çã♦ ✸✳✶ P❛r❛ q✉❛❧q✉❡r A∈Z[i]✱ ❝♦♠ ◆(A) =aI ❡a≥2✱ ❡①✐st❡ ✉♠ ♣r✐♠♦

●❛✉ss✐❛♥♦ P q✉❡ ❞✐✈✐❞❡ A✳

Pr♦✈❛✳ ❙❡❥❛

S ={a ∈N−{1}:∃ A∈Z[i], ❝♦♠ ◆(A) =aI ❡ Q∤A, ∀ ♣r✐♠♦ ●❛✉ss✐❛♥♦ Q}.

❊♥tã♦S =∅✳ ❉❡ ❢❛t♦✱ s❡ S 6=∅✱ ❡♥tã♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✱S ❝♦♥té♠

✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦✱ ❞✐❣❛♠♦s d ∈ S✳ ❙❡❥❛ D ∈ Z[i] t❛❧ q✉❡ ◆(D) = dI✳ ❈♦♠♦ D

❞✐✈✐❞❡ D t❡♠♦s q✉❡ D ♥ã♦ é ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦✳ ❆ss✐♠✱

D=BC, ❝♦♠ ◆(B) = bI, ◆(C) =cI ❡ 1< b, c < d.

▲♦❣♦✱ b /∈S✳ ◆❡st❡ ❝❛s♦✱ ❡①✐st❡ ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦ Pt❛❧ q✉❡ P❞✐✈✐❞❡ B✳ P♦r ❞❡✲

✜♥✐çã♦✱P❞✐✈✐❞❡ D✳ P♦rt❛♥t♦✱d /∈S✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

❡①✐st❡ ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦P q✉❡ ❞✐✈✐❞❡ A✳

❙❡❥❛ A ∈ Z[i]✳ ❉✐r❡♠♦s q✉❡ A é r❡❞✉tí✈❡❧ s♦❜r❡ Z[i] s❡ ❡❧❡ ♥ã♦ ❢♦r ✉♠ ♣r✐♠♦

●❛✉ss✐❛♥♦✳

❚❡♦r❡♠❛ ✸✳✶ ❙❡❥❛ A∈ Z[i] r❡❞✉tí✈❡❧✳ ❊♥tã♦ A ❝♦♥té♠ ✉♠ ❞✐✈✐s♦r ♣r✐♠♦ ●❛✉ss✐✲

❛♥♦ P t❛❧ q✉❡

◆(P)≤p◆(A).

Pr♦✈❛✳ ❙❡❥❛

S ={a ∈N−{1}:∃ A ∈Z[i], ❝♦♠ ◆(A) =aI ❡ Q|A, ❝♦♠ Q Pr✐♠♦ ●❛✉s✐❛♥♦}.

❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✶✱ S 6= ∅✳ ❆ss✐♠✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✱ S

❝♦♥té♠ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦✱ ❞✐❣❛♠♦s p ∈ S✳ ❙❡❥❛ P ∈ Z[i] t❛❧ q✉❡ ◆(P) = pI✳

❊♥tã♦ ❡①✐st❡ B∈Z[i] t❛❧ q✉❡ A=PB✳ ➱ ❝❧❛r♦ q✉❡ ◆(P)≤◆(B) ❡

◆(P)2 ≤◆(P)◆(B) = ◆(A).

P♦rt❛♥t♦✱ ◆(P)≤p

◆(A)✳

✸✳✶✳ ❋❆❚❖❘❆➬➹❖ Ú◆■❈❆

❚❡♦r❡♠❛ ✸✳✷ ◗✉❛❧q✉❡r A ∈Z[i]✱ ❝♦♠ ◆(A) = aI ❡ a≥2✱ ♣♦❞❡ s❡r ❢❛t♦r❛❞♦ ❝♦♠♦

✉♠ ♣r♦❞✉t♦ ✜♥✐t♦ ❞❡ ♣r✐♠♦s ●❛✉ss✐❛♥♦s✳ Pr♦✈❛✳ ❙❡❥❛

S ={a∈N− {1}:∃ A∈Z[i], ❝♦♠ ◆(A) =aI ❡ A6=P1· · ·Pn}.

❊♥tã♦S =∅✳ ❉❡ ❢❛t♦✱ s❡ S 6=∅✱ ❡♥tã♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✱S ❝♦♥té♠

✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦✱ ❞✐❣❛♠♦s b ∈ S✳ ❙❡❥❛ B ∈ Z[i] t❛❧ q✉❡ ◆(B) = bI✳ ❊♥tã♦✱

♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✶✱ B = PC✱ ♣❛r❛ ❛❧❣✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦ P ∈ Z[i]✳ ❆ss✐♠✱ 1 < c < b✱ ❝♦♠ ◆(C) = cI✳ ▲♦❣♦✱ c /∈ S ❡ C ∈ U(Z[i]) ♦✉ ❡①✐st❡♠ ♣r✐♠♦s

●❛✉ss✐❛♥♦s P1, . . . ,Pm ∈Z[i]t❛✐s q✉❡

C=P₁· · ·Pm

P♦rt❛♥t♦✱ B é ❛ss♦❝✐❛❞♦ ❛ P ♦✉

B =P₁P₁· · ·Pm,

♦ q✉❡ é ✐♠♣♦ssí✈❡❧✳

▲❡♠❛ ✸✳✶ ✭❊✉❝❧✐❞❡s✮ ❙❡❥❛♠ A,B,C∈ Z[i]✳ ❙❡ A | BC ❡ ♠❞❝(A,B) =I✱ ❡♥tã♦ A|C✳

Pr♦✈❛✳ ❚❡♠♦s q✉❡ ❡①✐st❡♠X,Y ∈Z[i] t❛✐s q✉❡

AX+BY=I.

❆ss✐♠✱

C=CI=C(AX+BY) = (AC)X+ (BC)Y.

❈♦♠♦ A|AC ❡ A|BC t❡♠♦s q✉❡A|C✳

❚❡♦r❡♠❛ ✸✳✸ ❙❡❥❛♠ P,P1, . . . ,Pn ∈Z[i] ♣r✐♠♦s ●❛✉ss✐❛♥♦s✳ ❙❡

P|P₁· · ·Pn,

❡♥tã♦ P é ❛ss♦❝✐❛❞♦ ❛ Pk✱ ♣❛r❛ ❛❧❣✉♠ k ∈ {1, . . . , n}✳

Pr♦✈❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡

P|P₁· · ·Pn,

♠❛s

P6=UkPk, ♦♥❞❡ Uk ∈ U(Z[i]) ❡ k= 1, . . . , n−1.

❊♥tã♦✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡✱

♠❞❝(P,Pk) =I, k= 1, . . . , n−1.

❆ss✐♠✱ ♣❡❧♦ ▲❡♠❛ ❄❄✱

♠❞❝(P,P1· · ·Pn−1) =I.

▲♦❣♦✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✱ P|Pn✳ P♦rt❛♥t♦✱ Pé ❛ss♦❝✐❛❞♦ ❛ Pn✳

✸✳✶✳ ❋❆❚❖❘❆➬➹❖ Ú◆■❈❆

❚❡♦r❡♠❛ ✸✳✹ ✭❋❛t♦r❛çã♦ Ú♥✐❝❛✮ ◗✉❛❧q✉❡r A ∈ Z[i]✱ ❝♦♠ ◆(A) = aI ❡ a ≥ 2✱

♣♦❞❡ s❡r ❢❛t♦r❛❞♦ ❞❡ ♠♦❞♦ ú♥✐❝♦ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ✜♥✐t♦ ❞❡ ♣r✐♠♦s ●❛✉ss✐❛♥♦s✱ ❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❡ ✉♥✐❞❛❞❡s✳

Pr♦✈❛✳ ❇❛st❛ ♣r♦✈❛r ❛ ✉♥✐❝✐❞❛❞❡✳ ❙✉♣♦♥❤❛♠♦s q✉❡ A ∈ Z[i]✱ ❝♦♠ ◆(A) = aI ❡

a≥2✱

A=P₁· · ·Pm ❡ A=Q1· · ·Qn.

❊♥tã♦

P₁· · ·Pm =Q1· · ·Qn ❡ P1 |Q1· · ·Qn.

◆♦t❡ q✉❡ m > 1✱ ♣♦✐s s❡ m = 1✱ ❡♥tã♦ A ❥á s❡r✐❛ ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦✳ ❆ss✐♠✱

♣❡❧♦ ❚❡♦r❡♠❛ ✸✳✸✱ P₁ é ❛ss♦❝✐❛❞♦ ❛ Qk✱ ♣❛r❛ ❛❧❣✉♠ k ∈ {1, . . . , n}✳ ❘❡✐♥❞❡①❛♥❞♦✱

s❡ ♥❡❝❡ssár✐♦✱ ❞❡ ♠♦❞♦ q✉❡Q1 =U1P1✱ ♣❛r❛ ❛❧❣✉♠U1 ∈ U(Z[i])✳ ▲♦❣♦✱ ♣❡❧❛ ❧❡✐ ❞♦

❝❛♥❝❡❧❛♠❡♥t♦✱

P₂· · ·Pm =U1Q2· · ·Qn.

❆❣♦r❛✱ ✈❛♠♦s ✉s❛r ✐♥❞✉çã♦ s♦❜r❡ max{m, n}✳ ❙❡ m > n✱ ❡♥tã♦

Pn+1· · ·Pm =U,

♦ q✉❡ é ✐♠♣♦ssí✈❡❧✳ ❙❡ m < n✱ ❡♥tã♦

U=Qm+1· · ·Qn,

♦ q✉❡ é ✐♠♣♦ssí✈❡❧✳ P♦rt❛♥t♦✱ m=n ❡

P2· · ·Pn

é ♥♦ ♠á①✐♠♦ ✉♠❛ r❡♦r❞❡♥❛çã♦ ❞❡

Q2· · ·Qn,

♦✉ s❡❥❛✱ ❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❡ ✉♥✐❞❛❞❡s✳

❊①❡♠♣❧♦✿ ❉❡t❡r♠✐♥❡ ❛ ❢❛t♦r❛çã♦ ❞❡A= 20I∈Z[i]✳ ⋄

❙♦❧✉çã♦✳ ❈♦♠♦ ◆(A) = 400I t❡♠♦s✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✸✳✶✱ q✉❡ ♦s ♣♦ssí✈❡✐s ❞✐✈✐s♦r❡s

♣r✐♠♦s ●❛✉ss✐❛♥♦s ❞❡ A ❞❡✈❡♠ t❡r ♥♦r♠❛ ◆(D) = 2I ♦✉ ◆(D) = 5I✳ ❈♦♠ ❛❧❣✉♥s

❝á❧❝✉❧♦s✱ q✉❡ ✈❡r❡♠♦s ♥❛ ♣ró①✐♠❛ s❡❝çã♦✱ ♦❜t❡♠♦s

A=−(I+J)4(I+ 2J)(I−2J),

✸✳✷✳ ❈❘■❚➱❘■❖❙ ❉❊ P❘■▼❆▲■❉❆❉❊

✸✳✷ ❈r✐tér✐♦s ❞❡ Pr✐♠❛❧✐❞❛❞❡

◆❡st❛ s❡çã♦ ✈❡r❡♠♦s q✉❡ ❛ ♠❡❧❤♦r ♠❛♥❡✐r❛ ❞❡ ✐❞❡♥t✐✜❝❛r ♦s ♣r✐♠♦s ●❛✉ss✐❛♥♦s é ❛tr❛✈és ❞❡ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡♠ Z✳

Pr♦♣♦s✐çã♦ ✸✳✷ ❙❡❥❛ P ∈ Z[i] ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦✳ ❊♥tã♦ P | pI✱ ♣❛r❛ ❛❧❣✉♠

♥ú♠❡r♦ ♣r✐♠♦ p∈Z✳

Pr♦✈❛✳ ❈♦♠♦ ◆(P) =aI✱ ♦♥❞❡ a∈Z✱ t❡♠♦s q✉❡

◆(P) = (p1· · ·pn)I

é s✉❛ ❢❛t♦r❛çã♦ ❡♠ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❙❡♥❞♦ ◆(P) = PPt✱ t❡r❡♠♦s P|(p1· · ·pn)I.

❆ss✐♠✱ ♣♦r ❞❡✜♥✐çã♦✱ P|pkI✱ ♣❛r❛ ❛❧❣✉♠k ∈ {1, . . . , n}✳

▲❡♠❛ ✸✳✷ ❙❡❥❛♠ A ∈ Z[i] ❡ p ∈ Z ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ ❙❡ ◆(A) =pI✱ ❡♥tã♦ A é

✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦✳

Pr♦✈❛✳ ❙❡❥❛♠ B,C∈Z[i] t❛✐s q✉❡ A=BC✳ ❊♥tã♦

pI=◆(A) = ◆(B)◆(C)⇒◆(B)∈ U(Z[i]) ♦✉ ◆(C)∈ U(Z[i]).

❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✶✱ B ∈ U(Z[i]) ♦✉ C∈ U(Z[i])✳ P♦rt❛♥t♦✱ A é ✉♠ ♣r✐♠♦

●❛✉ss✐❛♥♦✳

❖❜s❡r✈❡ q✉❡ ❛ r❡❝í♣r♦❝❛ ❞♦ ▲❡♠❛ ✸✳✶ é ❢❛❧s❛✱ ♣♦✐s ❝♦♠♦ ✈❡r❡♠♦s P = 3I é ✉♠

♣r✐♠♦ ●❛✉ss✐❛♥♦✱ ♠❛s ◆(P) = 9I ❡ 9 ♥ã♦ é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡♠Z✳

Pr♦♣♦s✐çã♦ ✸✳✸ ❙❡❥❛ p ∈ Z ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ ❊♥tã♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦

❡q✉✐✈❛❧❡♥t❡s:

✶✳ A=pI é r❡❞✉tí✈❡❧ s♦❜r❡ Z[i];

✷✳ A=PPt✱ ♣❛r❛ ❛❧❣✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦ _P∈Z[i];

✸✳ A= (a2_+b2_{)I✱ ♦✉ s❡❥❛✱ A é s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳}

Pr♦✈❛✳ (1 ⇒ 2) ❙✉♣♦♥❤❛♠♦s q✉❡ A s❡❥❛ r❡❞✉tí✈❡❧ s♦❜r❡ Z[i]✳ ❊♥tã♦ ❡①✐st❡♠

B,C∈Z[i] t❛✐s q✉❡

A=BC, ❝♦♠ I<◆(B),◆(C)<◆(A).

✸✳✷✳ ❈❘■❚➱❘■❖❙ ❉❊ P❘■▼❆▲■❉❆❉❊

❈♦♠♦

p2I =◆(A) = ◆(B)◆(C)

t❡♠♦s q✉❡ ◆(B) = pI✳ ❆ss✐♠✱ ♣❡❧♦ ▲❡♠❛ ✸✳✷✱ B = P é ✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦✳ P♦r

♦✉tr♦ ❧❛❞♦✱

C= 1

pP

t

A=Pt.

P♦rt❛♥t♦✱ A=PPt_{✱ ♣❛r❛ ❛❧❣✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦P✳}

(2⇒3)❙❡❥❛

P=

a −b

b a

∈Z[i]

✉♠ ♣r✐♠♦ ●❛✉ss✐❛♥♦✳ ❊♥tã♦

A=PPt = (a2+b2)I,

♦✉ s❡❥❛✱ A é s♦♠❛ ❞❡ ❞♦✐s q✉❛❞r❛❞♦s✳

(3⇒1)❙✉♣♦♥❤❛♠♦s q✉❡ A= (a2_+b2_{)I✳ ❊♥tã♦ A=PP}t_{✱ ❝♦♠}

P=

a −b

b a

∈Z[i].

❈♦♠♦ ◆(P) = ◆(Pt_{) ❡}

p2I =◆(A) = ◆(P)◆(Pt)

t❡♠♦s q✉❡ ◆(P) = pI✳ P♦rt❛♥t♦✱ A é r❡❞✉tí✈❡❧ s♦❜r❡ Z[i]✳

❚❡♦r❡♠❛ ✸✳✺ ✭❋❡r♠❛t✮ ❙❡❥❛♠a, p∈Z✱ ❝♦♠ p✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ ❙❡ ♠❞❝(a, p) = 1✱ ❡♥tã♦

p|ap−1_−1⇔ap−1 _{≡1 (mod p).}

❈♦♥❝❧✉❛ q✉❡

p|ap_−a⇔ap _{≡a (mod p).}

Pr♦✈❛✳ ❙❛❜❡♥❞♦ q✉❡ ♦ ❝♦♥❥✉♥t♦

Z•

p ={1, . . . , p−1}

♠✉♥✐❞♦ ❝♦♠ ❛s ♦♣❡r❛çõ❡s

a⊕b≡r (mod p) ❡ a⊙b≡r (mod p), ∀ a, b∈Z•

p,

❡♠ q✉❡ r é ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ♣♦r p✱ é ✉♠ ❝♦r♣♦✳ ❆ ❢✉♥çã♦ σ : Z•

p → Z•p ❞❡✜♥✐❞❛

❝♦♠♦

σ(x) =ax

✸✳✷✳ ❈❘■❚➱❘■❖❙ ❉❊ P❘■▼❆▲■❉❆❉❊

é ❝❧❛r❛♠❡♥t❡ ❜✐❥❡t♦r❛✳ ❆ss✐♠✱

Z•p =σ(Z

•

p) = {a, . . . , a(p−1)}.

▲♦❣♦✱

a·2a· · ·a(p−1)≡1·2· · ·(p−1) (mod p)⇒ap−1 _{≡1 (mod p),}

♣♦✐s ♠❞❝(k, p) = 1✱ ❝♦♠ k = 1, . . . , p−1✳ P♦rt❛♥t♦✱

ap−1 _{≡1 (mod p).}

❋✐♥❛❧♠❡♥t❡✱ s❡p|a✱ ❡♥tã♦

a≡0 (mod p)⇒ap ≡a (mod p),

q✉❡ é ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳

❚❡♦r❡♠❛ ✸✳✻ ✭❲✐❧s♦♥✮ ❙❡❥❛ p∈Z ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ ❊♥tã♦

p|(p−1)! + 1⇔(p−1)!≡ −1 (mod p).

Pr♦✈❛✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ♣♦❧✐♥ô♠✐♦

f(x) = xp−1−1−

p−1

Y

k=1

(x−k) = c0+c1x+· · ·+cp−2xp−2.

❊♥tã♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✸✳✺✱ ❛ ❡q✉❛çã♦

f(x)≡0 (mod p)

♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s p−1 s♦❧✉çõ❡s✱ ❛ s❛❜❡r✱ x∈Z•

p✳ ❆ss✐♠✱

cm ≡0 (mod p), m= 0, . . . , p−2.

❈♦♠♦

c0 =−1−(−1)p−1(p−1)!

t❡♠♦s q✉❡

(p−1)!≡ −1 (mod p),

q✉❡ é ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳

◆♦t❡ q✉❡ ♦ ❚❡♦r❡♠❛ ✸✳✻ ❢♦r♥❡❝❡ ✉♠ ❝r✐tér✐♦ ❞❡ ♣r✐♠❛❧✐❞❛❞❡✿ s❡ n∈Z✱ ❡♥tã♦ n é

✉♠ ♥ú♠❡r♦ ♣r✐♠♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

n|(n−1)! + 1,