❯♥✐✈❡%&✐❞❛❞❡ ❞❡ ❆✈❡✐%♦ ❉❡♣❛$%❛♠❡♥%♦ ❞❡ ▼❛%❡♠+%✐❝❛ ✷✵✶✺
▼!♥✐❝❛ ❈❛❧✈)*✐♦
❇-*.♦❧♦
❯♥✐✈❡%&✐❞❛❞❡ ❞❡ ❆✈❡✐%♦ ❉❡♣❛$%❛♠❡♥%♦ ❞❡ ▼❛%❡♠+%✐❝❛ ✷✵✶✺
▼!♥✐❝❛ ❈❛❧✈)*✐♦
❇-*.♦❧♦
■♥.❡✐*♦1 ●❛✉11✐❛♥♦1
❉✐..❡$%❛/0♦ ❛♣$❡.❡♥%❛❞❛ 1 ❯♥✐✈❡$.✐❞❛❞❡ ❞❡ ❆✈❡✐$♦ ♣❛$❛ ❝✉♠♣$✐♠❡♥%♦ ❞♦. $❡6✉✐.✐%♦. 1 ♦❜%❡♥/0♦ ❞♦ ❣$❛✉ ❞❡ ▼❡.%$❡ ❡♠ ▼❛%❡♠+%✐❝❛ ♣❛$❛ 9$♦❢❡..♦✲ $❡. $❡❛❧✐③❛❞♦ .♦❜ ❛ ♦$✐❡♥%❛/0♦ ❝✐❡♥%>✜❝❛ ❞♦ 9$♦❢❡..♦$ ❉♦✉%♦$ 9❛✉❧♦ ❏♦.A ❋❡$♥❛♥❞❡. ❆❧♠❡✐❞❛✱ 9$♦❢❡..♦$ ❆✉①✐❧✐❛$ ❞♦ ❉❡♣❛$%❛♠❡♥%♦ ❞❡ ▼❛%❡♠+%✐❝❛ ❞❛ ❯♥✐✈❡$.✐❞❛❞❡ ❞❡ ❆✈❡✐$♦✳♦ ❥"#✐ ✴ &❤❡ ❥✉#② ♣!❡#✐❞❡♥'❡ ✴ ♣!❡#✐❞❡♥' +#♦❢❡--♦#❛ ❉♦✉&♦#❛ ❆♥❞#❡✐❛ ❖❧✐✈❡✐#❛ ❍❛❧❧ !♦❢❡%%♦!❛ ❆%%♦❝✐❛❞❛ ❞♦ ❉❡♣❛!-❛♠❡♥-♦ ❞❡ ▼❛-❡♠1-✐❝❛ ❞❛ ❯♥✐✈❡!%✐❞❛❞❡ ❞❡ ❆✈❡✐!♦ ✈♦❣❛✐# ✴ ❡①❛♠✐♥❡!# ❝♦♠♠✐''❡❡ +#♦❢❡--♦# ❉♦✉&♦# ❏♦-8 +❡❞#♦ ▼✐#❛♥❞❛ ▼♦✉#:♦ +❛&#;❝✐♦ !♦❢❡%%♦! ❆✉①✐❧✐❛! ❞♦ ❉❡♣❛!-❛♠❡♥-♦ ❞❡ ▼❛-❡♠1-✐❝❛ ❞❛ ❯♥✐✈❡!%✐❞❛❞❡ ❞♦ ▼✐♥❤♦ +#♦❢❡--♦# ❉♦✉&♦# +❛✉❧♦ ❏♦-8 ❋❡#♥❛♥❞❡- ❆❧♠❡✐❞❛ ✭❖#✐❡♥&❛❞♦#✮ !♦❢❡%%♦! ❆✉①✐❧✐❛! ❞♦ ❉❡♣❛!-❛♠❡♥-♦ ❞❡ ▼❛-❡♠1-✐❝❛ ❞❛ ❯♥✐✈❡!%✐❞❛❞❡ ❞❡ ❆✈❡✐!♦
❛❣"❛❞❡❝✐♠❡♥)♦+ ✴ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥)+
❆❣"❛❞❡&♦ ❛♦ ("♦❢❡**♦" ❉♦✉-♦" (❛✉❧♦ ❆❧♠❡✐❞❛ ♣❡❧❛ ❢♦"&❛ ❡ ♠♦-✐✈❛&3♦✱ ❞❡*❞❡ ♦ ♣"✐♠❡✐"♦ ❞✐❛ ❡✱ ❡**❡♥❝✐❛❧♠❡♥-❡✱ ♣❡❧❛ ♣❛"-✐❧❤❛ ❞❛ ♣❛✐①3♦ ♣♦" -❡♦"✐❛ ❞♦* ♥9♠❡"♦*✳
❖❜"✐❣❛❞❛ ❛♦ ❈❛"❧♦* ▼✐❣✉❡❧✱ ♣♦✐* ❢♦✐ *❡♠ ❞9✈✐❞❛ ♦ ♠❡✉ ♠❛✐♦" ❛♣♦✐♦ ❡ ❛ ❝❤❛✈❡ ♣❛"❛ ?✉❡ -❡"♠✐♥❛**❡ ❡*-❡ -"❛❜❛❧❤♦✳
❛❧❛✈$❛%✲❝❤❛✈❡ ✐♥"❡✐$♦& ❣❛✉&&✐❛♥♦&✱ ♠,①✐♠♦ ❞✐✈✐&♦$ ❝♦♠✉♠✱ "❡♦$❡♠❛ ❞❛ ❞✐✈✐&1♦✱ ❛❧❣♦✲ $✐"♠♦ ❞❡ ❊✉❝❧✐❞❡&✱ ❢❛❝"♦$✐③❛71♦ 8♥✐❝❛✱ ♣$✐♠♦& ❣❛✉&&✐❛♥♦&✱ ❢✉♥71♦ ❞❡ ❊✉❧❡$✱ $❡❝✐♣$♦❝✐❞❛❞❡ ❜✐;✉❛❞$,"✐❝❛✳
❘❡%✉♠♦ ❯♠ ✐♥"❡✐$♦ ❣❛✉&&✐❛♥♦ > ✉♠ ♥8♠❡$♦ ❝♦♠♣❧❡①♦ ❞❛ ❢♦$♠❛ a + bi✱ ♦♥❞❡ a ❡ b &1♦ ❛♠❜♦& ✐♥"❡✐$♦&✳ ❖& ✐♥"❡✐$♦& ❣❛✉&&✐❛♥♦& ❢♦$❛♠ ♣❡❧❛ ♣$✐♠❡✐$❛ ✈❡③ ❡&"✉❞❛❞♦& ♣♦$ ❈❛$❧ ❋$✐❡❞$✐❝❤ ●❛✉&&✱ "❡♥❞♦ ✈❡$✐✜❝❛❞♦ ;✉❡ ❛ ❧❡✐ ❞❛ $❡❝✐♣$♦❝✐❞❛❞❡ ❜✐;✉❛❞$,"✐❝❛ ♣♦❞❡ &❡$ ❡①♣$❡&&❛ ❞❡ ✉♠ ♠♦❞♦ ♠❛✐& &✐♠✲ ♣❧❡& ✉"✐❧✐③❛♥❞♦ ❡&"❡& ♥8♠❡$♦&✱ ;✉❡ "E♠ ✉♠ ❝♦♠♣♦$"❛♠❡♥"♦ ♠✉✐"♦ &❡✲ ♠❡❧❤❛♥"❡ ❛♦ ❞♦& ♥8♠❡$♦& ✐♥"❡✐$♦&✱ ❤❛✈❡♥❞♦ ♥8♠❡$♦& ♣$✐♠♦&✱ ♠,①✐♠♦ ❞✐✈✐&♦$ ❝♦♠✉♠✱ ❛❧❣♦$✐"♠♦ ❞❡ ❊✉❝❧✐❞❡& ❡ ❢❛❝"♦$✐③❛71♦ 8♥✐❝❛ ❡♠ ♣$✐♠♦&✱ ❡♥"$❡ ♦✉"$♦&✳ ◆❡&"❡ "$❛❜❛❧❤♦ ✐$❡♠♦& ❡&"✉❞❛$ ❛& ♣$♦♣$✐❡❞❛❞❡& ❡ ❛ ❛$✐"✲ ♠>"✐❝❛ ❞♦& ✐♥"❡✐$♦& ❣❛✉&&✐❛♥♦& ❡ ❛❧❣✉♠❛& ❞❛& &✉❛& ❛♣❧✐❝❛7G❡&✱ ✐♥❝❧✉✐♥❞♦ ❛ ❧❡✐ ❞❛ $❡❝✐♣$♦❝✐❞❛❞❡ ❜✐;✉❛❞$,"✐❝❛✳
❑❡②✇♦%❞' ●❛✉##✐❛♥ ✐♥&❡❣❡)#✱ ❣)❡❛&❡#& ❝♦♠♠♦♥ ❞✐✈✐#♦)✱ ❊✉❝❧✐❞❡❛♥ ❛❧❣♦)✐&❤♠✱ ✉♥✐✲ 4✉❡ ❢❛❝&♦)✐③❛&✐♦♥✱ ●❛✉##✐❛♥ ♣)✐♠❡#✱ ❊✉❧❡) ❢✉♥❝&✐♦♥✱ ❜✐4✉❛❞)❛&✐❝ )❡❝✐♣)♦✲ ❝✐&②✳
❆❜'*%❛❝* ❆ ●❛✉##✐❛♥ ✐♥&❡❣❡) ✐# ❛ ❝♦♠♣❧❡① ♥✉♠❜❡) ♦❢ &❤❡ ❢♦)♠ a+bi ✇❤❡)❡ a ❛♥❞ b ❛)❡ ❜♦&❤ ✐♥&❡❣❡)#✳ ❚❤❡ ●❛✉##✐❛♥ ✐♥&❡❣❡)# ✇❡)❡ ✜)#& #&✉❞✐❡❞ ❜② ❈❛)❧ ❋)✐❡❞)✐❝❤ ●❛✉##✱ ❤❛✈✐♥❣ ♥♦&❡❞ &❤❛& &❤❡ ❧❛✇ ♦❢ ❜✐4✉❛❞)❛&✐❝ )❡❝✐♣)♦❝✐&② ❝❛♥ ❜❡ ❡①♣)❡##❡❞ ✐♥ ❛ #✐♠♣❧❡) ✇❛② ✉#✐♥❣ &❤❡#❡ ♥✉♠❜❡)#✱ ✇❤✐❝❤ ❤❛✈❡ ❛ ✈❡)② #✐♠✐❧❛) ❜❡❤❛✈✐♦) &♦ &❤❛& ♦❢ ✐♥&❡❣❡)#✱ ✇✐&❤ ♣)✐♠❡ ♥✉♠❜❡)#✱ ❣)❡❛✲ &❡#& ❝♦♠♠♦♥ ❞✐✈✐#♦)✱ ❊✉❝❧✐❞❡❛♥ ❛❧❣♦)✐&❤♠ ❛♥❞ ✉♥✐4✉❡ ❢❛❝&♦)✐③❛&✐♦♥ ✐♥&♦ ♣)✐♠❡#✱ ❛♠♦♥❣ ♦&❤❡)#✳ ■♥ &❤✐# ✇♦)❦ ✇❡ #&✉❞② &❤❡ ♣)♦♣❡)&✐❡# ❛♥❞ &❤❡ ❛)✐&❤♠❡&✐❝ ♦❢ ●❛✉##✐❛♥ ✐♥&❡❣❡)# ❛♥❞ #♦♠❡ ♦❢ ✐&# ❛♣♣❧✐❝❛&✐♦♥#✱ ✐♥❝❧✉❞✐♥❣ &❤❡ ❧❛✇ ♦❢ ❜✐4✉❛❞)❛&✐❝ )❡❝✐♣)♦❝✐&②✳
✏❆ ♠❛$❡♠&$✐❝❛ ) ❛ *❛✐♥❤❛ ❞❛. ❝✐/♥❝✐❛. ❡ ❛ ❛*✐$♠)$✐❝❛ ) ❛ *❛✐♥❤❛ ❞❛ ♠❛$❡♠&$✐❝❛✑ ❈❛"❧ ●❛✉&&
❈♦♥#❡%❞♦
❈♦♥#❡%❞♦ ✶ ✶ ■♥#)♦❞✉+,♦ ✸ ✷ /)❡❧✐♠✐♥❛)❡4 ✺ ✷✳✶ ❈❛%❧ ❋%✐❡❞%✐❝❤ ●❛✉// ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ◆♦34❡/ /♦❜%❡ ◆6♠❡%♦/ ■♥:❡✐%♦/ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸ ◆♦+7❡4 ❇94✐❝❛4 4♦❜)❡ ■♥#❡✐)♦4 ●❛✉44✐❛♥♦4 ✶✺ ✸✳✶ ◆6♠❡%♦/ ■♥:❡✐%♦/ ❈♦♠♣❧❡①♦/ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✷ ◆♦%♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✸ ❉✐✈✐/✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✹ ❚❡♦%❡♠❛ ❞❛ ❉✐✈✐/E♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✺ ❆❧❣♦%✐:♠♦ ❞❡ ❊✉❝❧✐❞❡/ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹ ❆)✐#♠?#✐❝❛ ❞♦4 ■♥#❡✐)♦4 ●❛✉44✐❛♥♦4 ✹✶ ✹✳✶ ❚❡♦%❡♠❛ ❋✉♥❞❛♠❡♥:❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✶❈❖◆❚❊%❉❖ ❈❖◆❚❊%❉❖ ✹✳✷ #$✐♠♦( ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✸ ❈♦♥❣$✉0♥❝✐❛( ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✺ ❆♣❧✐❝❛'(❡* ❞♦* ■♥/❡✐0♦* ●❛✉**✐❛♥♦* ✼✸ ✺✳✶ ❚❡$♥♦( #✐7❛❣8$✐❝♦( ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✺✳✷ ❆ ❡;✉❛<=♦ y2 = x3− 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✺✳✸ ❙♦♠❛ ❞❡ ❞♦✐( ;✉❛❞$❛❞♦( ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✺✳✹ ▲❡✐ ❞❛ ❘❡❝✐♣$♦❝✐❞❛❞❡ ❇✐;✉❛❞$D7✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✻ ❈♦♥❝❧✉*7♦ ✾✶ ❇✐❜❧✐♦❣0❛✜❛ ✾✸ ✷
✶✳ ■♥$%♦❞✉)*♦
❯♠ ✐♥$❡✐&♦ ❣❛✉++✐❛♥♦ , ✉♠ ♥-♠❡&♦ ❝♦♠♣❧❡①♦ ❞❛ ❢♦&♠❛ a + bi✱ ♦♥❞❡ a ❡ b +5♦ ❛♠❜♦+ ✐♥$❡✐&♦+✳ ❊+$❡+ ♥-♠❡&♦+✱ ❡+$✉❞❛❞♦+✱ ♣❡❧❛ ♣&✐♠❡✐&❛ ✈❡③✱ ♣♦& ❈❛&❧ ❋&✐❡❞&✐❝❤ ●❛✉++✱ $?♠ ✉♠ ❝♦♠♣♦&$❛♠❡♥$♦ ♠✉✐$♦ +❡♠❡❧❤❛♥$❡ ❛♦ ❞♦+ ♥-♠❡&♦+ ✐♥$❡✐&♦+✳ ●❛✉++ ✉$✐❧✐③♦✉ ❡+$❡+ ♥-♠❡&♦+ ♣❛&❛ ♣♦❞❡& ♣&♦✈❛& &❡+✉❧$❛❞♦+ ♥♦+ ✐♥$❡✐&♦+ ❝♦♠✉♥+✱ ♥♦♠❡❛❞❛♠❡♥$❡ ❛ ❧❡✐ ❞❛ &❡❝✐♣&♦❝✐❞❛❞❡ ❜✐@✉❛❞&A$✐❝❛✳
■♥"#♦❞✉'(♦ ◆♦ ❝❛♣%&✉❧♦ ✷✱ ✐,❡♠♦/ ❢❛③❡, ✉♠❛ ❝♦♥&❡①&✉❛❧✐③❛45♦ ❤✐/&7,✐❝❛ /♦❜,❡ ♦ ❢❛♠♦/♦ ♠❛&❡✲ ♠:&✐❝♦ ❈❛,❧ ●❛✉// ❡ ❛ ❞❡/❝♦❜❡,&❛ ❞❡/&❡/ ♥>♠❡,♦/✳ ❆//✐♠✱ ❝♦♠♦ ✉♠❛ ,❡❢❡,A♥❝✐❛ ❛ ♣,♦♣,✐❡❞❛❞❡/ ❡ ,❡/✉❧&❛❞♦/ ❞♦/ ♥>♠❡,♦/ ✐♥&❡✐,♦/ B✉❡ /❡,5♦ >&❡✐/ ♣❛,❛ ❝♦♠♣,❡❡♥❞❡,♠♦/ ❛ ❛♥❛❧♦❣✐❛✱ ❡♥&,❡ ✐♥&❡✐,♦/ ❣❛✉//✐❛♥♦/ ❡ ✐♥&❡✐,♦/ ❝♦♠✉♥/✱ B✉❡ /❡,: ❢❡✐&❛ ❛♦ ❧♦♥❣♦ ❞♦ &,❛❜❛❧❤♦✳
◆♦ ❝❛♣%&✉❧♦ ✸ ❞❡❞✐❝❛♠♦✲♥♦/ ❛♦ ❡/&✉❞♦ ❞❛/ ♥♦4E❡/ ❜:/✐❝❛/ /♦❜,❡ ✐♥&❡✐,♦/ ❣❛✉//✐❛♥♦/✿ ❞❡✜♥✐45♦ ❞❡ ✐♥&❡✐,♦ ❣❛✉//✐❛♥♦ ❡ /✉❛/ ♣,♦♣,✐❡❞❛❞❡/✱ ♥♦,♠❛✱ ❞✐✈✐/✐❜✐❧✐❞❛❞❡✱ &❡♦,❡♠❛ ❞❛ ❞✐✈✐/5♦✱ ♠:①✐♠♦ ❞✐✈✐/♦, ❝♦♠✉♠✱ ❛❧❣♦,✐&♠♦ ❞❡ ❊✉❝❧✐❞❡/ ❡ &❡♦,❡♠❛ ❞❡ ❇K③♦✉&✳ ❖ B✉❛,&♦ ❝❛♣%&✉❧♦ /❡,: ❞❡❞✐❝❛❞♦ M ❛,✐&♠K&✐❝❛ ❡♠ Z[i]✱ ❢♦❝❛♥❞♦ ♦ &❡♦,❡♠❛ ❢✉♥❞❛♠❡♥✲ &❛❧✱ ❛ ❝❛,❛❝&❡,✐③❛45♦ ❞♦/ ♣,✐♠♦/ ❣❛✉//✐❛♥♦/ ❡ ❛/ ❝♦♥❣,✉A♥❝✐❛/ ♣❛,❛ ✐♥&❡✐,♦/ ❣❛✉//✐❛✲ ♥♦/✳
❚❡,♠✐♥❛♠♦/ ❡/&❡ &,❛❜❛❧❤♦ ❝♦♠ ✉♠ ❝❛♣%&✉❧♦ /♦❜,❡ ❛♣❧✐❝❛4E❡/ ❞❡ Z[i] ♥❛ ❛,✐&♠K&✐❝❛ ❞❡ Z ♦♥❞❡ ✈❛♠♦/ ❡/&✉❞❛, ,❡/✉❧&❛❞♦/ /♦❜,❡ &❡,♥♦/ ♣✐&❛❣7,✐❝♦/ ♣,✐♠✐&✐✈♦/✱ ❛ ❝✉,✈❛ ❡❧%♣&✐❝❛ y2 = x3− 1✱ ❛ /♦♠❛ ❞❡ ❞♦✐/ B✉❛❞,❛❞♦/ ❡✱ ✜♥❛❧♠❡♥&❡✱ ❛ ❧❡✐ ❞❛ ,❡❝✐♣,♦❝✐❞❛❞❡
❜✐B✉❛❞,:&✐❝❛✳ ❊/&❡ ❝❛♣%&✉❧♦ /❡,✈❡ ♣❛,❛ ♠♦/&,❛, ❛ ❛❜,❛♥❣A♥❝✐❛ ❞❡ ,❡/✉❧&❛❞♦/ ♣❛,❛ ♦/ B✉❛✐/ ♦/ ✐♥&❡✐,♦/ ❣❛✉//✐❛♥♦/ /5♦ ✐♠♣♦,&❛♥&❡/✳ O♦, ❡①❡♠♣❧♦✱ ♦/ ✐♥&❡✐,♦/ ❣❛✉//✐❛♥♦/ ♣♦❞❡♠ &❡, ❛♣❧✐❝❛45♦ ♥❛ ❝,✐♣&♦❣,❛✜❛ ✭✈❡, ❬✼❪✮✳
✷✳ "#❡❧✐♠✐♥❛#❡*
◆❡"#❡ ❝❛♣'#✉❧♦ ✐,❡♠♦" ❢❛③❡, ✉♠❛ ❜,❡✈❡ ✐♥#,♦❞✉45♦ ❤✐"#7,✐❝❛ "♦❜,❡ ❛ ✈✐❞❛ ❡ ♦❜,❛ ❞❡ ❈❛,❧ ●❛✉""✱ ♣❛,❛ ♠❡❧❤♦, ❝♦♠♣,❡❡♥❞❡,♠♦" ♦ ;✉❡ ♦ ❧❡✈♦✉ ❛ ❞❡"❝♦❜,✐, ♦" ✐♥#❡✐,♦" ❣❛✉""✐❛♥♦"✳ ■,❡♠♦"✱ #❛♠❜?♠✱ ,❡❢❡,✐, ❛❧❣✉♠❛" ♥♦4@❡" ❜A"✐❝❛" ❡ ❛❧❣✉♥" ,❡"✉❧#❛❞♦" "♦❜,❡ ✐♥#❡✐,♦" ❝♦♠✉♥" ;✉❡ ♥♦" "❡,5♦ B#❡✐" ♣❛,❛ ❝♦♠♣,❡❡♥❞❡, ♠❡❧❤♦, ❛ ❛♥❛❧♦❣✐❛ ❡♥#,❡ ✐♥#❡✐,♦" ❝♦♠✉♥" ❡ ✐♥#❡✐,♦" ❣❛✉""✐❛♥♦"✳ ✺✷✳✶ ❈❛%❧ ❋%✐❡❞%✐❝❤ ●❛✉// 0%❡❧✐♠✐♥❛%❡/
✷✳✶ ❈❛%❧ ❋%✐❡❞%✐❝❤ ●❛✉//
❈❛"❧ ❋"✐❡❞"✐❝❤ ●❛✉,, ♥❛,❝❡✉ ❡♠ ❇"❛✉♥,❝❤✇❡✐❣✱ ❆❧❡♠❛♥❤❛✱ ❛ 30 ❞❡ ❆❜"✐❧ ❞❡ 1777✳ ❋✐❧❤♦ ❞❡ ❝❛♠♣♦♥❡,❡, ♣♦❜"❡,✱ ❝♦♠ ❛♣❡♥❛, 8"9, ❛♥♦, ❞❡ ✐❞❛❞❡ ●❛✉,, ♠♦,8"♦✉ ❛♣8✐❞:♦ ♣❛"❛ ♦♣❡"❛;<❡, ❛"✐8♠=8✐❝❛, ,❡♥❞♦✱ ♣♦" ✐,,♦✱ ✉♠ ❞♦, ❝❛,♦, ♠❛✐, ♣"❡❝♦❝❡, ❞❛ ❤✐,8>"✐❛ ❞❛ ♠❛8❡♠?8✐❝❛✳ ▼❡,♠♦ ,❡♠ ❝♦♥❞✐;<❡, ♣❛"❛ ♣❛❣❛" ♦, ❡,8✉❞♦, ❞♦ ✜❧❤♦✱ ❛ ,✉❛ ♠:❡ ❡ ♦ ,❡✉ 8✐♦✱ ❛♣❡,❛" ❞❛, ♦❜❥❡;<❡, ♣❛8❡"♥❛,✱ ♠❛8"✐❝✉❧❛"❛♠✲♥♦✱ ❝♦♠ ,❡8❡ ❛♥♦,✱ ♥✉♠❛ ❡,❝♦❧❛ ❧♦❝❛❧ ❞❡ ❡♥,✐♥♦ ♣D❜❧✐❝♦ ✳ ❋♦✐ ❛E F✉❡✱ ❛♦, ❞❡③ ❛♥♦,✱ ●❛✉,, ✐♥✐❝✐♦✉ ♦, ,❡✉, ❡,8✉❞♦, ❡♠ ❛"✐8♠=8✐❝❛ ♣♦✐,✱ ♣❛"❛ ❡,♣❛♥8♦ ❞♦ ,❡✉ ♠❡,8"❡ ❇✉88♥❡"✱ ❝♦♠♣❧❡8❛✈❛ ❝♦♠♣❧✐❝❛❞❛, ♦♣❡"❛;<❡,✳ ❙✉♣<❡✲,❡ F✉❡✱ ♥❡,8❛ ❛❧8✉"❛✱ ❥? ❤❛✈✐❛ ❞❡,❝♦❜❡"8♦ ❛ ❢>"♠✉❧❛ ❞❛ ,♦♠❛ ❞❡ ✉♠❛ ♣"♦❣"❡,,:♦ ❛"✐8♠=8✐❝❛✳ ❇✉88♥❡" 8✐♥❤❛✱ ♥❡,,❛ =♣♦❝❛✱ ✉♠ ❥♦✈❡♠ ❛,,✐,8❡♥8❡ ❞❡ 17 ❛♥♦,✱ ❏♦❤❛♥♥ ▼❛"8✐♥ ❇❛"8❡❧,✱ ❛♣❛✐①♦♥❛❞♦ ♣❡❧❛ ♠❛8❡♠?8✐❝❛✱ ❛ F✉❡♠ ❡♥8"❡❣♦✉ ❛ 8❛"❡❢❛ ❞❡ ❡♥,✐♥❛" ♦ ♣"❡❝♦❝❡ ●❛✉,,✳ ▼❛, ❝❡❞♦ ❇❛"8❡❧, ❝♦♠♣"❡❡♥❞❡✉ F✉❡ ♥❛❞❛ 8✐♥❤❛ ♣❛"❛ ❡♥,✐♥❛" ❛ ●❛✉,,✱ ✧♦ ❛❧✉♥♦ 8✐♥❤❛ ,✉♣❡"❛❞♦ ♦ ♠❡,8"❡✧✳ ❚❡♥❞♦ ❛♠✐❣♦, ✐♥✢✉❡♥8❡,✱ ❇❛"8❡❧, ❢❡③ ❝♦♠ F✉❡ ●❛✉,, ,❡ 8♦"♥❛,,❡ ❝♦♥❤❡❝✐❞♦ ❞♦ ❞✉F✉❡ ❞❡ ❇"❛✉♥,❝❤✇❡✐❣✱ ❈❛"❧ ❲✐❧❤❡❧♠ ❋❡"❞✐♥❛♥❞✱ F✉❡ ♦ ♣"♦8❡❣❡✉ ❛8= Q ,✉❛ ♠♦"8❡✱ ❣❛"❛♥8✐♥❞♦ "❡❝✉",♦, ♣❛"❛ F✉❡ 8✐✈❡,,❡ ♠❡✐♦, ❞❡ ,✉❜,✐,89♥❝✐❛ ❡✱ ❛,,✐♠✱ ❝♦♥8✐♥✉❛,,❡ ❛ ❡,8✉❞❛"✳ ❆♦, ❞♦③❡ ❛♥♦, ●❛✉,, ❥? ♦❧❤❛✈❛ ❝♦♠ ❞❡,❝♦♥✜❛♥;❛ ♣❛"❛ ♦, ❢✉♥❞❛♠❡♥8♦, ❞❛ ❣❡♦♠❡8"✐❛ ❡✉❝❧✐❞✐❛♥❛ ❡ ❛♦, ❞❡③❛,,❡✐, ❥? 8✐♥❤❛ 8✐❞♦ ♦ ♣"✐♠❡✐"♦ ✈✐,❧✉♠❜"❡ ❞❡ ✉♠❛ ❣❡♦♠❡8"✐❛ ❞✐❢❡"❡♥8❡ ❞❛ ❞❡ ❊✉❝❧✐❞❡,✳ ❊♠ 1792✱ ●❛✉,, ❡♥8"♦✉ ♣❛"❛ ♦ ❈♦❧❧❡❣✐✉♠ ❈❛"♦❧✐♥✉♠✱ ♦♥❞❡ ♣❡"♠❛♥❡❝❡✉ ♣♦" 8"9, ❛♥♦,✳ ❊,8✉❞♦✉ ❛, ♦❜"❛, ♠❛✐, ♥♦8?✈❡✐, ❞❡ ▲❡♦♥❤❛"❞ ❊✉❧❡"✱ ❏♦,❡♣❤✲▲♦✉✐, ❞❡ ▲❛❣"❛♥❣❡ ❡ ■,❛❛❝ ◆❡✇8♦♥✱ 8♦♠❛♥❞♦ ❛ ❞❡❝✐,:♦ ❞❡ ♣"❡❡♥❝❤❡" ♦, ✈❛③✐♦, ❡ ❝♦♠♣❧❡8❛" ♦ F✉❡ 8✐♥❤❛ ,✐❞♦ ❢❡✐8♦ ♣❡❧❛ ♠❡8❛❞❡✳ ❈♦♠♦ "❡,✉❧8❛❞♦ ❞❡,8❡ ,❡✉ ♣"♦❥❡8♦✱ ❛♦, ✈✐♥8❡ ❡ ✉♠ ❛♥♦,✱ ♣"♦❞✉③✐✉ ♦ ❧✐✈"♦ ✑❉✐#$✉✐#✐&✐♦♥❡# ❆+✐&❤♠❡&✐❝❛❡✑ F✉❡ ❝♦♥8=♠ ❣"❛♥❞❡, ❝♦♥8"✐❜✉✐;<❡, ♣❛"❛ ❛ ❆"✐8♠=8✐❝❛ ❡ ♣❛"❛ ❛ ➪❧❣❡❜"❛✳ ❋♦✐ ❝♦♥,✐❞❡"❛❞♦✱ ♣♦" ♠✉✐8♦,✱ ❛ ,✉❛ ♦❜"❛ ♣"✐♠❛ ❡ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✽✵✶✱ 8"9, ❛♥♦, ❛♣>, ❛ ,✉❛ ❝♦♥❝❧✉,:♦✳ ◆♦ ❧✐✈"♦ ✑❉✐,F✉✐,✐8✐♦♥❡, ❆"✐8❤♠❡8✐❝❛❡✑ ●❛✉,, ✐♥8"♦❞✉③ ❡ ❡,8✉❞❛ ❛, ❝♦♥❣"✉9♥❝✐❛, ❞♦ 8✐♣♦ xn ≡ a mod p✱ ✐,8♦ =✱ ❛, ❡F✉❛;<❡, xn= a ❡♠ Z p✳ ❯♠ ♣"♦❜❧❡♠❛ ♥❛8✉"❛❧ ♥❡,8❡ ❝♦♥8❡①8♦ = ,❛❜❡" ♣❛"❛ F✉❡ ✈❛❧♦"❡, ✐♥8❡✐"♦, ❞❡ a ❛ ❡F✉❛;:♦ 8❡♠ ,♦❧✉;:♦✳ ❊♠ ❜✉,❝❛ ❞❛ ,♦❧✉;:♦✱ ●❛✉,, "❡,♦❧✈❡✉ 8♦8❛❧♠❡♥8❡ ♦ ❝❛,♦ n = 2 ❡ ❞❡♠♦♥,8"♦✉ ✉♠❛ ♣"♦♣"✐❡❞❛❞❡ ♠❛"❛✈✐❧❤♦,❛✱ ❞❡8❡8❛❞❛ ❛♥8❡"✐♦"♠❡♥8❡ ♣♦" ❊✉❧❡" ❡ F✉❡ ▲❡❣❡♥❞"❡ 8❡♥8❛"❛ ♣"♦✈❛"✱ ❛ ▲❡✐ ❞❛ ❘❡❝✐♣+♦❝✐❞❛❞❡ ◗✉❛❞+5&✐❝❛ ✭✈❡" 8❡♦"❡♠❛ ✷✳✸✻✮✳ ✻!❡❧✐♠✐♥❛!❡( ✷✳✷ ◆♦-.❡( (♦❜!❡ ◆0♠❡!♦( ■♥2❡✐!♦( ❋♦✐ ♥❡%&❛ ❛❧&✉*❛ +✉❡ ♦ ✐♥&✐&✉❧❛*❛♠ ❞❡ ✑♣"#♥❝✐♣❡ ❞❛ ♠❛+❡♠,+✐❝❛✑ ✱ ♣♦✐% ❢♦✐ ♦ ♣*✐✲ ♠❡✐*♦ ❛ ♣*♦✈❛* ❡%&❛ ❧❡✐ +✉❡ ❞❡%✐❣♥♦✉ ❝♦♠♦ ✑❛ ❥7✐❛ ❞❛ ❛*✐&♠8&✐❝❛✑✱ ♦ ✑&❤❡♦*❡♠❛ ❛✉*❡✉♠✑ ♦✉ ✑&❡♦*❡♠❛ ❞❡ ♦✉*♦✑✳
●❛✉%% ❢❡③ ❝♦♥&*✐❜✉✐>?❡% ✐♠♣♦*&❛♥&❡% ♣❛*❛ ♦ ❝❛%♦ n = 4 ❡✱ ❡♠❜♦*❛ ♥@♦ &❡♥❤❛ ♣*♦✈❛❞♦ ❛ ❧❡✐ ❞❛ *❡❝✐♣*♦❝✐❞❛❞❡ ❜✐+✉❛❞*A&✐❝❛✱ ♣❡*❝❡❜❡✉ +✉❡ ♦ %❡✉ ❡%&✉❞♦ %❡*✐❛ ♠❛✐% %✐♠♣❧❡% ✉&✐❧✐③❛♥❞♦ ♦% ♥B♠❡*♦% ✐♥&❡✐*♦% ❝♦♠♣❧❡①♦%✱ ♥B♠❡*♦% ❞❛ ❢♦*♠❛ a + bi✱ ♦♥❞❡ a ❡ b %@♦ ✐♥&❡✐*♦%✳ ❊%&❡% ♥B♠❡*♦%✱ ❞❡%✐❣♥❛❞♦% ♣♦* ✐♥+❡✐"♦. ❣❛✉..✐❛♥♦.✱ ♣♦%%✉❡♠ ♣*♦♣*✐❡❞❛✲ ❞❡% %❡♠❡❧❤❛♥&❡% E% ❞♦% ♥B♠❡*♦% ✐♥&❡✐*♦%✳ ◆♦ %❡✉ &*❛❜❛❧❤♦ ✑❚❤❡♦"② ♦❢ ❇✐6✉❛❞"❛+✐❝ ❘❡.✐❞✉❡.✑✱ ♣✉❜❧✐❝❛❞♦ ❡♠ 1825✱ ●❛✉%% ❛♣*❡%❡♥&❛ ❡%&❡% ♥B♠❡*♦% ❡ ❛% %✉❛% ♣*♦♣*✐❡✲ ❞❛❞❡%✳
●❛✉%% ❞❡✐①♦✉ ❝♦♥&*✐❜✉✐>?❡% *❡❧❡✈❛♥&❡% ❡♠ ✈A*✐❛% A*❡❛% ❞❛ ▼❛&❡♠A&✐❝❛ ❝♦♠♦✱ ♣♦* ❡①❡♠♣❧♦✱ ♦ ❢❛♠♦%♦ ❚❡♦*❡♠❛ ❋✉♥❞❛♠❡♥&❛❧ ❞❛ ➪❧❣❡❜*❛✱ ❛ ❞✐%&*✐❜✉✐>@♦ ♥♦*♠❛❧ ❞❡ ●❛✉%% ♥❛ ❊%&❛&J%&✐❝❛✱ ❛ ❣❡♦♠❡&*✐❛ ❞❛% %✉♣❡*❢J❝✐❡% ❡ ❣❡♦♠❡&*✐❛ ♥@♦ ❡✉❝❧✐❞❡❛♥❛ ♥❛ ●❡✲ ♦♠❡&*✐❛ ❡✱ ♥❛ ❋J%✐❝❛✱ ❝♦♥&*✐❜✉✐✉ ❝♦♠ ♦ ❡%&✉❞♦ ❞♦ ♠❛❣♥❡&✐%♠♦✳
▼❛% ❞❡ &♦❞♦ ❡%&❡ ✉♥✐✈❡*%♦✱ ●❛✉%% ♥✉♥❝❛ ❡%❝♦♥❞❡✉ ❛ %✉❛ ♣*❡❢❡*K♥❝✐❛ %✐♥&❡&✐③❛❞❛ ♥❛ %❡❣✉✐♥&❡ ❢*❛%❡✿ ✑❆ ♠❛+❡♠,+✐❝❛ 9 ❛ "❛✐♥❤❛ ❞❛. ❝✐:♥❝✐❛. ❡ ❛ ❛"✐+♠9+✐❝❛ 9 ❛ "❛✐♥❤❛ ❞❛ ♠❛+❡♠,+✐❝❛✑ ✳
❆ ✈✐❞❛ ❞❡ ●❛✉%%✱ ❞❡ ❝*✐❛♥>❛ ♣*♦❞J❣✐♦ ❛ ❣*❛♥❞❡ ♠❛&❡♠A&✐❝♦✱ &❡*♠✐♥♦✉ ❛ 23 ❞❡ ❏❛♥❡✐*♦ ❞❡ 1855✱ ❞❡✐①❛♥❞♦ ✉♠ ❣*❛♥❞❡ ❧❡❣❛❞♦ ♣❛*❛ ❛% ❝✐K♥❝✐❛%✱ ❡♠ ❡%♣❡❝✐❛❧ ♣❛*❛ ❛ ▼❛&❡♠A✲ &✐❝❛✳
❆♣7% 43 ❛♥♦% ❞❛ %✉❛ ♠♦*&❡ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ✉♠ ❞✐A*✐♦ ❝♦♠ 146 ❛♥♦&❛>?❡% ❞❡ ●❛✉%%✱ ❝♦♠ ♣❡*♠✐%%@♦ ❞♦% %❡✉% ♥❡&♦%✱ ❡ +✉❡ ✱%❡❣✉♥❞♦ *❡❣✐%&♦% ❡♥❝♦♥&*❛❞♦%✱ ●❛✉%% ♥@♦ &❡*✐❛ ❞✐✈✉❧❣❛❞♦ &❛✐% ❞❡%❝♦❜❡*&❛% ♣♦* ❝♦♥%✐❞❡*A✲❧❛% ✐♥❢❡*✐♦*❡% E% ❛♥&❡*✐♦*❡%✳
✷✳✷ ◆♦$%❡' '♦❜)❡ ◆*♠❡)♦' ■♥.❡✐)♦'
◆❡%&❛ %❡❝>@♦ ✈❛♠♦% *❡❝♦*❞❛* ✈A*✐♦% *❡%✉❧&❛❞♦% %♦❜*❡ ♥B♠❡*♦% ✐♥&❡✐*♦% ♣♦%✐&✐✈♦% +✉❡ %❡*@♦ ✉&✐❧✐③❛❞♦% ❛♦ ❧♦♥❣♦ ❞♦ &*❛❜❛❧❤♦✳ ❆ ♠❛✐♦*✐❛ ❞❛% ❞❡♠♦♥%&*❛>?❡% ❞♦% *❡%✉❧&❛❞♦% ♣♦❞❡♠ %❡* ❡♥❝♦♥&*❛❞❛% ♥♦ ❧✐✈*♦ ❞♦% ❛✉&♦*❡% ❍❛*❞② ✫ ❲*✐❣❤& ❬✶✷❪✱ ♥♦% ❛♣♦♥&❛♠❡♥&♦% ❞❡ ❆❧♠❡✐❞❛ ❬✶❪ ❡ ❞❡ ❋❧♦*❡♥&✐♥♦ ❬✽❪✳
❯♠❛ ✈❡③ +✉❡ %@♦ *❡%✉❧&❛❞♦% ❞✐%♣❡*%♦% ❞❛ &❡♦*✐❛ ❞♦% ♥B♠❡*♦%✱ ❛ ♦*❞❡♠ ❡♠ +✉❡ ♦% ❛♣*❡%❡♥&❛♠♦% %❡❣✉❡ ❛ ❡%&*✉&✉*❛ ❞❡%&❡ &*❛❜❛❧❤♦ %♦❜*❡ ♦% ✐♥&❡✐*♦% ❣❛✉%%✐❛♥♦%✳
✷✳✷ ◆♦$%❡' '♦❜)❡ ◆*♠❡)♦' ■♥.❡✐)♦' 0)❡❧✐♠✐♥❛)❡' ❖! "❡!✉❧&❛❞♦! *✉❡ !❡ !❡❣✉❡♠ !-♦✱ ♣"✐♥❝✐♣❛❧♠❡♥&❡✱ 3&❡✐! ♣❛"❛ ♦ ❝❛♣4&✉❧♦ ✸✳
❖ ❝♦♥❥✉♥&♦ ❞♦! ♥3♠❡"♦! ✐♥&❡✐"♦!✱ "❡♣"❡!❡♥&❛❞♦ ♣❡❧❛ ❧❡&"❛ Z✱ ✐♥❝❧✉✐ &♦❞♦! ♦! ♥3♠❡"♦! ✐♥&❡✐"♦! ♣♦!✐&✐✈♦!✱ ✐♥&❡✐"♦! ♥❡❣❛&✐✈♦! ❡ ♦ ③❡"♦✳
❉❡✜♥✐%&♦ ✷✳✶✳ ❉✐✈✐,✐❜✐❧✐❞❛❞❡✳
❙❡❥❛♠ a ❡ b ∈ Z✳ ❙❡ a 6= 0 ❡ &❡ ❡①✐&)✐* ✉♠ ✐♥)❡✐*♦ c )❛❧ /✉❡ b = ac✱ ❞✐③❡♠♦& /✉❡ a ❞✐✈✐❞❡ b✱ ❡ ❞❡♥♦)❛♠♦& ♣♦* a | b✳
❙❡ a ♥&♦ ❞✐✈✐❞❡ b ❞❡♥♦)❛♠♦& ♣♦* a ∤ b✳
❉❡✜♥✐%&♦ ✷✳✷✳ ◆2♠❡4♦ ♣4✐♠♦ ❡ ◆2♠❡4♦ ❈♦♠♣♦,7♦✳
❆ /✉❛❧/✉❡* ✐♥)❡✐*♦ ♠❛✐♦* /✉❡ 1 ❝✉❥♦& 6♥✐❝♦& ❞✐✈✐&♦*❡& ♣♦&✐)✐✈♦& &❡❥❛♠ ❡❧❡ ♣*8♣*✐♦ ❡ 1✱ ❝❤❛♠❛♠♦& ♥2♠❡4♦ ♣4✐♠♦✳ ❯♠ ✐♥)❡✐*♦ ♠❛✐♦* /✉❡ 1 /✉❡ ♥;♦ &❡❥❛ ♣*✐♠♦ < ✉♠ ♥2♠❡4♦ ❝♦♠♣♦,7♦✳
❉❡✜♥✐%&♦ ✷✳✸✳ ▼;①✐♠♦ ❉✐✈✐,♦4 ❈♦♠✉♠✳
❙❡❥❛♠ a ❡ b ❞♦✐& ✐♥)❡✐*♦& )❛✐& /✉❡ ♣❡❧♦ ♠❡♥♦& ✉♠ ❞❡❧❡& < ♥;♦ ♥✉❧♦✳ ❉❡&✐❣♥❛♠♦& ♣♦* ♠;①✐♠♦ ❞✐✈✐,♦4 ❝♦♠✉♠ ♦ ♠❛✐♦* ❡❧❡♠❡♥)♦ ❞♦ ❝♦♥❥✉♥)♦ ❞♦& ❞✐✈✐&♦*❡& ❝♦♠✉♥& ❞❡ a ❡ b ❡ ❞❡♥♦)❛♠♦& ♣♦* (a, b)✳
❙❡ ❝♦♥!✐❞❡"❛"♠♦! &";! ♥3♠❡"♦! ✐♥&❡✐"♦! a, b ❡ c ❞❡♥♦&❛♠♦! ♦ !❡✉ ♠<①✐♠♦ ❞✐✈✐!♦" ❝♦♠✉♠ ♣♦" (a, b, c)✳
❉❡✜♥✐%&♦ ✷✳✹✳ ?4✐♠♦, ❡♥74❡ ,✐✳
❙❡❥❛♠ a ❡ b ✐♥)❡✐*♦&✱ ♦♥❞❡ ♣❡❧♦ ♠❡♥♦& ✉♠ ❞❡❧❡& < ♥;♦ ♥✉❧♦✳ ❙❡ (a, b) = 1✱ ❞✐③❡♠♦& /✉❡ a ❡ b &;♦ ♣4✐♠♦, ❡♥74❡ ,✐✳
❚❡♦4❡♠❛ ✷✳✺✳ ❚❡♦4❡♠❛ ❞❛ ❉✐✈✐,&♦ ❡♠ Z✳ ❙❡❥❛♠ a ❡ b ∈ Z✱ ♦♥❞❡ b > 0✳
❊♥);♦✱ ❡①✐&)❡♠ ❡ &;♦ 6♥✐❝♦& ♦& ♥6♠❡*♦& ✐♥)❡✐*♦& q ❡ r /✉❡ &❛)✐&❢❛③❡♠ ❛& ❝♦♥❞✐AB❡&✿ a= bq + r,♦♥❞❡ 0 ≤ r < b.
❖& ✐♥)❡✐*♦& q ❡ r &;♦ ❞❡&✐❣♥❛❞♦& ♣♦* /✉♦❝✐❡♥)❡ ❡ *❡&)♦✱ *❡&♣❡)✐✈❛♠❡♥)❡✳ ❚❡♦4❡♠❛ ✷✳✻✳ ❚❡♦4❡♠❛ ❞❛ ❉✐✈✐,&♦ ▼♦❞✐✜❝❛❞♦ ❡♠ Z✳
❙❡❥❛♠ a ❡ b ∈ Z✱ ♦♥❞❡ b > 0✳
❊♥);♦✱ ❡①✐&)❡♠ ❞♦✐& ♥6♠❡*♦& ✐♥)❡✐*♦& q ❡ r /✉❡ &❛)✐&❢❛③❡♠ ❛& ❝♦♥❞✐AB❡&✿ a= bq + r,♦♥❞❡ r ≤ b
2.
❖& ✐♥)❡✐*♦& q ❡ r &;♦ ❞❡&✐❣♥❛❞♦& ♣♦* /✉♦❝✐❡♥)❡ ❡ *❡&)♦✱ *❡&♣❡)✐✈❛♠❡♥)❡✳ ✽
!❡❧✐♠✐♥❛!❡( ✷✳✷ ◆♦-.❡( (♦❜!❡ ◆0♠❡!♦( ■♥2❡✐!♦( ◆♦"❡✲%❡ &✉❡ ♥)♦ * ❡①✐❣✐❞❛ ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ &✉♦❝✐❡♥"❡ ❡ ❞♦ 1❡%"♦✱ ♣♦❞❡♥❞♦ ❡%"❡ 4❧"✐♠♦ %❡1 ♥❡❣❛"✐✈♦✳ ❖❜"❡$✈❛'(♦ ✷✳✼✳ ◆♦ "❡♦$❡♠❛ ❞❛ ❞✐✈✐*+♦ ♦ $❡*"♦ , ♥+♦ ♥❡❣❛"✐✈♦ ❡ ❧✐♠✐"❛❞♦ ❛❝✐♠❛ ♣♦$ b ❡ ♣❛$❛ ♦ ✈❛❧♦$ ❞❡ q "♦♠❛♠♦* *❡♠♣$❡ ♦ ♠❛✐♦$ ✐♥"❡✐$♦ ♠❡♥♦$ ♦✉ ✐❣✉❛❧ ❛ a b✱ ✐*"♦ ,✱ q = a b ✳ ◆♦ "❡♦$❡♠❛ ❞❛ ❞✐✈✐*+♦ ♠♦❞✐✜❝❛❞♦ ♦ $❡*"♦ ♣♦❞❡ *❡$ ♥❡❣❛"✐✈♦ ❡ "♦♠❛♠♦* ♣❛$❛ q ♦ ✐♥"❡✐$♦ ♠❛✐* ♣$6①✐♠♦ ❞❡ a b✳ 8♦$ ✈❡③❡* a ♣♦❞❡ ❡*"❛$ ❡♥"$❡ ❞♦✐* ♠:❧"✐♣❧♦* ❞❡ b✱ ♦ ;✉❡ ❢❛③ ❝♦♠ ;✉❡ ♦ ;✉♦❝✐❡♥"❡ ❡ ♦ $❡*"♦ ♥+♦ *❡❥❛♠ :♥✐❝♦*✳ 8♦$ ❡①❡♠♣❧♦✱ ❝♦♥*✐❞❡$❡♠♦* a = 21 ❡ b = 6✱ ❡♥"+♦ a ❡*"> ❡♥"$❡ 3b = 18 ❡ 4b = 24✱ ❞❡ ♠♦❞♦ ;✉❡ ♣♦❞❡♠♦* ❡*❝$❡✈❡$ 21 = 6 × 3 + 3 ou 21 = 6 × 4 − 3 "❡♥❞♦ ❛**✐♠ ❞✉❛* ❡*❝♦❧❤❛* ♣❛$❛ ♦ $❡*"♦✿ 3 ❡ −3✳ ❆❧❣♦$✐1♠♦ ❞❡ ❊✉❝❧✐❞❡" ❙❡❥❛♠ a ❡ b ❞♦✐% ✐♥"❡✐1♦% ♣♦%✐"✐✈♦%✳ ;❡❧♦ ❛❧❣♦1✐"♠♦ ❞❛ ❞✐✈✐%)♦✱ ❡①✐%"❡♠ ❞♦✐% ✐♥"❡✐1♦% q0 ❡ r0"❛✐% &✉❡ a = q0b+r0✱ ♦♥❞❡ 0 ≤ r0 < b✳ ❙❡ r0 6= 0 ♣♦❞❡♠♦% ✉"✐❧✐③❛1 ♦ ❛❧❣♦1✐"♠♦ ❞❛ ❞✐✈✐%)♦ ♣❛1❛ ♦% ✐♥"❡✐1♦% b ❡ r0✳ ❊♥")♦✱ ❡①✐%"❡♠ q1 ❡ r1 "❛✐% &✉❡ b = q1r0+ r1✱ ♦♥❞❡ 0 ≤ r1 < r0✳ ;1♦❝❡❞❡♥❞♦ ❞❡%"❛ ❢♦1♠❛ ♦❜"❡♠♦% ✉♠❛ %❡&✉@♥❝✐❛ ❞❡ ✐♥"❡✐1♦% ♥)♦ ♥❡❣❛"✐✈♦% r0, r1, ...rk "❛✐% &✉❡ r0 > r1 > ... > rk>0✳ ❊%"❡ ♣1♦❝❡%%♦✱ ❞❡%✐❣♥❛❞♦ ♣♦1 ❛❧❣♦$✐1♠♦ ❞❡ ❊✉❝❧✐❞❡"✱ "❡1♠✐♥❛ ❛♦ ✜♠ ❞❡ ✉♠ ♥4♠❡1♦ ✜♥✐"♦ ❞❡ ♣❛%%♦% ❡ ♦ 4❧"✐♠♦ 1❡%"♦✱ &✉❡ ❞❡♥♦"❛♠♦% ♣♦1 rk+1✱ * ♥✉❧♦✳ ❚❡♦$❡♠❛ ✷✳✽✳ ❚❡♦$❡♠❛ ❞♦ ❆❧❣♦$✐1♠♦ ❞❡ ❊✉❝❧✐❞❡"✳ ❖ :❧"✐♠♦ $❡*"♦ ♥+♦ ♥✉❧♦ ♦❜"✐❞♦ ♣❡❧♦ ❛❧❣♦$✐"♠♦ ❞❡ ❊✉❝❧✐❞❡* , rk✱ ;✉❡ , ✐❣✉❛❧ ❛♦ ♠>①✐♠♦ ❞✐✈✐*♦$ ❝♦♠✉♠ ❞❡ a ❡ b✱ ✐*"♦ ,✱ rk = (a, b)✳ ❆ ❞❡♠♦♥%"1❛C)♦ ❞❡%"❡ "❡♦1❡♠❛ ♣♦❞❡ %❡1 ❡♥❝♦♥"1❛❞❛ ❡♠ ❬✶❪✳ ❚❡♦$❡♠❛ ✷✳✾✳ ❚❡♦$❡♠❛ ❞❡ ❇;③♦✉1✳ ❙❡❥❛♠ a, b ❡ d ∈ Z✱ ♥+♦ ♥✉❧♦*✳ ❙❡ d = (a, b)✱ ❡♥"+♦ ❡①✐*"❡♠ ✐♥"❡✐$♦* x ❡ y "❛✐* ;✉❡ d = ax + by✳ ✾
✷✳✷ ◆♦$%❡' '♦❜)❡ ◆*♠❡)♦' ■♥.❡✐)♦' 0)❡❧✐♠✐♥❛)❡' ❖! "❡!✉❧&❛❞♦! *✉❡ !❡ !❡❣✉❡♠ !-♦ ✐♠♣♦"&❛♥&❡! ♣❛"❛ ♦ ❝❛♣2&✉❧♦ ✹✳
❊!&❡! ♣"✐♠❡✐"♦! "❡!✉❧&❛❞♦! !-♦✱ ❡!!❡♥❝✐❛❧♠❡♥&❡✱ ♣❛"❛ ❛♣♦✐♦ 7 !❡❝8-♦ ✹✳✶✳ ❚❡♦#❡♠❛ ✷✳✶✵✳ ❙❡❥❛♠ p ✉♠ ♣'✐♠♦ ❡♠ Z ❡ a ❡ b ✐♥+❡✐'♦,✳ ❙❡ p | ab✱ ❡♥+/♦ p | a ♦✉ p | b✳ ❚❡♦#❡♠❛ ✷✳✶✶✳ ❙❡❥❛♠ p ✉♠ ♣'✐♠♦ ❡♠ Z ❡ a1, a2, ..., ak ✐♥+❡✐'♦,✳ ❙❡ p | a1a2...ak✱ ❡♥+/♦ p | aj✱ ♣❛'❛ ❛❧❣✉♠ 1 ≤ j ≤ k✳ ❚❡♦#❡♠❛ ✷✳✶✷✳ ❚❡♦#❡♠❛ ❋✉♥❞❛♠❡♥.❛❧ ❞❛ ❆#✐.♠2.✐❝❛✳ ❙❡❥❛ n ∈ Z✱ ♦♥❞❡ n > 1✱ +❛❧ 3✉❡ n= p1p2...pr = q1q2...qs ♦♥❞❡ p1, p2, ..., pr, q1, q2, ..., qs ,/♦ ♣'✐♠♦,✳ ❊♥+/♦✱ r = s ❡ ❛, ❞✉❛, ❢❛❝+♦'✐③❛89❡, ❞❡ n ,/♦ ✐❣✉❛✐,✱ ❝♦♠ ❛ ♣♦,,:✈❡❧ ❡①❝❡8/♦ ❞❛ ♦'❞❡♠ ❞♦, ❢❛❝+♦'❡,✳ ❚❡♦#❡♠❛ ✷✳✶✸✳ ❙❡❥❛♠ a, b ❡ c ✐♥+❡✐'♦, ♣♦,✐+✐✈♦,✱ ♦♥❞❡ (a, b) = 1✳ ❙❡ ab = cn✱ ♦♥❞❡ n = ✉♠ ✐♥+❡✐'♦ ♣♦,✐+✐✈♦✱ ❡♥+/♦ ❡①✐,+❡♠ ✐♥+❡✐'♦, ♣♦,✐+✐✈♦, d ❡ f +❛✐, 3✉❡ a = dn ❡ b = fn✳
❖! !❡❣✉✐♥&❡! "❡!✉❧&❛❞♦! !-♦✱ ❡!!❡♥❝✐❛❧♠❡♥&❡✱ ♣❛"❛ ❛♣♦✐♦ 7 !❡❝8-♦ ✹✳✷✳ ❉❡✜♥✐78♦ ✷✳✶✹✳ ❈♦♥❣#✉<♥❝✐❛✳ ❙❡❥❛♠ a ❡ b ✐♥+❡✐'♦, ❡ n ✉♠ ✐♥+❡✐'♦ ♣♦,✐+✐✈♦✳ ❙❡ n | a − b ✱ ❞✐③❡♠♦, 3✉❡ a = ❝♦♥❣#✉❡♥.❡ ❝♦♠ b ♠=❞✉❧♦ n ❡ ❞❡♥♦+❛♠♦, ♣♦' a ≡ b mod n✳ >♦' ❞❡✜♥✐8/♦ ❞❡ ❞✐✈✐,✐❜✐❧✐❞❛❞❡✱ ,❡ a ≡ b mod n✱ ❡♥+/♦ ❡①✐,+❡ ✉♠ ✐♥+❡✐'♦ k +❛❧ 3✉❡ a = b + kn✳ >#♦♣♦@✐78♦ ✷✳✶✺✳ ❙❡❥❛♠ m ❡ n ✐♥+❡✐'♦, ♣♦,✐+✐✈♦, ❡ a, b, c ❡ d ∈ Z✳ ❊♥+/♦✿ • a ≡ b mod m ,❡ ❡ ,B ,❡ a + c ≡ b + c mod m
• ,❡ a ≡ b mod m ❡ c ≡ d mod m✱ ❡♥+/♦ a + c ≡ b + d mod m ❡ ac≡ bd mod m
• ,❡ n | m ❡ a ≡ b mod m✱ ❡♥+/♦ a ≡ b mod n • ,❡ ab ≡ ac mod n ❡ (a, n) = d✱ ❡♥+/♦ b ≡ c mod nd
!❡❧✐♠✐♥❛!❡( ✷✳✷ ◆♦-.❡( (♦❜!❡ ◆0♠❡!♦( ■♥2❡✐!♦( !♦♣♦$✐&'♦ ✷✳✶✻✳ ❙❡❥❛ n ✉♠ &✉❛❞(❛❞♦✳ ❊♥-.♦✱ n ≡ 0 mod 4 ♦✉ n ≡ 1 mod 4✳ !♦♣♦$✐&'♦ ✷✳✶✼✳ ❙❡❥❛♠ a ❡ b ❞♦✐1 ✐♥-❡✐(♦1✳ ❙❡ a2 + b2 ≡ 0 mod 2✱ ❡♥-.♦ a ≡ b mod 2✱ ✐1-♦ 2✱ 1.♦ ❛♠❜♦1 ♣❛(❡1 ♦✉ ❛♠❜♦1 5♠♣❛(❡1✳ ❉❡♠♦♥1-(❛7.♦✳ ❙❡ a2+ b2 ≡ 0 mod 2✱ ❡♥$%♦ 2 | a2 + b2✳ ■)$♦ ❡*✉✐✈❛❧❡ ❛ ❞✐③❡2 *✉❡ a2+ b2 3 ♣❛2✳ ❈♦♠♦✱ ❛♣❡♥❛)✱ ❛ )♦♠❛ ❞❡ ❞♦✐) ♥7♠❡2♦) ♣❛2❡) ♦✉ ❞❡ ❞♦✐) ♥7♠❡2♦) 8♠♣❛2❡) 3 ♣❛2✱ $❡♠♦) *✉❡ a2 ❡ b2 ♦✉ )%♦ ❛♠❜♦) ♣❛2❡) ♦✉ ❛♠❜♦) 8♠♣❛2❡)✳ ❈♦♠♦ ♦ *✉❛❞2❛❞♦ ❞❡ ✉♠ ♥7♠❡2♦ ♣❛2 3 ♣❛2 ❡ ♦ *✉❛❞2❛❞♦ ❞❡ ✉♠ ♥7♠❡2♦ 8♠♣❛2 3 8♠♣❛2✱ $❡♠♦) *✉❡ a ❡ b )%♦ )✐♠✉❧$❛♥❡❛♠❡♥$❡ ♣❛2❡) ♦✉ )✐♠✉❧$❛♥❡❛♠❡♥$❡ 8♠♣❛2❡)✳ ❚❡♦!❡♠❛ ✷✳✶✽✳ ❙❡ n ≡ 3 mod 4✱ ❡♥-.♦ n ♥.♦ 2 ❛ 1♦♠❛ ❞❡ ❞♦✐1 &✉❛❞(❛❞♦1✳ ❚❡♦!❡♠❛ ✷✳✶✾✳ ❚❡♦!❡♠❛ ❞❡ ❋❡!♠❛5✳ ❯♠ ♥9♠❡(♦ ♣(✐♠♦ p 2 ❛ 1♦♠❛ ❞❡ ❞♦✐1 &✉❛❞(❛❞♦1 1❡ ❡ 1: 1❡ p = 2 ♦✉ p ≡ 1 mod 4✳ !♦♣♦$✐&'♦ ✷✳✷✵✳ ❙❡ ❞♦✐1 ♥9♠❡(♦1 ♥❛-✉(❛✐1 1.♦ 1♦♠❛1 ❞❡ ❞♦✐1 &✉❛❞(❛❞♦1✱ ❡♥-.♦ ♦ 1❡✉ ♣(♦❞✉-♦✱ -❛♠❜2♠✱ ♦ 2✳ ❚❡♦!❡♠❛ ✷✳✷✶✳ ❋❡!♠❛5✲❊✉❧❡!✳ ❙❡❥❛ n ✉♠ ✐♥-❡✐(♦ ♣♦1✐-✐✈♦✳ ❊♥-.♦✱ n 2 ✉♠❛ 1♦♠❛ ❞❡ ❞♦✐1 &✉❛❞(❛❞♦1 1❡ ❡ 1: 1❡ -♦❞♦ ♦ ❞✐✈✐1♦( ♣(✐♠♦ ❞❡ n✱ &✉❡ 2 ❝♦♥❣(✉❡♥-❡ ❝♦♠ 3 ♠:❞✉❧♦ 4✱ ❛♣❛(❡❝❡ ✉♠ ♥9♠❡(♦ ♣❛( ❞❡ ✈❡③❡1 ♥❛ ❢❛❝-♦(✐③❛7.♦ ❡♠ ♣(✐♠♦1 ❞❡ n✳ ❚❡♦!❡♠❛ ✷✳✷✷✳ ❲✐❧$♦♥✳ ❙❡ p 2 ♣(✐♠♦✱ ❡♥-.♦ (p − 1)! ≡ −1 mod p✳ ❖) 2❡)✉❧$❛❞♦) *✉❡ )❡ )❡❣✉❡♠ )%♦✱ ❡))❡♥❝✐❛❧♠❡♥$❡✱ ♣❛2❛ ❛♣♦✐♦ = )❡❝>%♦ ✹✳✸✳ ❉❡✜♥✐&'♦ ✷✳✷✸✳ ■♥✈❡!5✐❜✐❧✐❞❛❞❡ ♠C❞✉❧♦ m ❙❡❥❛♠ a ❡ m ∈ Z✱ ♦♥❞❡ m 2 ♥.♦ ♥✉❧♦✳ ❙❡ ❡①✐1-❡ b ∈ Z -❛❧ &✉❡ ab ≡ 1 mod m✱ ❞✐③❡♠♦1 &✉❡ a 2 ✐♥✈❡!5D✈❡❧ ♠C❞✉❧♦ m✳ ❚❛❧ b✱ &✉❛♥❞♦ ❡①✐1-❡✱ 2 ❞❡1✐❣♥❛❞♦ ♣♦( ✐♥✈❡!$♦ ❞❡ a ❡ ❞❡♥♦-❛✲1❡ ♣♦( a−1 mod m✳ !♦♣♦$✐&'♦ ✷✳✷✹✳ ❙❡❥❛♠ a ❡ m ∈ Z✱ ♦♥❞❡ m 2 ♥.♦ ♥✉❧♦✳ ❊♥-.♦✱ a 2 ✐♥✈❡(-5✈❡❧ ♠:❞✉❧♦ m 1❡ ❡ 1: 1❡ (a, m) = 1✳ ❖ ✐♥✈❡(1♦ 2 9♥✐❝♦ ♠:❞✉❧♦ m✳ ✶✶
✷✳✷ ◆♦$%❡' '♦❜)❡ ◆*♠❡)♦' ■♥.❡✐)♦' 0)❡❧✐♠✐♥❛)❡' ❉❡✜♥✐%&♦ ✷✳✷✺✳ ❘❡,-❞✉♦ ♠1❞✉❧♦ m✳ ❙❡❥❛♠ a, b ❡ m ∈ Z✱ ♦♥❞❡ m 6= 0✱ )❛✐+ ,✉❡ a ≡ b mod m✳ ❉✐③❡♠♦+ ,✉❡ b 1 ✉♠ 3❡,-❞✉♦ ❞❡ a ♠1❞✉❧♦ m ♦✉ a 1 ✉♠ 2❡+3❞✉♦ ❞❡ b ♠4❞✉❧♦ m✳ ❉❡✜♥✐%&♦ ✷✳✷✻✳ ❉❛❞♦ ✉♠ ✐♥)❡✐2♦ n > 1✱ ❞❡♥♦)❛♠♦+ ♦ ❝♦♥❥✉♥)♦ {0, 1, ..., n − 1} ❞❡ 2❡+3❞✉♦+ ♠4❞✉❧♦ n✱ ✐+)♦ 1✱ ♦ ❝♦♥❥✉♥)♦ ❞♦+ 2❡+)♦+ ♣♦++3✈❡✐+ ❞❡ ❞✐✈✐+9❡+ ❞❡ ♥:♠❡2♦+ ✐♥)❡✐2♦+ ♣♦2 n✱ ♣♦2 Zn✳ ❈♦♠♦ )♦❞♦ ♦ ✐♥)❡✐2♦ ♣2♦❞✉③ ✉♠ 2❡+)♦ ❛♦ +❡2 ❞✐✈✐❞✐❞♦ ♣♦2 n✱ Zn )❡♠ ✉♠ 2❡♣2❡+❡♥)❛♥)❡ ♣❛2❛ ❝❛❞❛ ✐♥)❡✐2♦✳ ❉❡♥♦)❛♠♦+ ♣♦2 Z× n ♦ ❝♦♥❥✉♥)♦ ❞♦+ ✐♥)❡✐2♦+ ♣♦+✐)✐✈♦+ ♠❡♥♦2❡+ ♦✉ ✐❣✉❛✐+ ❛ n✱ ,✉❡ +=♦ ♣2✐♠♦+ ❝♦♠ n✳ ❉❡✜♥✐♠♦+ Z× 1 = {1}✳ ❚❡♦3❡♠❛ ✷✳✷✼✳ 8❡9✉❡♥♦ ❚❡♦3❡♠❛ ❞❡ ❋❡3♠❛;✳ ❙❡ p 1 ♣2✐♠♦✱ ❡♥)=♦ ap ≡ a mod p✱ ♣❛2❛ ,✉❛❧,✉❡2 ✐♥)❡✐2♦ a✳ ■!"♦ $✱ !❡❥❛♠ a ❡ p ∈ Z✱ ♦♥❞❡ p $ ♣-✐♠♦✱ !❡ a 6≡ 0 mod p✱ ❡♥"/♦ ap−1≡ 1 mod p✳ ❉❡✜♥✐%&♦ ✷✳✷✽✳ ❋✉♥%&♦ ❞❡ ❊✉❧❡3✳ ❙❡❥❛ n ∈ Z✱ ♦♥❞❡ n ≥ 1✳ ❖ ♥:♠❡2♦ ❞❡ ✐♥)❡✐2♦+ ♣♦+✐)✐✈♦+ ♠❡♥♦2❡+ ♦✉ ✐❣✉❛✐+ ❛ n ,✉❡ +=♦ ♣2✐♠♦+ ❝♦♠ n 1 ❞❡♥♦)❛❞♦ ♣♦2 φ(n)✱ ♦✉ +❡❥❛✱ #Z× n = φ(n)✳ ❊+)❛ ❢✉♥B=♦ ❞❡ n 1 ❝❤❛♠❛❞❛ ❢✉♥%&♦ ❞❡ ❊✉❧❡3✳ ❙❡ n = 1✱ φ(1) = 1✳ ❙❡ a ≡ b mod n ❡ a $ ♣-✐♠♦ ❝♦♠ n ❡♥"/♦✱ b✱ "❛♠❜$♠✱ $ ♣-✐♠♦ ❝♦♠ n✳ ❙❡ p $ ♣-✐♠♦✱ ❡♥"/♦ 4✉❛❧4✉❡- ✐♥"❡✐-♦ ♣♦!✐"✐✈♦ ♠❡♥♦- 4✉❡ p $ ♣-✐♠♦ ❝♦♠ p✱ ♣♦-"❛♥"♦✱ φ(p) = p − 1✳ ❚❡♦3❡♠❛ ✷✳✷✾✳ ❚❡♦3❡♠❛ ❞❡ ❊✉❧❡3✳ ❙❡❥❛♠ a ❡ n ✐♥)❡✐2♦+✳ ❙❡ (a, n) = 1✱ ❡♥)=♦ aφ(n) ≡ 1 mod n✳ ❉❡✜♥✐%&♦ ✷✳✸✵✳ ❋✉♥%&♦ ♠✉❧;✐♣❧✐❝❛;✐✈❛✳ ❙❡ ❛ ❢✉♥B=♦ f(n) ❡+)D ❞❡✜♥✐❞❛ ♣❛2❛ )♦❞♦+ ♦+ ✐♥)❡✐2♦+ ♣♦+✐)✐✈♦+✱ ❞✐③❡♠♦+ ,✉❡ f(n) 1 ♠✉❧;✐♣❧✐❝❛;✐✈❛✱ ✐+)♦ 1✱ ♣❛2❛ ,✉❛❧,✉❡2 ♣❛2 ❞❡ ✐♥)❡✐2♦+ ♣♦+✐)✐✈♦+ m ❡ n✱ )❛✐+ ,✉❡ (m, n) = 1✱ )❡♠✲+❡ ,✉❡ f(mn) = f (m)f (n) ❚❡♦3❡♠❛ ✷✳✸✶✳ ❆ ❢✉♥B=♦ φ(n) 1 ♠✉❧)✐♣❧✐❝❛)✐✈❛✳ ✶✷
!❡❧✐♠✐♥❛!❡( ✷✳✷ ◆♦-.❡( (♦❜!❡ ◆0♠❡!♦( ■♥2❡✐!♦( ❖! "❡!✉❧&❛❞♦! !❡❣✉✐♥&❡!✱ !♦❜"❡ "❡❝✐♣"♦❝✐❞❛❞❡ 1✉❛❞"2&✐❝❛✱ !❡"3♦ ♥❡❝❡!!2"✐♦! ♣❛"❛ ❛ !❡❝43♦ ✺✳✹✳ ❉❡✜♥✐%&♦ ✷✳✸✷✳ ❘❡,-❞✉♦ ◗✉❛❞234✐❝♦✳ ❙❡❥❛♠ p ✉♠ ♣'✐♠♦ ✐♥+❡✐'♦ ,♠♣❛' ❡ a ✉♠ ✐♥+❡✐'♦✳ ❙❡ ❡①✐/+❡ ✉♠ ✐♥+❡✐'♦ x 0✉❡ ✈❡'✐✜❝❛ ❛ ❝♦♥❣'✉5♥❝✐❛ x2 ≡ a mod p✱ ❞✐③❡♠♦/ 0✉❡ a 9 2❡,-❞✉♦ 6✉❛❞234✐❝♦ ❞❡ p✳ ❉❡✜♥✐%&♦ ✷✳✸✸✳ ❙-♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞2❡✳ ❙❡❥❛♠ p ✉♠ ♣'✐♠♦ ✐♥+❡✐'♦ ,♠♣❛' ❡ a ✉♠ ✐♥+❡✐'♦ +❛❧ 0✉❡ p ∤ a✳ ❉❡✜♥✐♠♦/ ♦ ,-♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞2❡ ❞❛ /❡❣✉✐♥+❡ ♠❛♥❡✐'❛✿ a p = ( 1, /❡ a 9 ✉♠ '❡/,❞✉♦ 0✉❛❞'=+✐❝♦ ❞❡ p −1, /❡ a 9 ✉♠ ♥>♦ '❡/,❞✉♦ 0✉❛❞'=+✐❝♦ ❞❡ p ❚❡♦2❡♠❛ ✷✳✸✹✳ ❈2✐4@2✐♦ ❞❡ ❊✉❧❡2✳ ❙❡❥❛♠ p ✉♠ ♣'✐♠♦ ,♠♣❛' ❡ a ✉♠ ✐♥+❡✐'♦ +❛❧ 0✉❡ p ∤ a✳ ❊♥+>♦✱ a p ≡ ap−12 mod p B2♦♣♦,✐%&♦ ✷✳✸✺✳ B2♦♣2✐❡❞❛❞❡, ❞♦ ❙-♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞2❡✳ ❙❡❥❛♠ p ✉♠ ♣'✐♠♦ ,♠♣❛' ❡ a ❡ b ✐♥+❡✐'♦/ +❛✐/ ❞❡ p ∤ a ❡ p ∤ b✳ ❊♥+>♦✱ ✈❡'✐✜❝❛♠✲/❡ ❛/ ♣'♦♣'✐❡❞❛❞❡/ /❡❣✉✐♥+❡/✿ • a p =b p✱ /❡ a ≡ b mod p • 1p = 1 • −1 p = (−1)p−12 • 2p = (−1)p2−18 • ap b p =abp ✶✸
✷✳✷ ◆♦$%❡' '♦❜)❡ ◆*♠❡)♦' ■♥.❡✐)♦' 0)❡❧✐♠✐♥❛)❡' ❆ ❞❡♠♦♥&'(❛*+♦ ❞❛& ♣(♦♣(✐❡❞❛❞❡& ❛♥'❡(✐♦(❡& ♣♦❞❡ &❡( ✈✐&'❛ ❡♠ ❬✶✷❪ ❡ ❬✶❪✳
❚❡♦#❡♠❛ ✷✳✸✻✳ ▲❡✐ ❞❛ ❘❡❝✐♣#♦❝✐❞❛❞❡ ◗✉❛❞#23✐❝❛ ❞❡ ●❛✉55✳ ❙❡❥❛♠ p ❡ q ♣&✐♠♦) *♠♣❛&❡) ❞✐),✐♥,♦)✳
❙❡ p ≡ 1 mod 4 ♦✉ q ≡ 1 mod 4✱ ❡♥,1♦ p 2 ✉♠ &❡)*❞✉♦ 3✉❛❞&4,✐❝♦ ❞❡ q )❡ ❡ )6 )❡ q 2 ✉♠ &❡)*❞✉♦ 3✉❛❞&4,✐❝♦ ❞❡ p✳
❙❡ p ≡ q ≡ 3 mod 4✱ ❡♥,1♦ p 2 ✉♠ &❡)*❞✉♦ 3✉❛❞&4,✐❝♦ ❞❡ q )❡ ❡ )6 )❡ q 2 ✉♠ ♥1♦ &❡)*❞✉♦ 3✉❛❞&4,✐❝♦ ❞❡ p✳ ■),♦ 2✱ p q q p = (−1)p−12 q−12
◆♦ ❧✐✈(♦ ✑❉✐&8✉✐&✐'✐♦♥❡& ❆(✐'❤♠❡'✐❝❛❡✑ ❬✶✶❪✱ ●❛✉&& ❛♣(❡&❡♥'♦✉ &❡'❡ ❞❡♠♦♥&'(❛*>❡& ❞❡&'❡ '❡♦(❡♠❛ ❬✶✸❪ ❡ ❛'@ ❛♦& ❞✐❛& ❞❡ ❤♦❥❡ ❢♦(❛♠ ♣✉❜❧✐❝❛❞❛& ✷✹✻ ❞❡♠♦♥&'(❛*>❡& ❬✷✸❪✳
✸✳ ◆♦$%❡' ❇)'✐❝❛' '♦❜.❡ ■♥1❡✐.♦'
●❛✉''✐❛♥♦'
◆❡"#❡ ❝❛♣'#✉❧♦ ✈❛♠♦" ❡"#✉❞❛. ❛" ♣.♦♣.✐❡❞❛❞❡" ❜1"✐❝❛" ❞♦" ✐♥#❡✐.♦" ❣❛✉""✐❛♥♦"✱ #❡♥❞♦ ❡♠ ❝♦♥#❛ ❛ "✉❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦" ✐♥#❡✐.♦"✳
✸✳✶ ◆$♠❡'♦) ■♥,❡✐'♦) ❈♦♠♣❧❡①♦) ◆♦23❡) ❇5)✐❝❛) )♦❜'❡ ■♥,❡✐'♦) ●❛✉))✐❛♥♦)
✸✳✶ ◆$♠❡'♦) ■♥,❡✐'♦) ❈♦♠♣❧❡①♦)
◆❡"#❛ "❡❝&'♦ ✈❛♠♦" ✈❡+ ❛" ❞❡✜♥✐&0❡" ✐♥✐❝✐❛✐" ♣❛+❛ ♦" ✐♥#❡✐+♦" ❣❛✉""✐❛♥♦" 4✉❡ "❡+'♦ ❛ ❜❛"❡ ❞♦ ❡"#✉❞♦ ❞❡"#❡" ♥6♠❡+♦" ✑❡"♣❡❝✐❛✐"✑ ❝+✐❛❞♦" ♣♦+ ●❛✉""✳ :❛+❛ ♠❡❧❤♦+ ❝♦♠♣+❡❡♥❞❡+ ♦" ✐♥#❡✐+♦" ❞❡ ●❛✉"" ❞❡✈❡♠♦" +❡❧❡♠❜+❛+ 4✉❡ ✉♠ ♥!♠❡$♦ ❝♦♠♣❧❡①♦ = ✉♠ ♥6♠❡+♦ z 4✉❡ ♣♦❞❡ "❡+ ❡"❝+✐#♦ ♥❛ ❢♦+♠❛ z = a + bi✱ ❡♠ 4✉❡ a ❡ b "'♦ ♥6♠❡+♦" +❡❛✐" ❡ i ❞❡♥♦#❛ ❛ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥@+✐❛✳ ❊"#❛ #❡♠ ❛ ♣+♦♣+✐❡❞❛❞❡ i2 = −1✱ "❡♥❞♦ 4✉❡ a = ❞❡"✐❣♥❛❞♦ ♣♦+ ♣❛+#❡ +❡❛❧ ❡ b ♣♦+ ♣❛+#❡ ✐♠❛❣✐♥@+✐❛ ❞❡ z✳ ❖" ♥6♠❡+♦" ❝♦♠♣❧❡①♦" ❢♦+♠❛♠ ✉♠ ❝♦+♣♦ C = {x + yi : x, y ∈ R} ♠✉♥✐❞♦ ❞❡ ❛❞✐&'♦ ❡ ❞❡ ♠✉❧#✐♣❧✐❝❛&'♦✳ ❆ "❡❣✉✐♥#❡ ❞❡✜♥✐&'♦ ❞❡ ✐♥#❡✐+♦" ❣❛✉""✐❛♥♦" = ❛ 4✉❡ ❢♦✐ ❞❛❞❛ ✐♥✐❝✐❛❧♠❡♥#❡ ♣♦+ ●❛✉"" ✭✈❡+ ❬✾❪✮✳ ❉❡✜♥✐-.♦ ✸✳✶✳ ◆!♠❡$♦ ■♥4❡✐$♦ ❈♦♠♣❧❡①♦✳ ❙❡❥❛ a + bi✱ ♦♥❞❡ i2 = −1✳ ❙❡ a ❡ b )*♦ ✐♥,❡✐-♦)✱ ❡♥,*♦ ♦ ♥.♠❡-♦ a + bi ❞❡✈❡ )❡-❞❡)✐❣♥❛❞♦ ♣♦- ♥!♠❡$♦ ✐♥4❡✐$♦ ❝♦♠♣❧❡①♦✳ ❖❜8❡$✈❛-.♦ ✸✳✷✳ ●❛✉)) -❡❢❡-✐✉ ❡),❡) ♥.♠❡-♦) ❝♦♠♦ ✉♠ ❝❛)♦ ❡)♣❡❝✐❛❧ ❞♦) ♥.♠❡-♦) ❝♦♠♣❧❡①♦)✳ ❊),❡) ♥.♠❡-♦) ✜❝❛-❛♠ ❝♦♥❤❡❝✐❞♦) ♣♦- ✐♥4❡✐$♦8 ❞❡ ●❛✉88 ♦✉ ✐♥4❡✐$♦8 ❣❛✉88✐❛♥♦8 ❞❡✈✐❞♦ ❛ ,❡- )✐❞♦ ●❛✉)) ♦ )❡✉ ✧✐♥✈❡♥,♦-✧✭✈❡- ❬✾❪✮✳ ❉❡✜♥✐-.♦ ✸✳✸✳ ❈♦♥❥✉♥4♦ ❞♦8 ■♥4❡✐$♦8 ●❛✉88✐❛♥♦8✳ Z[i] = {a + bi : a, b ∈ Z} B ✉♠ )✉❜❝♦♥❥✉♥,♦ ❞♦) ♥.♠❡-♦) ❝♦♠♣❧❡①♦)✱ ❞❡)✐❣♥❛❞♦ ♣♦-❝♦♥❥✉♥4♦ ❞♦8 ✐♥4❡✐$♦8 ❣❛✉88✐❛♥♦8✳ ❖ ❝♦♥❥✉♥,♦ ❞♦) ✐♥,❡✐-♦) ❣❛✉))✐❛♥♦) ♣♦))✉✐ ❛ ❡),-✉,✉-❛ ❛❧❣B❜-✐❝❛ ❞❡ ✉♠ ❛♥❡❧✳ ❉❡ "❡❣✉✐❞❛✱ ✈❛♠♦" +❡❢❡+✐+ ❛❧❣✉♠❛" ❞❡✜♥✐&0❡" ✐♠♣♦+#❛♥#❡" ♣❛+❛ ✐♥#❡✐+♦" ❣❛✉""✐❛♥♦"✱ ❛♥@❧♦❣❛" K" ❞♦" ♥6♠❡+♦" ❝♦♠♣❧❡①♦" ✭✈❡+ ❬✹❪✮✳ ❉❡✜♥✐-.♦ ✸✳✹✳ ❆❞✐-.♦ ❡♠ Z[i]✳ ❙❡❥❛♠ α = a + bi ❡ β = c + di ∈ Z[i]✱ ❛ )✉❛ ❛❞✐E*♦ B ❞❡✜♥✐❞❛ ♣♦-✿ α+ β = (a + bi) + (c + di) = (a + c) + (b + d)i ❉❡✜♥✐-.♦ ✸✳✺✳ ▼✉❧4✐♣❧✐❝❛-.♦ ❡♠ Z[i]✳❙❡❥❛♠ α = a + bi ❡ β = c + di ∈ Z[i]✱ ♦ )❡✉ ♣-♦❞✉,♦ B ❞❡✜♥✐❞♦ ♣♦-✿ α· β = (a + bi) · (c + di) = (ac − bd) + (ad + bc)i ❉❡♥♦,❛♠♦) ♦ ♣-♦❞✉,♦ ❞❡ α ♣♦- β ❝♦♠♦ α · β ♦✉ αβ✳
◆♦"#❡% ❇'%✐❝❛% %♦❜,❡ ■♥/❡✐,♦% ●❛✉%%✐❛♥♦% ✸✳✶ ◆5♠❡,♦% ■♥/❡✐,♦% ❈♦♠♣❧❡①♦% ❉❡✜♥✐%&♦ ✸✳✻✳ ❈♦♥❥✉❣❛❞♦ ❞❡ ✉♠ ✐♥2❡✐3♦ ❣❛✉44✐❛♥♦✳ ❖ ❝♦♥❥✉❣❛❞♦ ❞❡ ✉♠ ✐♥'❡✐(♦ ❣❛✉,,✐❛♥♦✱ α = a + bi✱ . ♦ ♥/♠❡(♦ (❡♣(❡,❡♥'❛❞♦ ♣❡❧♦ ✐♥'❡✐(♦ ❣❛✉,,✐❛♥♦ α = a − bi✳ 53♦♣♦4✐%&♦ ✸✳✼✳ ❙❡❥❛♠ α ❡ β ∈ Z[i]✳ ❖, ,❡✉, ❝♦♥❥✉❣❛❞♦, '6♠ ❛, ,❡❣✉✐♥'❡, ♣(♦♣(✐❡❞❛❞❡,✿ • α + β = α + β • α − β = α − β • αβ = αβ ❉❡♠♦♥,'(❛9:♦✳ ❈♦♥#✐❞❡'❡♠♦# α = a + bi ❡ β = c + di✱ ♦♥❞❡ a, b, c ❡ d ∈ Z✳ α+ β = α + β ⇔ a + bi + c + di = a + bi + c + di ⇔ a + c + (b + d)i = a − bi + (c − di) ⇔ a + c + (−b − d)i = a + c + (−b − d)i α− β = α − β ⇔ a + bi − (c + di) = a + bi − c + di ⇔ a − c + (b − d)i = a − bi − (c − di) ⇔ a − c + (−b + d)i = a − c + (−b + d)i αβ = αβ
⇔ (a + bi)(c + di) = a + bi c + di ⇔ (ac − bd) + (ad + bc)i = (a − bi)(c − di)
⇔ (ac − bd) + (−ad − bc)i = (ac − bd) + (−ad − bc)i
❉❡✜♥✐%&♦ ✸✳✽✳ ❯♥✐❞❛❞❡ ❞❡ Z[i]✳ ❯♠ ♥/♠❡(♦ α ∈ Z[i] ❞✐③✲,❡ ✐♥✈❡('?✈❡❧ ,❡ ♣♦,,✉✐ ✐♥✈❡(,♦✱ ♦✉ ,❡❥❛✱ ,❡ ❡①✐,'❡ α′ ∈ Z[i] '❛❧ A✉❡ αα′ = 1✳ ❊♠ Z[i]✱ ✉♠ ♥/♠❡(♦ ✐♥✈❡('?✈❡❧ ❞❡,✐❣♥❛✲,❡ ♣♦( ✉♥✐❞❛❞❡✳ ✶✼
✸✳✷ ◆♦%♠❛ ◆♦()❡+ ❇-+✐❝❛+ +♦❜%❡ ■♥3❡✐%♦+ ●❛✉++✐❛♥♦+
✸✳✷ ◆♦%♠❛
❊♠ Z[i] ♣❛$❛ %❡$♠♦( ❛ ♥♦*+♦ ❞♦ %❛♠❛♥❤♦ ❞❡ ✉♠ ♥/♠❡$♦ ❞❡%❡$♠✐♥❛♠♦( ❛ (✉❛ ♥♦$♠❛✱ ❡♥2✉❛♥%♦ 2✉❡ ❡♠ C ❞❡%❡$♠✐♥❛♠♦( ♦ (❡✉ ✈❛❧♦$ ❛❜(♦❧✉%♦✳ ❉❡✜♥✐%&♦ ✸✳✾✳ ◆♦,♠❛ ❡♠ Z[i]✳ ❙❡❥❛ α = a + bi ∈ Z[i]✳ ❆ &✉❛ ♥♦,♠❛ ( N(α) = αα = (a + bi)(a− bi) = a2+ b2 ■(%♦ 8✱ ❝♦♥(✐❞❡$❛♥❞♦ a + bi ✉♠ ♥/♠❡$♦ ❝♦♠♣❧❡①♦✱ ❛ (✉❛ ♥♦$♠❛ 8 ♦ 2✉❛❞$❛❞♦ ❞♦ (❡✉ ✈❛❧♦$ ❛❜(♦❧✉%♦✳ ❊♠ ❧✐♥❣✉❛❣❡♠ (✐♠❜<❧✐❝❛ %❡♠♦( 2✉❡✿ |a + bi| =√a2+ b2 N(a + bi) = a2 + b2 = |a + bi|2❖❜1❡,✈❛%&♦ ✸✳✶✵✳ ❙❡ α = a + bi✱ ♦♥❞❡ a ❡ b ∈ R✱ ✐&.♦ (✱ &❡ α ∈ C✱ ❛ &✉❛ ♥♦/♠❛ .❛♠❜(♠ &❡ ❡&❝/❡✈❡ ❝♦♠♦ N(α) = a2+ b2✳
❊①❡♠♣❧♦ ✸✳✶✶✳ N(3 + 4i) = 32+ 42 = 25✳
❖❜1❡,✈❛%&♦ ✸✳✶✷✳ ❈♦♥&✐❞❡/❡♠♦& α = a + bi ∈ Z[i]✳ ❙❡ b = 0✱ ❡♥.5♦ α = a ❡ ❛ &✉❛ ♥♦/♠❛ ( N(α) = N(a) = a2✳ ❊♠ ♣❛/.✐❝✉❧❛/✱ N(1) = 1✳
❖❜1❡,✈❛%&♦ ✸✳✶✸✳ 9♦/ ❞❡✜♥✐;5♦ ❞❡ ♥♦/♠❛ ❞❡ ✉♠ ✐♥.❡✐/♦ ❣❛✉&&✐❛♥♦ &❛❜❡♠♦& =✉❡ ❛ ♥♦/♠❛ ( ✉♠ ✐♥.❡✐/♦ ♥5♦ ♥❡❣❛.✐✈♦✳ ◆♦ ❡♥.❛♥.♦✱ ♥5♦ ♣♦❞❡♠♦& ❛✜/♠❛/ ♦ /❡❝?♣/♦❝♦✱ ✐&.♦ (✱ =✉❡ ✉♠ ✐♥.❡✐/♦ ♥5♦ ♥❡❣❛.✐✈♦ ( ♥♦/♠❛ ❞❡ ❛❧❣✉♠ ✐♥.❡✐/♦ ❣❛✉&&✐❛♥♦✳ ❉❡ ❢❛.♦✱ ✉♠❛ ♥♦/♠❛ ❡&❝/❡✈❡✲&❡ ❝♦♠♦ &♦♠❛ ❞❡ ❞♦✐& =✉❛❞/❛❞♦&✱ ♠❛& ♥❡♠ .♦❞♦& ♦& ✐♥.❡✐/♦& ♣♦&✐.✐✈♦& &❡ ❡&❝/❡✈❡♠ ❝♦♠♦ .❛❧✳ 9♦/ ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦& ❛✜/♠❛/ =✉❡ ♥❡♥❤✉♠ ✐♥.❡✐/♦ ❣❛✉&&✐❛♥♦ .❡♠ ♥♦/♠❛ 3 ♣♦✐& ❡&.❡ ♥E♠❡/♦ ♥5♦ ( ❛ &♦♠❛ ❞❡ ❞♦✐& =✉❛❞/❛❞♦&✱ ♣❡❧♦ .❡♦/❡♠❛ ✷✳✷✶✳ ▼❛✐& I ❢/❡♥.❡ ✐/❡♠♦& ✈❡/ ♠❛✐& /❡&✉❧.❛❞♦& I❝❡/❝❛ ❞❛ &♦♠❛ ❞❡ ❞♦✐& =✉❛❞/❛❞♦&✳
:,♦♣♦1✐%&♦ ✸✳✶✹✳ ❙❡❥❛♠ α ∈ Z[i] ❡ α ♦ &❡✉ ❝♦♥❥✉❣❛❞♦✳ ❊♥.5♦✱ N(α) = N(α)✳ ❉❡♠♦♥&./❛;5♦✳ ❈♦♥(✐❞❡$❡♠♦( α = a + bi✱ ♦♥❞❡ a ❡ b ∈ Z✳ ❚❡♠♦( 2✉❡ N(α) = N (a − bi) = a2+ (−b)2 = a2+ b2 = N (α) ✶✽
◆♦"#❡% ❇'%✐❝❛% %♦❜,❡ ■♥/❡✐,♦% ●❛✉%%✐❛♥♦% ✸✳✷ ◆♦,♠❛ ❚❡♦#❡♠❛ ✸✳✶✺✳ ❙❡❥❛♠ α ❡ β ∈ Z[i]✳
❊♥()♦✱ ❛ ♥♦,♠❛ (❡♠ ❛ -❡❣✉✐♥(❡ ♣,♦♣,✐❡❞❛❞❡
N(αβ) = N (α)N (β)
❉❡♠♦♥-(,❛4)♦✳ ❈♦♥#✐❞❡'❡♠♦# α = a + bi ❡ β = c + di✱ ♦♥❞❡ a, b, c ❡ d ∈ Z✳ ❊♥,-♦✱ ♦ #❡✉ ♣'♦❞✉,♦ 0 αβ = (ac − bd) + (ad + bc)i✳
❈❛❧❝✉❧❛♥❞♦ N(α)N(β) ❡ N(αβ) ♦❜,❡♠♦#✿
N(α)N (β) = (a2+ b2)(c2+ d2) = (ac)2+ (ad)2 + (bc)2+ (bd)2
N(αβ) = (ac − bd)2+ (ad + bc)2
= (ac)2− 2abcd + (bd)2+ (ad)2+ 2abcd + (bc)2 = (ac)2+ (ad)2+ (bc)2+ (bd)2 ▲♦❣♦✱ N(α)N(β) = N(αβ)✳ ❖❜,❡#✈❛./♦ ✸✳✶✻✳ ❖ ❢❛(♦ ❞❛ ♥♦,♠❛ ✈❡,✐✜❝❛, ♦ (❡♦,❡♠❛ ✸✳✶✺ ♠♦-(,❛✲♥♦- >✉❡ ♦ ♣,♦❞✉(♦ ❞❛ -♦♠❛ ❞❡ ❞♦✐- >✉❛❞,❛❞♦- ? ❛ -♦♠❛ ❞❡ ❞♦✐- >✉❛❞,❛❞♦-✱ ,❡-✉❧(❛❞♦ >✉❡ ❡♠ Z ✭✈❡, ♣,♦♣♦-✐4)♦ ✷✳✷✵✮ ? ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✐❞❡♥(✐❞❛❞❡ ❞❡ ❉✐♦♣❤❛♥(✉- ✭✈❡, ❬✶✾❪✮✳ ❖❜,❡#✈❛./♦ ✸✳✶✼✳ ◗✉❛♥❞♦ α ❡ β -)♦ >✉❛✐->✉❡, ♥J♠❡,♦- ❝♦♠♣❧❡①♦- (❡♠♦- >✉❡ N α β = N(α) N(β) ❊♠ ♣❛,(✐❝✉❧❛,✱ -❡ αβ = 1✱ ❡♥()♦ N(α) = 1 N (β)✳ ◗✉❛♥❞♦ ❛♣❧✐❝❛♠♦# ♥♦'♠❛# ❡♠ ✐♥,❡✐'♦# ❣❛✉##✐❛♥♦# ♣♦❞❡♠♦# '❡❝♦''❡' 9 ✐♥✈❡',✐❜✐❧✐❞❛❞❡ ❡♠ Z✱ ♣♦❞❡♥❞♦✱ ❛##✐♠✱ ❞❡,❡'♠✐♥❛' ♦# ✐♥,❡✐'♦# ❣❛✉##✐❛♥♦# ;✉❡ #-♦ ✐♥✈❡',<✈❡✐#✳ ❈♦#♦❧4#✐♦ ✸✳✶✽✳ ❖- J♥✐❝♦- ✐♥(❡✐,♦- ❣❛✉--✐❛♥♦- >✉❡ -)♦ ✐♥✈❡,(L✈❡✐- -)♦ 1, −1, i ❡ − i ❉❡ ❢❛,♦✱ ♣♦❞❡♠♦# ❞✐③❡' ;✉❡ α 0 ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ Z[i] #❡ ❡ #@ #❡ N(α) = 1✳
❉❡♠♦♥-(,❛4)♦✳ ✭⇐✮ ❋❛❝✐❧♠❡♥,❡ ✈❡'✐✜❝❛♠♦# ;✉❡ 1✱ −1✱ i ❡ −i ,E♠ ✐♥✈❡'#♦# ❡♠ Z[i]✱ ♣♦✐# 1 ❡ −1 #-♦ ♦# #❡✉# ♣'@♣'✐♦# ✐♥✈❡'#♦# ❡ i ❡ −i #-♦ ✐♥✈❡'#♦# ✉♠ ❞♦ ♦✉,'♦✳
✸✳✸ ❉✐✈✐%✐❜✐❧✐❞❛❞❡ ◆♦-.❡% ❇0%✐❝❛% %♦❜2❡ ■♥5❡✐2♦% ●❛✉%%✐❛♥♦% ✭⇒✮ ❘❡❝✐♣'♦❝❛♠❡♥,❡✱ .✉♣♦♥❤❛♠♦. 1✉❡ α ∈ Z[i] 2 ✐♥✈❡',4✈❡❧✳ 7❡❧❛ ❞❡✜♥✐:;♦ ✸✳✽✱ ,❡♠♦. 1✉❡ ❡①✐.,❡ β ∈ Z[i] ,❛❧ 1✉❡ αβ = 1✳ ◗✉❡'❡♠♦. ♠♦.,'❛' 1✉❡ ♦. @♥✐❝♦. ✈❛❧♦'❡. 1✉❡ α ♣♦❞❡ ,♦♠❛' .;♦ 1, −1, i ❡ −i✳ ❆♣❧✐❝❛♥❞♦ ❛ ♥♦'♠❛ ❡♠ ❛♠❜♦. ♦. ❧❛❞♦. ❞❛ ❡1✉❛:;♦ αβ = 1✱ ♦❜,❡♠♦. N(α)N(β) = 1✳ ❈♦♠♦ N(α) ❡ N(β) .;♦ ✐♥,❡✐'♦. ♥;♦ ♥❡❣❛,✐✈♦. ♦❜,❡♠♦. N(α) = 1✳ ❈♦♥.✐❞❡'❛♥❞♦ α = a + bi✱ ♦♥❞❡ a ❡ b ∈ Z✱ ♦❜,❡♠♦. a2 + b2 = 1✳ ❙❡♥❞♦ ❡.,❛ ✉♠❛ ❡1✉❛:;♦ ❡♠ Z2 ♦❜,❡♠♦. ❛. .♦❧✉:F❡. (1, 0), (−1, 0), (0, 1) ❡ (0, −1)✳ ▲♦❣♦✱ α ∈ {1, −1, i, −i}✳ ❉❡✜♥✐%&♦ ✸✳✶✾✳ ◆-♠❡/♦0 ❛00♦❝✐❛❞♦0 ❡♠ Z[i]✳ ❙❡❥❛♠ α ❡ β ∈ Z[i]✳ ❙❡ α ❡ β ❢♦(❡♠ ✐❣✉❛✐, ❛ ♠❡♥♦, ❞❡ ✉♠❛ ✉♥✐❞❛❞❡✱ ✐,0♦ 1✱ α = uβ✱ ♦♥❞❡ u ∈ {−1, 1, i, −i}✱ ❞✐③❡♠♦, 3✉❡ α ❡ β ,4♦ ❛00♦❝✐❛❞♦0✳ ❖❜0❡/✈❛%&♦ ✸✳✷✵✳ ❈❧❛(❛♠❡♥0❡✱ ,❡ α ❡ β ∈ Z[i] ,4♦ ❛,,♦❝✐❛❞♦, ❡♥04♦ N(α) = N(β)✳ ◆♦ ❡♥0❛♥0♦✱ ❞❛❞♦, ρ ❡ µ ∈ Z[i] 0❛✐, 3✉❡ N(ρ) = N(µ)✱ ♥❡♠ ,❡♠♣(❡ 1 ✈❡(❞❛❞❡ 3✉❡ ρ ❡ µ ,4♦ ❛,,♦❝✐❛❞♦,✳ ➱ ♦ ❝❛,♦ ❞❡ ✐♥0❡✐(♦, ❣❛✉,,✐❛♥♦, ❝♦♥❥✉❣❛❞♦,✱ ♣♦✐, ❡♠❜♦(❛ 0❡♥❤❛♠ ❛ ♠❡,♠❛ ♥♦(♠❛✱ ♥4♦ ,4♦ ❛,,♦❝✐❛❞♦,✱ ❝♦♠ ❡①❝❡?4♦ ❞♦, ✐♥0❡✐(♦, ❣❛✉,,✐❛♥♦, ❝♦♠ ♣❛(0❡ (❡❛❧ ❡ ♣❛(0❡ ✐♠❛❣✐♥@(✐❛ ✐❣✉❛✐,✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ ✐♥0❡✐(♦, ❣❛✉,,✐❛♥♦, 3✉❡ ♥4♦ ,4♦ ❝♦♥❥✉❣❛❞♦, ♥❡♠ ❛,,♦❝✐❛❞♦,✱ ♠❛, 0B♠ ✐❣✉❛❧ ♥♦(♠❛ 1 ♦ ❝❛,♦ ❞❡ 7 + 4i ❡ 8 + i✱ ♣♦✐, ❛♠❜♦, 0B♠ ♥♦(♠❛ 65 ❡✱ ♦❜✈✐❛♠❡♥0❡✱ ♥4♦ ,4♦ ❝♦♥❥✉❣❛❞♦, ♥❡♠ ❛,,♦❝✐❛❞♦,✳
✸✳✸ ❉✐✈✐%✐❜✐❧✐❞❛❞❡
❆ ♥♦'♠❛ ❞❡ ✉♠ ✐♥,❡✐'♦ ❣❛✉..✐❛♥♦ ♥;♦ ♥✉❧♦ 2 ✉♠ ✐♥,❡✐'♦ ♣♦.✐,✐✈♦✱ ♣♦' ✐..♦ ♦♣,❛♠♦. ♣♦' ,'❛❜❛❧❤❛' ❝♦♠ ❛ ♥♦'♠❛ ❡♠ ✈❡③ ❞♦ ✈❛❧♦' ❛❜.♦❧✉,♦✱ ♦ 1✉❡ ♥♦. ♣❡'♠✐,✐'I ❝♦♥❝❧✉✐' '❡.✉❧,❛❞♦. ✐♠♣♦',❛♥,❡. .♦❜'❡ ❛ ❞✐✈✐.✐❜✐❧✐❞❛❞❡ ❡♠ Z[i]✱ ❛ ♣❛',✐' ❞❛. ♣'♦♣'✐❡❞❛❞❡. ❞❛ ❞✐✈✐.✐❜✐❧✐❞❛❞❡ ❡♠ Z✳ ❉❡✜♥✐%&♦ ✸✳✷✶✳ ❉✐✈✐0✐❜✐❧✐❞❛❞❡ ❡♠ Z[i]✳ ❙❡❥❛♠ α ❡ β ∈ Z[i]✱ ♦♥❞❡ β 6= 0✳ ❉✐③❡♠♦, 3✉❡ β ❞✐✈✐❞❡ α ❡ ❡,❝(❡✈❡♠♦, β | α ,❡ ❡①✐,0❡ γ ∈ Z[i] 0❛❧ 3✉❡ α = βγ✳ ◆❡,0❡ ❝❛,♦✱ ❞❡,✐❣♥❛♠♦, β ♣♦( ❞✐✈✐0♦/ ♦✉ ❢❛❝;♦/ ❞❡ α✳ </♦♣♦0✐%&♦ ✸✳✷✷✳ ❙❡❥❛♠ α, β ❡ γ ∈ Z[i] ♥4♦ ♥✉❧♦,✳ ❙❡ γ | α ❡ γ | β✱ ❡♥04♦ γ | α + β ❡ γ | α − β✳ ✷✵◆♦"#❡% ❇'%✐❝❛% %♦❜,❡ ■♥/❡✐,♦% ●❛✉%%✐❛♥♦% ✸✳✸ ❉✐✈✐%✐❜✐❧✐❞❛❞❡ ❉❡♠♦♥%&'❛)*♦✳ ❡❧❛ ❞❡✜♥✐()♦ ✸✳✷✶✱ 0❡ γ | α ❡①✐02❡ µ ∈ Z[i] 2❛❧ 3✉❡ α = µγ ❡ 0❡ γ | β ❡①✐02❡ µ′ ∈ Z[i] 2❛❧ 3✉❡ β = µ′ γ✳ ❙❡ ❝♦♥0✐❞❡7❛7♠♦0 ❛ 0♦♠❛ ❞❡ α ❝♦♠ β ♦❜2❡♠♦0✿ α+ β = µγ + µ′ γ = (µ + µ′ )γ ❈♦♠♦✱ ♣❡❧❛ ❞❡✜♥✐()♦ ✸✳✹✱ µ + µ′ ∈ Z[i] ❝♦♥❝❧✉✐♠♦0 3✉❡ γ | α + β✳ ❈♦♠ ♦ ♠❡0♠♦ 2✐♣♦ ❞❡ 7❛❝✐♦❝>♥✐♦ ♣7♦✈❛♠♦0 3✉❡ γ | α − β✳ ❊①❡♠♣❧♦ ✸✳✷✸✳ ❈♦♥%✐❞❡'❡♠♦% ♦ ✐♥&❡✐'♦ ❣❛✉%%✐❛♥♦ 10 + 5i✳ ❈♦♠♦ 10 + 5i = (3 + 4i)(2 − i) ❞✐③❡♠♦% 2✉❡ 3 + 4i | 10 + 5i✳ ❊①❡♠♣❧♦ ✸✳✷✹✳ 3❛'❛ ✈❡'✐✜❝❛' %❡ 3 + 4i | 10 − 5i ❢❛③❡♠♦% ❛ ❞✐✈✐%*♦✱ '❛❝✐♦♥❛❧✐③❛♥❞♦ ♦ ❞❡♥♦♠✐♥❛❞♦'✳ 10 − 5i 3 + 4i = (10 − 5i)(3 − 4i) (3 + 4i)(3 − 4i) = 10 − 55i 25 = 2 5 − 11 5 i
❊%&❡ ♥;♠❡'♦ ♥*♦ ♣❡'&❡♥❝❡ ❛ Z[i] ♣♦✐% ❛% ♣❛'&❡% '❡❛❧ ❡ ✐♠❛❣✐♥='✐❛ %*♦ 2
5 ❡ − 11 5 ✱ '❡%♣❡&✐✈❛♠❡♥&❡✱ 2✉❡ ♥*♦ %*♦ ♥;♠❡'♦% ✐♥&❡✐'♦%✳ 3♦'&❛♥&♦✱ 3 + 4i ♥*♦ ❞✐✈✐❞❡ 10 − 5i ❡♠ Z[i]✳ ❚❡♦,❡♠❛ ✸✳✷✺✳ ❙❡❥❛ α ∈ Z[i] ♥*♦ ♥✉❧♦✳ ◗✉❛❧2✉❡' ❞✐✈✐%♦' ❞❡ α✱ ❝✉❥❛ ♥♦'♠❛ A 1 ♦✉ N(α)✱ A ✉♠❛ ✉♥✐❞❛❞❡ ♦✉ ✉♠ ❛%%♦❝✐❛❞♦ ❞❡ α✳ ❉❡♠♦♥%&'❛)*♦✳ ❈♦♥0✐❞❡7❡♠♦0 3✉❡ β ∈ Z[i] @ ✉♠ ❞✐✈✐0♦7 ❞❡ α✱ ✐02♦ @✱ β | α✳ ❙❡ N(β) = 1✱ ❡♥2)♦ β = ±1 ♦✉ β = ±i✳ ♦72❛♥2♦✱ β @ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ Z[i]✳ ❙❡ N(β) = N(α)✱ ❝♦♥0✐❞❡7❡♠♦0 γ ∈ Z[i] 2❛❧ 3✉❡ α = βγ✳ ❆♣❧✐❝❛♥❞♦ ❛ ♥♦7♠❛✱ ❡♠ ❛♠❜♦0 ♦0 ♠❡♠❜7♦0 ❞❛ ✐❣✉❛❧❞❛❞❡✱ ♦❜2❡♠♦0 N(α) = N(α)N(γ)✳ ■02♦ @✱ 1 = N(γ)✳ ▲♦❣♦✱ γ = ±1 ♦✉ γ = ±i✳ ♦72❛♥2♦✱ β = ±α ♦✉ β = ±iα✱ ♦✉ 0❡❥❛✱ β @ ❛00♦❝✐❛❞♦ ❞❡ α✳ ❖❜1❡,✈❛34♦ ✸✳✷✻✳ ❖ 2✉❡ ♦ &❡♦'❡♠❛ ✸✳✷✺ ♥♦% ❞✐③ A 2✉❡✱ ♦% ;♥✐❝♦% ✐♥&❡✐'♦% ❣❛✉%%✐❛♥♦% 2✉❡ ❞✐✈✐❞❡♠ α ❡ 2✉❡ &F♠ ♥♦'♠❛ ✐❣✉❛❧ ❛ N(α) %*♦ α, −α, iα ❡ −iα✳
❙❛❧✐❡♥&❛♠♦% 2✉❡ ♦ &❡♦'❡♠❛ ♥*♦ ❞✐③ 2✉❡ ♦% ;♥✐❝♦% ✐♥&❡✐'♦% ❣❛✉%%✐❛♥♦% 2✉❡ &F♠ ♥♦'♠❛ N(α) %*♦ ♦% ❛%%♦❝✐❛❞♦% ❞❡ α ✭✈❡' ♦❜%❡'✈❛)*♦ ✸✳✷✵✮✳
✸✳✸ ❉✐✈✐%✐❜✐❧✐❞❛❞❡ ◆♦-.❡% ❇0%✐❝❛% %♦❜2❡ ■♥5❡✐2♦% ●❛✉%%✐❛♥♦% ❚❡♦#❡♠❛ ✸✳✷✼✳ ❙❡❥❛♠ α = a + bi ∈ Z[i] ❡ c ✉♠ ✐♥(❡✐)♦✳ ❊♥(-♦✱ c | α ❡♠ Z[i] /❡ ❡ /0 /❡ c | a ❡ c | b ❡♠ Z✳ ❉❡♠♦♥/()❛2-♦✳ ✭⇒✮ ❈♦♥%✐❞❡)❡♠♦% +✉❡ c | a + bi ❡♠ Z[i]✳ .❡❧❛ ❞❡✜♥✐23♦ ✸✳✷✶✱ ❡①✐%9❡ m + ni ∈ Z[i] 9❛❧ +✉❡ a + bi = c(m + ni)✳ ❉♦♥❞❡✱ a = cm ❡ b = cn✱ ♦✉ %❡❥❛✱ c | a ❡ c | b ❡♠ Z✳ ✭⇐✮ .❡❧❛ ❞❡✜♥✐23♦ ✷✳✶✱ %❡ c | a ❡♠ Z✱ ❡♥93♦ ❡①✐%9❡ x ∈ Z 9❛❧ +✉❡ a = xc✳ ❙❡ c | b ❡♠ Z✱ ❡♥93♦ ❡①✐%9❡ y ∈ Z 9❛❧ +✉❡ b = yc✳ ▼✉❧9✐♣❧✐❝❛♥❞♦ ❛♠❜♦% ♦% ♠❡♠❜)♦% ♣♦) i ♦❜9❡♠♦% bi = yci✳ .♦)9❛♥9♦✱ a + bi = xc + yci = (x + yi)c✳ ❖✉ %❡❥❛✱ c | a + bi ❡♠ Z[i]✳ ❖❜,❡#✈❛./♦ ✸✳✷✽✳ ❙❡ ❝♦♥/✐❞❡)❛)♠♦/ b = 0✱ ♣❡❧♦ (❡♦)❡♠❛ ✸✳✷✼✱ ❝♦♥❝❧✉✐♠♦/ :✉❡ ❛ ❞✐✈✐/✐❜✐❧✐❞❛❞❡ ❡♥()❡ ✐♥(❡✐)♦/ ❝♦♠✉♥/ ♥-♦ ♠✉❞❛ :✉❛♥❞♦ /❡ ()❛❜❛❧❤❛ ❡♠ Z[i]✱ ✐/(♦ >✱ ♣❛)❛ a ❡ c ∈ Z✱ c | a ❡♠ Z[i] /❡ ❡ /0 /❡ c | a ❡♠ Z✳ ❊♠❜♦)❛ ❡/(❛ ♣)♦♣)✐❡❞❛❞❡ ❞❡ Z /❡ ♠❛♥(❡♥❤❛ ❡♠ Z[i] ♥-♦ :✉❡) ❞✐③❡) :✉❡ ✐//♦ ❛❝♦♥(❡2❛ ❝♦♠ (♦❞❛/ ❛/ ♣)♦♣)✐❡❞❛❞❡/ ❞♦/ ♥@♠❡)♦/ ✐♥(❡✐)♦/✱ (❛❧ ❝♦♠♦ ✐)❡♠♦/ ✈❡) ❛♦ ❧♦♥❣♦ ❞♦ ()❛❜❛❧❤♦✳ ❖ 9❡♦)❡♠❛ %❡❣✉✐♥9❡ ♠♦%9)❛✲♥♦% +✉❡ ❛9)❛✈E% ❞❛ ♠✉❧9✐♣❧✐❝✐❞❛❞❡ ❞❛ ♥♦)♠❛ ♣♦❞❡♠♦% 9)❛♥%❢♦)♠❛) )❡❧❛2G❡% ❞❡ ❞✐✈✐%✐❜✐❧✐❞❛❞❡ ❡♠ Z[i] ❡♠ )❡❧❛2G❡% ❞❡ ❞✐✈✐%✐❜✐❧✐❞❛❞❡ ❡♠ Z✳ ❚❡♦#❡♠❛ ✸✳✷✾✳ ❙❡❥❛♠ α ❡ β ∈ Z[i] ♥-♦ ♥✉❧♦/✳ ❙❡ β | α ❡♠ Z[i]✱ ❡♥(-♦ N(β) | N(α) ❡♠ Z✳ ❉❡♠♦♥/()❛2-♦✳ ❈♦♥%✐❞❡)❡♠♦% α ❡ β ∈ Z[i] 9❛✐% +✉❡ β | α✳ ❊♥93♦✱ ♣❡❧❛ ❞❡✜♥✐23♦ ✸✳✷✶✱ ❡①✐%9❡ γ ∈ Z[i] 9❛❧ +✉❡ α = βγ✳ ❆♣❧✐❝❛♥❞♦ ❛ ♥♦)♠❛ ❛ ❛♠❜♦% ♦% ❧❛❞♦% ❞❛ ❡+✉❛23♦✱ ♦❜9❡♠♦% N(α) = N(β)N(γ)✳ ❊%9❛ ❡+✉❛23♦ ❡%9J ❡♠ Z✱ ❧♦❣♦ ♣❡❧❛ ❞❡✜♥✐23♦ ✷✳✶✱ N(β) | N(α) ❡♠ Z✳ ❖❜,❡#✈❛./♦ ✸✳✸✵✳ ❖ (❡♦)❡♠❛ ✸✳✷✾ ♠♦/()❛ :✉❡✱ ❛ ❞✐✈✐/✐❜✐❧✐❞❛❞❡ ❞❛/ ♥♦)♠❛/ ❡♠ Z )❡/✉❧(❛ ❞❛ ❞✐✈✐/✐❜✐❧✐❞❛❞❡ ❡♠ Z[i]✱ ✐/(♦ >✱ /❡❥❛♠ α ❡ β ∈ Z[i]✱ ❡♥(-♦ β| α ⇒ N(β) | N(α) ❖ )❡❝D♣)♦❝♦ > ❣❡)❛❧♠❡♥(❡ ❢❛❧/♦✳ F♦) ❡①❡♠♣❧♦✱ /❡ ❝♦♥/✐❞❡)❛)♠♦/ α = 10 − 5i ❡ β = 3 + 4i✱ ❞♦♥❞❡ N(α) = 125 = 25 × 5 ❡ N(β) = 25✱ (❡♠♦/ :✉❡ N(β) | N(α) ❡♠ Z✳ ▼❛/✱ ♥♦ ❡①❡♠♣❧♦ ✸✳✷✹✱ ✈❡)✐✜❝K♠♦/ :✉❡ 3 + 4i ♥-♦ ❞✐✈✐❞❡ 10 − 5i✳ F♦)(❛♥(♦✱ N(β) | N(α) 6⇒ β | α ✷✷
◆♦"#❡% ❇'%✐❝❛% %♦❜,❡ ■♥/❡✐,♦% ●❛✉%%✐❛♥♦% ✸✳✸ ❉✐✈✐%✐❜✐❧✐❞❛❞❡ ❈♦♥❝❧✉&♠♦(✱ *❛♠❜-♠✱ .✉❡ ♦ *❡♦0❡♠❛ ✸✳✷✾ ♥♦( ✐♥❞✐❝❛ ✉♠❛ ❢♦0♠❛ (✐♠♣❧❡( ❡ 09♣✐❞❛ ❞❡ ✈❡0✐✜❝❛0 (❡ ✉♠ ✐♥*❡✐0♦ ❣❛✉((✐❛♥♦ ♥=♦ ❞✐✈✐❞❡ ♦✉*0♦✱ ✐(*♦ -✱ (❡ N(β) ∤ N (α) ⇒ β ∤ α >♦0 ❡①❡♠♣❧♦✱ (❡ 2 + 5i ❞✐✈✐❞✐((❡ 11 + 2i ❡♠ Z[i]✱ ❡♥*=♦ ❛( (✉❛( ♥♦0♠❛( *❛♠❜-♠ (❡ ❞✐✈✐❞✐0✐❛♠ ❡♠ Z✳ ◆♦ ❡♥*❛♥*♦✱ N(2 + 5i) = 29✱ N(11 + 2i) = 125 ❡ 29 ♥=♦ ❞✐✈✐❞❡ 125 ❡♠ Z✱ ❧♦❣♦ 2 + 5i ♥=♦ ❞✐✈✐❞❡ 11 + 2i ❡♠ Z[i]✳ >♦0*❛♥*♦✱ ❛ ✈❡0✐✜❝❛A=♦ ❞❛ ❞✐✈✐(✐❜✐❧✐❞❛❞❡ ❞❛ ♥♦0♠❛ ❡♠ Z - ✉♠❛ ❝♦♥❞✐A=♦ ♥❡❝❡((90✐❛✱ ♠❛( ♥=♦ (✉✜❝✐❡♥*❡✱ ♣❛0❛ ❛ ❞✐✈✐(✐❜✐❧✐❞❛❞❡ ❡♠ Z[i]✳ ❯♠ ♠*♦❞♦ ✐♥❢❛❧&✈❡❧ ❞❡ ✈❡0✐✜❝❛0 ❛ ❞✐✈✐(✐❜✐❧✐❞❛❞❡ ❡♥*0❡ ❞♦✐( ✐♥*❡✐0♦( ❣❛✉((✐❛♥♦( -✈❡0✐✜❝❛0 (❡ ♦ (❡✉ .✉♦❝✐❡♥*❡ ♣❡0*❡♥❝❡ ❛ Z[i]✱ ❞❡♣♦✐( ❞❡ 0❛❝✐♦♥❛❧✐③❛0 ♦ (❡✉ ❞❡♥♦♠✐♥❛❞♦0✱ ❝♦♠♦ ✜③-♠♦( ♥♦ ❡①❡♠♣❧♦ ✸✳✷✹✳ ❈♦"♦❧$"✐♦ ✸✳✸✶✳ ❯♠ ✐♥*❡✐0♦ ❣❛✉((✐❛♥♦ *❡♠ ♥♦0♠❛ ♣❛0 (❡ ❡ (E (❡ - ✉♠ ♠F❧*✐♣❧♦ ❞❡ 1 + i✳
❉❡♠♦♥(*0❛A=♦✳ ✭⇐✮ ❙❡❥❛♠ α ❡ β ∈ Z[i] '❛✐) *✉❡ α = β(1+i)✱ ✐)'♦ .✱ α . ✉♠ ♠/❧'✐♣❧♦ ❞❡ 1 + i✳ ❆))✐♠✱ N(α) = N (β(1 + i)) = N (β)N (1 + i), ♣❡❧♦ '❡♦5❡♠❛ ✸✳✶✺ = 2N (β) ▲♦❣♦✱ α '❡♠ ♥♦5♠❛ ♣❛5✳ ✭⇒✮ ❘❡❝✐♣5♦❝❛♠❡♥'❡✱ )✉♣♦♥❤❛♠♦) *✉❡ α = a + bi ∈ Z[i] '❡♠ ♥♦5♠❛ ♣❛5✳
?♦5'❛♥'♦✱ a2+ b2 ≡ 0 mod 2✳ ▲♦❣♦✱ ♣❡❧❛ ♣5♦♣♦)✐@A♦ ✷✳✶✼✱ a ≡ b mod 2✱ ✐)'♦ .✱ a ❡
b 'D♠ ❛ ♠❡)♠❛ ♣❛5✐❞❛❞❡✳ ❈♦♠♦ *✉❡5❡♠♦) ♣5♦✈❛5 *✉❡ 1+i | a+bi✱ ♣♦❞❡♠♦) ❝♦♥)✐❞❡5❛5 *✉❡ ❡①✐)'❡♠ m ❡ n ∈ Q '❛✐) *✉❡ a + bi = (1 + i)(m + ni)✳ ❖ *✉❡ ❡*✉✐✈❛❧❡ ❛ a+ bi = (m − n) + (m + n)i ■)'♦ .✱ a = m − n ❡ b = m + n✳ ❈♦♥)✐❞❡5❛♥❞♦ ❛ )✉❛ )♦♠❛ ❡ ❛ )✉❛ ❞✐❢❡5❡♥@❛ ♦❜'❡♠♦) a+ b = 2m e a − b = −2n ✷✸
