BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC CẦN THƠ PHẠM MỸ HẠNH NHÓM Q P LUẬN VĂN THẠC SĨ TOÁN HỌC Chuyên ngành: Đại số Lý thuyết số Mã số: 60 46 05 NGƯỜI HƯỚNG DẪN: PGS.TS BÙI XUÂN HẢI THÀNH PHỐ CẦN THƠ 11/2009 ✶ ợ t ỗ ố ỵ tt õ tr ởt t r➜t ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ✤➣ ❣➦♣ ❦❤â ❦❤➠♥ ❦❤✐ ự t tr số trữợ ỵ tt õ r t s s ▲❛❣r❛♥❣❡ sû ❞ö♥❣ ♥❤â♠ ❤♦→♥ ✈à ✤➸ t➻♠ ♥❣❤✐➺♠ ✤❛ t❤ù❝✳ ❙❛✉ ✤â tr♦♥❣ ❝→❝ ❜➔✐ ❜→♦✱ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè ❝õ❛ ▲❡♦♥❤❛r❞ ❊✉❧❡r✱ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss✱ ◆✐❡❧s ❍❡♥❞r✐❦ ❆❜❡❧ ✈➔ ❊✈❛r✐st❡ ●❛❧♦✐s✱ ♥❤ú♥❣ t❤✉➟t ♥❣ú tr♦♥❣ ỵ tt õ t r ỵ tt ♥❤â♠ ❝ơ♥❣ ✤÷đ❝ ❤➻♥❤ t❤➔♥❤ tø ❤➻♥❤ ❤å❝ ✈➔♦ ❦❤♦↔♥❣ ỳ t tứ ỵ tt số ố t ỵ tt õ ữủ t❤➔♥❤ ♥❤÷ ♠ët ♥❤→♥❤ ✤ë❝ ❧➟♣ ❝õ❛ ✤↕✐ sè ✭♥❤ú♥❣ ♥❣÷í✐ ❝â ❝ỉ♥❣ tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥➔② ♣❤↔✐ ❦➸ ✤➳♥ ❧➔ ❋❡r❞✐♥❛♥❞ ●❡♦r❣ ❋r♦❜❡♥✐✉s✱ ▲❡♦♣♦❧❞ ❑r♦♥❡❝❦❡r✱ ❊♠✐❧❡ ▼❛t❤✐❡✉ ✈✴✈✳ ✳ ✳ ✮✳ ◆❤✐➲✉ ❦❤→✐ ♥✐➺♠ ❝õ❛ ✤↕✐ sè ✤➣ ✤÷đ❝ ①➙② ❞ü♥❣ ❧↕✐ tø ❦❤→✐ ♥✐➺♠ ♥❤â♠ ✈➔ ✤➣ ❝â ♥❤✐➲✉ ❦➳t q✉↔ ♠ỵ✐ ✤â♥❣ ❣â♣ ❝❤♦ sü ♣❤→t tr✐➸♥ ❝õ❛ ♠ët ♥❣➔♥❤ q✉❛♥ trå♥❣ tr♦♥❣ t♦→♥ ❤å❝✳ ❍✐➺♥ ♥❛② ỵ tt õ ởt t tr tr số õ ự tr t ỵ t❤✉②➳t ❤➔♠✱ ♠➟t ♠➣ ❤å❝✱ ❝ì ❤å❝ ❧÷đ♥❣ tû ✈➔ ♥❤✐➲✉ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❝ì ❜↔♥ ❦❤→❝✳ ❇➔✐ t♦→♥ ❝ì ỵ tt õ t tt ❤➺ t❤è♥❣ ♥❤â♠ ✈ỵ✐ sü ❝❤➼♥❤ ①→❝ ✤➳♥ ♠ët ✤➥♥❣ ❝➜✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ tr➯♥ ❝→❝ ♥❤â♠✳ ❚r➯♥ t❤ü❝ t➳✱ ✈✐➺❝ ❧✐➺t ❦➯ ❤➳t ❝→❝ ❤➺ tố õ ổ t t ỵ t❤✉②➳t ♥❤â♠ ✈➝♥ t✐➳♣ tư❝ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♣❤→t tr✐➸♥✳ ✣➦❝ ❜✐➺t ✤è✐ ✈ỵ✐ ❧ỵ♣ ♥❤â♠ ❛❜❡♥ ✈ỉ ❤↕♥ ✈➝♥ ✤❛♥❣ ❧➔ ❝❤õ ✤➲ ✤❛♥❣ ✤÷đ❝ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐ ♥❣❤✐➯♥ ❝ù✉✳ ❱➻ t❤➳ ❝❤ó♥❣ tỉ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐ ♥❤â♠ Qp ✱ ♠ët ✈➼ ❞ö ✈➲ ♥❤â♠ ❛❜❡♥ ✈æ ❤↕♥ ✤➸ ❤✐➸✉ t❤➯♠ ♠ët sè t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❧ỵ♣ ♥❤â♠ ♥➔②✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❤❛✐ ❝❤÷ì♥❣ s❛✉✿ ❈❤÷ì♥❣ ✶✿ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ♠ët sè ♥❤â♠ q✉❡♥ t❤✉ë❝ ♥❤÷ ♥❤â♠ ①②❝❧✐❝ ✈➔ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣✱ ♥❤â♠ ①♦➢♥ õ ổ õ ữủ õ t tỵ ỡ t tỵ ỡ s ữỡ ữỡ ố ợ ữỡ ❝❤ó♥❣ tỉ✐ ✤➦t trå♥❣ t➙♠ ✈➔♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥❤â♠ Qp ♠ët ✈➼ ❞ö ✈➲ ♥❤â♠ ổ ỗ tớ ợ t sỡ ữủ ❞ö ♠ð rë♥❣ ❝õ❛ ♥❤â♠ ♥➔②✱ ✤â ❧➔ ♥❤â♠ ❆ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ ❆ ❂ ❇ ✰ ❈ ợ B = Qp ì {0} C = {0} × Qq ✈➔ D = 1/r, 1/r ✳ ❉♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ♥➯♥ ❝❤ó♥❣ tỉ✐ ❦❤ỉ♥❣ ✤✐ s➙✉ ❤ì♥ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ♠ð rë♥❣ ❝õ❛ ♥❤â♠ Qp ❝ơ♥❣ ♥❤÷ ❝❤÷❛ tr➻♥❤ ❜➔② ✤÷đ❝ ❝→❝ ù♥❣ ❞ö♥❣ ❝õ❛ ♥❤â♠ ♥➔② tr♦♥❣ ♠ët sè ❧➽♥❤ ✈ü❝ ❦❤♦❛ ❤å❝ ❦ÿ t❤✉➟t ❤✐➺♥ ♥❛②✳ ▼➦❝ ❞ò ❜↔♥ t❤➙♥ ✤➣ õ ố ữ ợ sỹ t ❦✐➳♥ t❤ù❝ ❝â ❤↕♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ✈➝♥ ❝á♥ ♥❤✐➲✉ s❛✐ sât✳ ❊♠ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü t❤ỉ♥❣ ❝↔♠ ✈➔ ỳ ỵ qỵ qỵ t ổ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥✦ ❈➛♥ ❚❤ì✱ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✵✾ P❤↕♠ ▼ÿ ❍↕♥❤ ✷ ▲❮■ ❈❷▼ ❒◆ ▲í✐ ✤➛✉ t✐➯♥ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❚❤➛② ❇ị✐ ❳✉➙♥ ❍↔✐ ✤➣ t➟♥ t➙♠ ❝❤➾ ❞➝♥ tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ q✉❛✱ ✤➦❝ ❜✐➺t tr♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ tr trồ ỷ ỡ qỵ ❚❤➛②✱ ❈ỉ t❤✉ë❝ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥ ❤å❝ ❚r÷í♥❣ ✣↕✐ ỹ P ỗ ũ ợ qỵ ổ ổ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❈➛♥ ❚❤ì ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ỳ tự qỵ tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ❳✐♥ tr➙♥ trå♥❣ ỡ qỵ ổ õ õ ỵ tr trå♥❣ ❝↔♠ ì♥ P❤á♥❣ ✣➔♦ t↕♦ ✈➔ ❑❤♦❛ ❑❤♦❛ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❈➛♥ ❚❤ì ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✤➸ tỉ✐ ❤♦➔♥ t➜t ❝❤÷ì♥❣ tr➻♥❤ ❤å❝ t➟♣✳ ũ tr trồ ỡ ỗ ✈➔ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥ ✈➔ ❤é trđ tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ tỉ✐ t❤❡♦ ❤å❝ ❝❤÷ì♥❣ tr➻♥❤ ❝❛♦ ❤å❝ ð ❚r÷í♥❣ ✣↕✐ ❤å❝ ❈➛♥ ❚❤ì✳ P❤↕♠ ▼ÿ ❍↕♥❤ ▼ö❝ ❧ö❝ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ✶✳✻ ✶✳✼ ✶✳✽ ✶✳✾ ✶✳✶✵ ✶✳✶✶ ✶✳✶✷ ✶✳✶✸ ✶✳✶✹ ◆❤â♠ ❛❜❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ ❛❜❡♥ tü ❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ tü ❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ õ ữợ t s ỗ t tự õ ◆❤â♠ ❦❤æ♥❣ ①♦➢♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉➔♥ ❝→❝ ♥❤â♠ ❝♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ ①②❝❧✐❝ ✲ ◆❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ ❝❤✐❛ ✤÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ ❝♦♥ t❤✉➛♥ tó② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ ✤ì♥ t❤✉➛♥ tó② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➼❝❤ trü❝ t✐➳♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ t❤➦♥❣ ❞÷ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ ❤♦♣❢ ✲ ◆❤â♠ ❝♦❤♦♣❢ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ◆❤â♠ Qp ✈➔ ♠ð rë♥❣ ❝õ❛ ♥❤â♠ ♥➔② ✷✳✶ ✷✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◆❤â♠ Qp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ð rë♥❣ ❝õ❛ ♥❤â♠ Qp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✺ ✺ ✼ ✽ ✶✶ ✶✸ ✶✺ ✶✽ ✷✵ ✷✷ ✷✾ ✸✵ ✸✶ ✸✶ ✸✸ ✸✺ ✸✺ ✹✺ ▼Ư❈ ▲Ư❈ ✹ ❇❷◆● ❑Þ ❍■➏❯ N✱ Z✱ Q✱ R✿ ❚➟♣ ❤ñ♣ sè tü ♥❤✐➯♥✱ sè ♥❣✉②➯♥✱ sè ❤ú✉ t➾✱ sè t❤ü❝✳ H ✂ A✿ ❍ ❧➔ ♥❤â♠ ❝♦♥ t õ Kerf ỗ Imf ỗ G = H ✿ ◆❤â♠ ● ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♥❤â♠ ❍✳ S ✿ ◆❤â♠ s✐♥❤ ❜ð✐ t➟♣ ❙✳ ⑤●⑤✿ ❈➜♣ ❝õ❛ ♥❤â♠ ●✳ H ≤ G✿ ❍ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ ●✳ P✿ ❚➟♣ ❝→❝ sè ♥❣✉②➯♥ tè✳ ❛⑤❜✿ P❤➛♥ tû ữợ t õ ●✳ A × B ✿ ❚➼❝❤ trü❝ t✐➳♣ ❝õ❛ ♥❤â♠ ❆ ✈➔ ❇✳ A ⊕ B ✿ ❚ê♥❣ trü❝ t✐➳♣ ❝õ❛ ♥❤â♠ ❆ ✈➔ ❇✳ G[p]✿ {x ∈ G|pm x = 0, m ∈ N}✳ Z(p∞ )✿ ◆❤â♠ ♣✲Pr✉❢❢❡r✳ Qp ✿ { a a | ∈ Q, n ∈ Z} ợ ởt số tố trữợ p n pn a a Qp ✿ { | ∈ Q, (b, p) = 1, p ∈ P }✳ b b ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ◆❤â♠ ❛❜❡♥ ◆❤â♠ ❛❜❡♥ ❧➔ ❧ỵ♣ ♥❤â♠ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ✈➔ ❦ÿ t❤✉➟t✳ ▼ët sè ♥❤â♠ ❛❜❡♥ tữớ ữ õ số ợ t ❝ë♥❣ ✭Z✱ ✰✮✱ ♥❤â♠ ❝→❝ sè ❤ú✉ t✛ ✈ỵ✐ ♣❤➨♣ t♦→♥ ❝ë♥❣ ✭Q✱ ✰✮✱ ❤❛② ♥❤â♠ ❝→❝ sè ❤ú✉ t✛ ❦❤→❝ ❦❤ỉ♥❣ ✈ỵ✐ ♣❤➨♣ t♦→♥ ♥❤➙♥ ✭Q∗ ✱ ✳ ✮✱ ♥❤â♠ ❝→❝ sè t❤ü❝ ✈ỵ✐ ♣❤➨♣ t♦→♥ ❝ë♥❣ ✭R✱ ✰✮ ❤♦➦❝ ♥❤â♠ ❝ë♥❣ ❝→❝ ♠❛ tr➟♥ ❝➜♣ ♥ ❤➺ sè t❤ü❝ (Mn (R), +) ✈✴✈✳✳✳ ❈→❝ ♥❤â♠ ❛❜❡♥ ❝â t❤➸ ữủ t t ợ ợ tự t ❧ỵ♣ ❝→❝ ♥❤â♠ ①♦➢♥ ✭t♦rs✐♦♥ ❣r♦✉♣✮✱ tù❝ ❧➔ ♠å✐ ♣❤➛♥ tû ❝õ❛ ♥❤â♠ ♥➔② ✤➲✉ ❝â ❝➜♣ ❤ú✉ ❤↕♥✳ ❱➼ ❞ư✿ (Z(2), +) ❤♦➦❝ ♣✲♥❤â♠✳ ▲ỵ♣ t❤ù ❤❛✐ ❧➔ ❧ỵ♣ ❝→❝ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ✭t♦rs✐♦♥ ✲ ❢r❡❡ ❣r♦✉♣s ✮✱ ✤â ❧➔ ♥❤â♠ ♠➔ ❦❤æ♥❣ ❝â ♣❤➛♥ tû ♥➔♦ ❝õ❛ ♥â ✭♥❣♦↕✐ trø ♣❤➛♥ tû ✤ì♥ ✈à✮ ❝â ❝➜♣ ❤ú✉ ❤↕♥✳ ❱➼ ❞ö✿ ◆❤â♠ ❝ë♥❣ ❝→❝ sè ♥❣✉②➯♥ ✭Z✱ ✰ ✮❀ ◆❤â♠ ❝ë♥❣ ❝→❝ sè ❤ú✉ t✛ ✭Q✱ ✰ ✮✳ ❈✉è✐ ũ ợ ỗ õ õ ỳ tû ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❝â ❝➜♣ ❤ú✉ ❤↕♥ ✈➔ ♥❤ú♥❣ ♣❤➛♥ tû ❝â ❝➜♣ ✈æ ❤↕♥✳ ◆❤ú♥❣ ♥❤â♠ ♥➔② ❣å✐ ❧➔ ♥❤â♠ ❛❜❡♥ ❤é♥ ❤đ♣ ✭♠✐①❡❞ ❣r♦✉♣ ✮✳ ❱➼ ❞ư✿ ◆❤â♠ Z2 ⊕ Z✱ ❤♦➦❝ ♥❤â♠ ♥❤➙♥ ❝→❝ sè t❤ü❝ ❦❤→❝ ❦❤æ♥❣ ❞♦ ♠å✐ ♣❤➛♥ tû ❦❤→❝ ♣❤➛♥ tû ✶ ✈➔ ✲✶ ❝õ❛ ♥❤â♠ ♥➔② ✤➲✉ ❝â ❝➜♣ ✈æ ❤↕♥✳ ✣è✐ ✈ỵ✐ ♥❤â♠ t❤✉ë❝ ❧ỵ♣ t❤ù ♥❤➜t ✈➔ ❧ỵ♣ t❤ù ❤❛✐ ❝â ❝→❝ t➼♥❤ ❝❤➜t ❝❤✉♥❣ ♥❤÷ s❛✉✿ ◆❤â♠ ❝♦♥ ①♦➢♥ ❝õ❛ ● ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ♥❤â♠ ♥➔②✳ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ✈✐➺❝ ♣❤➙♥ t➼❝❤ ❧ỵ♣ ❝→❝ õ ộ ủ t ợ õ ữ s ▼ët ♥❤â♠ ❛❜❡♥ ❤é♥ ❤ñ♣ ● ❝â t❤➸ ❝❤➾ ❝â t❤➸ ❝❤✐❛ t❤➔♥❤ ♥❤â♠ t→❝❤ ✤÷đ❝ ✭s♣❧✐tt✐♥❣✮ ❤❛② ♥❤â♠ ❦❤ỉ♥❣ t→❝❤ ✤÷đ❝ ✭♥♦♥s♣❧✐tt✐♥❣ ❣r♦✉♣✮ t✉ý t❤✉ë❝ ✈➔♦ ♥❤â♠ ❝♦♥ ①♦➢♥ ❝õ❛ ● ❝â ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ● ổ ố ợ ợ õ t ữủ ❝❤➫ r❛✮✱ ♥➳✉ t❛ ❜✐➳t t➼♥❤ ❝❤➜t ❝õ❛ t✭❆✮ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❆✴t✭❆✮ t❤➻ s➩ ❜✐➳t ✤÷đ❝ t➼♥❤ ❝❤➜t A = t(A) ì (A/t(A)) ợ t✭❆✮ ❧➔ ♥❤â♠ ❝❤ù❛ t➜t ❝↔ ♣❤➛♥ tû ①♦➢♥ ❝õ❛ ❆✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ❆ ❦❤æ♥❣ ❝â ♥❤✐➲✉ t➼♥❤ ❝❤➜t ❦❤→❝ ❤ì♥ s♦ ✈ỵ✐ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ t✭❆✮ ✈➔ ❆✴t✭❆✮✳ ❚✉② ♥❤✐➯♥✱ ♥➳✉ ♥❤â♠ ❆ ❦❤ỉ♥❣ t→❝❤ ✤÷đ❝ ✭❤❛② t✭❆✮ ❦❤æ♥❣ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ❆✮✱ ❦❤✐ ✤â ♥❤â♠ ❆ s➩ ❝â ♥❤✐➲✉ t➼♥❤ ❝❤➜t ❦❤→❝ ❤ì♥ s♦ ✈ỵ✐ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ t✭❆✮ ✈➔ ❆✴t✭❆✮✳ ❱➻ ✈➟②✱ ✤è✐ ✈ỵ✐ ❜❛ ❧ỵ♣ ❝õ❛ ❝→❝ ♥❤â♠ ❛❜❡♥ ♥➯✉ tr ự ữớ t ú ỵ ♥❤â♠ ①♦➢♥✱ ♥❤â♠ ❦❤ỉ♥❣ ①♦➢♥ ✈➔ ♥❤â♠ ❦❤ỉ♥❣ t→❝❤ ✤÷đ❝✳ ✺ ❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✻ ▼➦t ❦❤→❝✱ ♠é✐ ♥❤â♠ t→❝❤ ✤÷đ❝ ✭❤❛② ♣❤➙♥ t➼❝❤ ✤÷đ❝✮ ❆ ❝â t ữủ ữợ A = t(A) F õ t ỗ ❆ ✈➔ A/t(A) ∼ = F ✳ ❱➻ t❤➳✱ ∼ ❆✴t✭❆✮ ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ tr♦♥❣ ❆✳ ▼➦❝ ❦❤→❝ ❞♦ A/t(A) = F ❞➝♥ ✤➳♥ ❋ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ✈➻ ❆✴t✭❆✮ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ❱➟② ❆ ❝â ♠ët ❤↕♥❣ tû trü❝ t✐➳♣ ❦❤ỉ♥❣ ①♦➢♥ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ◆➳✉ ● ❧➔ ♠ët ♣✲♥❤â♠ ❛❜❡♥ t❤➻ ● ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥❤â♠ ♣✲♥❣✉②➯♥ sì✳ ❈❤♦ ● ởt õ ỳ ợ ộ ữợ tè ♣ ❝õ❛ ⑤●⑤✱ ✤➦t✿ Gp = {x ∈ G|pm x = 0, m ∈ N} ❑❤✐ ✤â✱ Gp ❧➔ ♠ët ♣✲♥❤â♠ ❝♦♥ ❙②❧♦✇ ❝õ❛ ●✳ ❍ì♥ ♥ú❛✱ ♥❤â♠ ❝♦♥ Gp ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ t❤➔♥❤ ♣❤➛♥ ♣ ✲ ♥❣✉②➯♥ ❝õ❛ ●✳ ◆❣♦➔✐ r❛✱ ♠å✐ ♣✲ ♥❤â♠ ❛❜❡♥ ● ❤ú✉ ❤↕♥ ✤÷đ❝ ❣å✐ ❧➔ ♣✲♥❤â♠ ❛❜❡♥ ❝➜♣ ❱➼ ❞ư✿ ◆❤â♠ ❝ë♥❣ Z9 ❝→❝ sè ♥❣✉②➯♥ ♠♦❞✉❧♦ ✾ ❧➔ ♥❤â♠ ♣✲ ♥❣✉②➯♥ ✈➻ ❝➜♣ ❝õ❛ ❝→❝ ♣❤➛♥ tû ❝õ❛ Z9 ❧➔ ❧✉ÿ t❤ø❛ ❝õ❛ sè ♥❣✉②➯♥ tè ✸✳ ▼å✐ ♥❤â♠ ❛❜❡♥ ❤ú✉ ❤↕♥ ✤➲✉ ❧➔ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ t❤➔♥❤ ♣❤➛♥ ♣✲♥❣✉②➯♥ sì✱ tù❝ G = p Gp ỵ ởt ♥❤â♠ ❛❜❡♥ ①♦➢♥ ❜➜t ❦ý ❝â t❤➸ ♣❤➙♥ t➼❝❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t t❤➔♥❤ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ♣ ✲ ♥❤â♠ ❝♦♥ ♥❣✉②➯♥ t❤❡♦ ❝→❝ sè ♥❣✉②➯♥ tè ❦❤→❝ ♥❤❛✉✳ ❈❤ù♥❣ ♠✐♥❤ ●✐↔ sû ❆ ❧➔ ♠ët ♥❤â♠ t ý ỵ t tt ❝→❝ ♣❤➛♥ tû ❝õ❛ ❆ ❝â ❝➜♣ ❧➔ ❧✉ÿ t❤ø❛ ♠ët sè ♥❣✉②➯♥ tè ♣ q✉❛ Ap ✳ ❑❤✐ ✤â t➜t ❝↔ ❝→❝ ♥❤â♠ ❝♦♥ Ap ❝õ❛ ❆ t❤❡♦ ❝→❝ sè ♥❣✉②➯♥ tè ❦❤→❝ ♥❤❛✉ t↕♦ t❤➔♥❤ ♠ët tê♥❣ trü❝ t✐➳♣ tr♦♥❣ ♥❤â♠ ❆✳ ▼➦t ❦❤→❝✱ ♠ët ♣❤➛♥ tû ① tý ỵ õ n = p1 pα2 pαk k ✱ t❤➻ x = x1 x2 xr , xi ∈ Ap ✳ ❉♦ ✤â✱ ♠é✐ ♣❤➛♥ tû ❝õ❛ ❆ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ tê♥❣ trü❝ t✐➳♣ ❝õ❛ t➜t ❝↔ ❝→❝ ♥❤â♠ ♥❣✉②➯♥ tr♦♥❣ ❆ ♥➯♥ A = ⊕p Ap ✳ ❇ê ✤➲ ✶✳✶✳✸✳ ◆➳✉ ● ❧➔ ♥❤â♠ ❛❜❡♥ ✈➔ A ≤ G ❦❤✐ ✤â ❝→❝ ✤✐➲✉ ❦❤➥♥❣ ✤à♥❤ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿ ❛✮ ❆ ❧➔ ❤↕♥❣ tû trỹ t ỗ t õ B G ♠➔ A ∩ B = ✈➔ A + B = G ỗ t õ B G ♠➔ ♠é✐ ♣❤➛♥ tû g ∈ G ❝â ❜✐➸✉ t ữợ ợ a A, b B ỗ t ỗ s : G/A G t ◦ s = 1G/A ✈ỵ✐ ν : G → G/A ỗ tỹ ỗ t π : G → A t❤ä❛ π(a) = a, ∀a ∈ A✳ ✶✳✷✳ ◆❍➶▼ ❆❇❊◆ ❚Ü ❉❖ ❈❤ù♥❣ ♠✐♥❤ ❛✮ s✉② r❛ ❞✮ ❤✐➸♥ ♥❤✐➯♥✳ ❜✮ s✉② r❛ ❛✮ ❤✐➸♥ ♥❤✐➯♥✳ ◆❣÷đ❝ ❧↕✐✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✭❛✮ s✉② r❛ ✭❜✮✳ ◆➳✉ ❆ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ● t t tỗ t õ õ ● ♠➔ A ∩ B = 0, A + B = G s✉② r❛ ✈ỵ✐ ♠å✐ ♠é✐ ♣❤➛♥ tû g G õ ữợ ✰ ❜ ✈ỵ✐ a ∈ A ✈➔ b ∈ B ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ sü ❜✐➸✉ ❞✐➵♥ ♥➔② ❧➔ ❞✉② ♥❤➜t✱ ✈➻ ♥➳✉ ❣ ❂ ❛✬ ✰ ❜✬ ✈ỵ✐ a ∈ A, b ∈ B ✳ ❑❤✐ ✤â✱ a − a = b − b ∈ A ∩ B = 0✳ ❙✉② r❛✱ ❛ ❂ ❛✬ ✈➔ ❜ ❂ ❜✬✳ ❍❛② sü ❜✐➸✉ ❞✐➵♥ ❝õ❛ ❣ ❧➔ ❞✉② ♥❤➜t✳ ❜✮ s✉② r❛ ❞✮✳ ❚❤➟t ✈➟②✱ ♥➳✉ ♠é✐ ♣❤➛♥ tû g G õ t ữợ ❣ ❂ ❛ ✰ ❜ ✈ỵ✐ a ∈ A ✈➔ b ∈ B ✳ ❳➨t →♥❤ ①↕ π : G → A t❤♦↔ π(a) = a, ∀a ∈ A✳ ❑✐➸♠ tr ữủ ởt ỗ s r tỗ t õ B G ♠➔ ♠é✐ ♣❤➛♥ tû g ∈ G ❝â ❜✐➸✉ t ữợ ợ a A, b B t ỗ : G G/A ỗ tỹ ♥❤✐➯♥✱ t❤➻ ∀g ∈ G, ν(g) = g + A = a + b + A = b + A ✈ỵ✐ a ∈ A, b ∈ B ✳ ❙✉② r❛✱ tỗ t ỗ s : G/A G t s✭❣ ✰ ❆✮ ❂ ❣✳ ❑❤✐ ✤â✱ ν ◦ s = 1G/A ✳ ❝✮ s✉② r❛ ❛✮✳ ❚❤➟t ✈➟②✱ ♥➳✉ tỗ t ỗ s : G/A G t ν ◦ s = 1G/A ✈ỵ✐ ν : G → G/A ỗ tỹ t õ ❝♦♥ ❝❤✉➞♥ t➢❝ ❝õ❛ ●✱ ❤ì♥ t❤➳ ❆ ❝á♥ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ●✳ ❞✮ s✉② r❛ ❛✮✳ t ỗ : G G tọ (a) = a, ∀a ∈ A ✈➔ π (x) = ♥➳✉ ① ❦❤æ♥❣ t❤✉ë❝ ❆✳ ❚❛ ❝â✱ G = A ker |A = ỗ t❤í✐ π ❧➔ t♦➔♥ ❝➜✉ ✈➔ kerπ = kerπ ✳ ❱➟② ❆ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ●✳ ✶✳✷ ◆❤â♠ ❛❜❡♥ tü ❞♦ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ▼ët ♥❤â♠ ❛❜❡♥ ❋ ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ ❛❜❡♥ tü ❞♦ ✭❢r❡❡ ❛❜❡❧✐❛♥ ❣r♦✉♣✮ ♥➳✉ ♥â ❧➔ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ♥❤â♠ ổ tỗ t t X F ❝→❝ ♣❤➛♥ tû ❝â ❝➜♣ ✈ỉ ❤↕♥✱ ✤÷đ❝ ❣å✐ ❧➔ ❝ì sð ❝õ❛ ❋ ✈ỵ✐ F = x∈X x ✱ F ∼ Z✳ = ◆➳✉ ❳ ❧➔ ♠ët ❝ì sð ❝õ❛ ♠ët ♥❤â♠ ❛❜❡♥ tü ❞♦ ❋ t❤➻ ✈ỵ✐ ộ u F tỗ t t ởt ❞↕♥❣ ❜✐➸✉ ❞✐➵♥ u = hh mx x ✈ỵ✐ x ∈ X, mx ∈ Z✳ ◆➳✉ X = ∅ t❤➻ F = {0}✱ ❤❛② ♥❤â♠ {0} ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ ❛❜❡♥ tü ❞♦ ✈ỵ✐ ❝ì sð ❧➔ t➟♣ ré♥❣✳ ◆❣♦➔✐ r❛✱ ❝ì sð ❝õ❛ ♠ët ♥❤â♠ ❛❜❡♥ tü ❞♦ ❳ ❧➔ ♠ët t➟♣ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ❱➼ ❞ö✿ ◆❤â♠ ❝ë♥❣ ❝→❝ sè ♥❣✉②➯♥ ✭Z✱ ✰ ✮ ❧➔ ♥❤â♠ ❛❜❡♥ tü ❞♦✳ ◆❤â♠ G = Z ⊕ Z ❧➔ ♥❤â♠ ❛❜❡♥ tü ❞♦✳ G = {(a, b)|a, b ∈ Z}✳ ✼ ❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✽ ◆❤➟♥ ①➨t✿ ▼å✐ ♥❤â♠ ❛❜❡♥ tü ❞♦ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ✈➔ ♠å✐ ♥❤â♠ ❛❜❡♥ ❦❤æ♥❣ ①♦➢♥ ❤ú✉ ❤↕♥ s✐♥❤ ❧➔ ♥❤â♠ ❛❜❡♥ tü ❞♦✳ ◆❤â♠ ✭Q✱ ✰✮ ❦❤æ♥❣ ❧➔ ♥❤â♠ ❛❜❡♥ tü ❞♦ ✈➻ Q ❦❤ỉ♥❣ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♥❤â♠ Z ỵ ởt õ tü ❞♦ ✈ỵ✐ ❝ì sð ❳ ✈➔ ● ❧➔ ♠ët ♥❤â♠ ❜➜t ❦ý✳ ●✐↔ sû f : X → G ởt tũ ỵ õ tỗ t t ởt ỗ : F G ♠ð rë♥❣ ❝õ❛ ❢ t❤♦↔ ϕ(x) = f (x), ∀x ∈ X ✳ ❈❤ù♥❣ ♠✐♥❤ ◆➳✉ u ∈ F ✈➻ ❳ ❧➔ ♠ët ❝ì sð ❝õ❛ ♥❤â♠ ❛❜❡♥ tü ❞♦ ❋ ♥➯♥ ✈ỵ✐ ♠é✐ u ∈ F ❝â ❜✐➸✉ ❞✐➵♥ t ữợ u = mx x tứ õ ϕ : u → mx f (u) ❧➔ ♠ët ❤➔♠ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tèt✳ ❑✐➸♠ tr❛ ✤÷đ❝ ϕ ❧➔ ♠ët ỗ rở t❤➻ ϕ = f ✳ ❚➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ϕ ữủ ỗ tr ũ ♠ët t➟♣ ❝→❝ ♣❤➛♥ tû s✐♥❤ ♣❤↔✐ ❣✐è♥❣ ♥❤❛✉✳ ❉♦ õ tỗ t t ỗ : F → G ❧➔ ♠ð rë♥❣ ❝õ❛ ❤➔♠ f : X → G t❤ä❛ ϕ(x) = f (x), ∀x ∈ X ✳ ❇ê ✤➲ ✶✳✷✳✸✳ ▼é✐ ♥❤â♠ ❛❜❡♥ ● ❧➔ ♥❤â♠ t❤÷ì♥❣ ❝õ❛ ♠ët ♥❤â♠ ❛❜❡♥ tü ❞♦✱ ❤❛② ♠ët ♥❤â♠ t ý ổ ợ õ tữỡ ❝õ❛ ♠ët ♥❤â♠ ❛❜❡♥ tü ❞♦✳ ❈❤ù♥❣ ♠✐♥❤ ●å✐ ❋ ❧➔ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ⑤●⑤ ❧➛♥ t❤➔♥❤ ♣❤➛♥ Z✱ F ∼ = |G| Z ✈➔ ❣å✐ xg ❧➔ ♣❤➛♥ tû s✐♥❤ ❝õ❛ t❤➔♥❤ ♣❤➛♥ t❤ù ❣ ❝õ❛ Z tr♦♥❣ tê♥❣ trü❝ t✐➳♣✱ ✈ỵ✐ g ∈ G✳ ❚❛ ❝â ❋ ❧➔ ♥❤â♠ ❛❜❡♥ tü ❞♦ ✈ỵ✐ ❝ì sð ❧➔ X = {xg |g ∈ G}✳ ✣à♥❤ ♥❣❤➽❛ ❤➔♠ f : X → G ❜ð✐ f (xg ) = g ✈ỵ✐ g G ỵ tỗ t ỗ : F G rở ❝õ❛ ❢✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❢ t❤➻ ❢ ❧➔ ♠ët t♦➔♥ ❝➜✉ ♥➯♥ ϕ ❧➔ t♦➔♥ ❝➜✉ ❞♦ ✤â✱ G = F/ker t ỵ tự t ◆❤➟♥ ①➨t✿ ❱ỵ✐ ❳ ❧➔ ♠ët t➟♣ ❤đ♣ ❜➜t ❦ý trữợ ổ ỹ ữủ ởt õ tỹ ❞♦ ❋ ♥❤➟♥ ❳ ❧➔♠ ❝ì sð✳ ●✐↔ sû ❋ ✈➔ ❋✬ ❧➔ ❤❛✐ ♥❤â♠ ❛❜❡♥ tü ❞♦ ✈ỵ✐ ❝→❝ ❝ì sð t÷ì♥❣ ù♥❣ ❧➔ ❳ ✈➔ ❳✬✳ ❑❤✐ ✤â✱ F ∼ = F ⇔ |X| = |X |✳ ✶✳✸ ◆❤â♠ tü ❞♦ ❈❤♦ ❳ ❧➔ ♠ët t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣✱ ❝❤å♥ ❨ ❧➔ t➟♣ t❤ä❛ ⑤❳⑤ ❂ ⑤❨⑤ ✈➔ X Y = õ tỗ t s →♥❤ ϕ : X −→ Y, ∀x ∈ X ✱ ✤➦t x−1 = ϕ(x)✳ ❚❛ ❝â Y = ϕ(X) = {x−1 |x ∈ X} : X −1 ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ▼é✐ ♣❤➛♥ tû ❝õ❛ X ∪ X −1 ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝❤ú ❝→✐ tr♦♥❣ ❜↔♥❣ ❝❤ú ❝→✐ ❳✳ ▼ët tø tr♦♥❣ ❜↔♥❣ ❝❤ú ❝→✐ ❳ ❧➔ ♠ët ❞➣② ❤ú✉ ❤↕♥ ❝â ❞↕♥❣ w = xε11 xε22 xεnn , xi ∈ X, εi ∈ {1, −1}✳ ✶✳✸✳ ◆❍➶▼ ❚Ü ❉❖ ❚r♦♥❣ t➟♣ ❤ñ♣ ❝→❝ tø t❛ ✤à♥❤ ♥❣❤➽❛ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ w ∼ u ♥➳✉ ✉ ❝â t❤➸ ♥❤➟♥ ✤÷đ❝ tø ✇ q✉❛ ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ ữợ tỹ t ợt tứ ❞↕♥❣ xε x−ε ✈ỵ✐ x ∈ X, ε ∈ {−1, 1}✳ ◗✉❛♥ ❤➺ tr➯♥ ❧➔ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ t tt tứ ợ tữỡ ✤÷ì♥❣ ❝❤ù❛ tø ✇✳ ▼ët tø ✤÷đ❝ ❣å✐ ❧➔ rót ❣å♥ ✤÷đ❝ ♥➳✉ ♥â ❝❤ù❛ tø ❝♦♥ ❞↕♥❣ xε x−ε ✈ỵ✐ x ∈ X, ε ∈ {−1, 1}✳ ❚ø ❦❤ỉ♥❣ rót ❣å♥ ✤÷đ❝ ❣å✐ ❧➔ tø rót ❣å♥✳ ▼å✐ tø tữỡ ữỡ ợ tứ rút [w]r ❧➔ ❞↕♥❣ rót ❣å♥ ❝õ❛ ✇✳ ●å✐ ❋✭❳✮ ❧➔ t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ tø rót ❣å♥ ✈➔ ∅ ❧➔ tø ❦❤æ♥❣ ❝❤ù❛ ❝❤ú ❝→✐ ♥➔♦✱ e = [∅]r ✭♣❤➛♥ tû ✤ì♥ ✈à ❝õ❛ ❋✭❳✮✮✳ −ε2 −εn ◆➳✉ w = xε11 xε22 xεnn , xi ∈ X, εi ∈ {1, −1} t❤➻ w−1 = x−ε x2 xn , xi ∈ −1 −1 X, εi ∈ {1, −1}✳ ❱➟② [w]r = [w ]r ✳ ❑❤✐ ✤â ❋✭❳✮ ❧➔ ♠ët õ ợ t ữủ [w]r [u]r = [wu]r ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✷✳ ❈❤♦ ❳ ❧➔ t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣✱ ❦❤✐ ✤â ♥❤â♠ ❋✭❳✮ ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ tü ❞♦✭❢r❡❡ ❣r♦✉♣✮ ✈ỵ✐ ❝ì sð ❳✳ ❱➼ ❞ư✿ ◆❤â♠ ❝→❝ sè ♥❣✉②➯♥ ✈ỵ✐ ♣❤➨♣ t♦→♥ ❝ë♥❣ ❧➔ ♥❤â♠ tü ❞♦✳ ◆❤➟♥ ①➨t✿ ◆❤â♠ ❛❜❡♥ tü ❞♦ ❦❤æ♥❣ ❤➥♥ ❧➔ ♥❤â♠ tü ❞♦ ♥❣♦↕✐ trø ✷ tr÷í♥❣ ❤đ♣ ❧➔ ♥❤â♠ t➛♠ t❤÷í♥❣ ❤♦➦❝ ❧➔ ♥❤â♠ ①②❝❧✐❝ ✈ỉ ❤↕♥✳ ❈→❝ ♥❤â♠ ❛❜❡♥ ❦❤→❝ ❦❤æ♥❣ ♣❤↔✐ ♥❤â♠ tü ❞♦ ✈➻ ♥❤â♠ tü ❞♦ ❛❜ ❦❤→❝ ✈ỵ✐ ♥❤â♠ tü ❞♦ ❜❛ ✈ỵ✐ ❛✱ ❜ ❧➔ ❝→❝ ♣❤➛♥ tû ❦❤→❝ ♥❤❛✉ ❝õ❛ ❝ì sð✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✸✳ ●✐↔ sû w = xε11 xε22 xεnn , xi ∈ X, εi ∈ {1, −1} ❧➔ tø rót ❣å♥✳ ❚❛ ♥â✐ ✇ ❧➔ tø rót ❣å♥ t✉➛♥ ❤♦➔♥ ♥➳✉ xn = x1 ❤♦➦❝ ♥➳✉ xn = x1 = a t❤➻ εn = −ε1 ✳ ❇ê ✤➲ ✶✳✸✳✹✳ tứ rút õ t t ữợ ❞↕♥❣ w = uvu−1 tr♦♥❣ ✤â ✈ ❧➔ tø rót ❣å♥ t✉➛♥ ❤♦➔♥✳ ❍➺ q✉↔ ✶✳✸✳✺✳ ❈❤♦ ✇✱ ✉ ❧➔ ❝→❝ tø rót ❣å♥ ❦❤✐ ✤â✱ ❛✮ ◆➳✉ w = e t❤➻ wn = e, ∀n ≥ 1✳ ❜✮ ◆➳✉ w = u t❤➻ wm = um , ∀m ≥ 1✳ ❈❤ù♥❣ ♠✐♥❤ ❛✮ ❚❤❡♦ ❇ê ✤➲ ✶✳✸✳✹ t❤➻ ♠å✐ tứ rút õ t t ữợ w = uvu−1 tr♦♥❣ ✤â ✈ ❧➔ tø rót ❣å♥ t✉➛♥ ❤♦➔♥ ✈➔ v = e✳ ❑❤✐ ✤â✱ ∀n ≥ t❤➻ wn = (uvu−1 )n = uv n u−1 ✳ ▼➦t ❦❤→❝✱ ❞♦ v = e ♥➯♥ wn = e✳ ❜✮ ❚❛ ❣✐↔ t❤✉②➳t r➡♥❣ w = e ✈➔ u = e t❤➻ w = rvr−1 ✈➔ u = sts−1 ✈ỵ✐ ✈ ✈➔ t ❧➔ ❝→❝ tø rót ❣å♥ t✉➛♥ ❤♦➔♥✱ wm = rv m r−1 ✈➔ um = stm s−1 ✳ ◆➳✉ r = s t❤➻ wm = um ✳ ◆➳✉ r ❂ s t❤➻ v = t ✭❞♦ w = u✮✱ s✉② r❛✱ v m = tm ✳ ❉♦ ✤â✱ wm = um , ∀m ≥ 1✳ ✾ ✷✳✶✳ ◆❍➶▼ QP ❉♦ ✤â✱ ❞➔♥ ❝→❝ ♥❤â♠ ❝♦♥ ▲✭●✮ ❧➔ ❞➔♥ ♣❤➙♥ ♣❤è✐✳ ❚➼♥❤ ❝❤➜t ✶✵✳ ❉➔♥ ❝→❝ ♥❤â♠ ❝♦♥ L(Qp ) ❧➔ ❞➔♥ ♣❤➙♥ ♣❤è✐✳ ❈❤ù♥❣ ♠✐♥❤ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✾ t❤➻ Qp ❧➔ ♥❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❇ê ✤➲ ✷✳✶✳✸ t❤➻ L(Qp ) ❧➔ ❞➔♥ ♣❤➙♥ ♣❤è✐✳ ❚➼♥❤ ❝❤➜t ✶✶✳ Qp ❧➔ ♥❤â♠ ✤ì♥ t❤✉➛♥ tỵ ự s ự Q õ õ ỡ t tỵ t ❣✐↔ sû ❍ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ Q ✈➔ ❇ ❧➔ ♥❤â♠ ❝♦♥ t❤✉➛♥ tó② ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ❍✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❇ ❂ ❍✳ ❈❤♦ x ∈ H t❤➻ B + x ∈ Q/B ✱ ❦❤✐ ✤â ❇ ✰ ① ❝â ❝➜♣ ❤ú✉ ❤↕♥✱ ❣✐↔ sû ❧➔ ♥ ✭t❤❡♦ ❇ê ✤➲ ✷✳✶✳✷✮✳ ❙✉② r❛✱ ♥✭❇ ✰ ①✮ ❂ ✵ ❤❛② nx ∈ B ✳ ❱➻ ✈➟②✱ nx nH B t t tỵ tr♦♥❣ ❍ ❞➝♥ ✤➳♥ nx ∈ nB ✳ ❙✉② r❛ nx = nb, b ∈ B ✳ ❉♦ ✤â✱ ♥✭① ✲ ❜✮ ❂ ✵✳ ❚ø ✤✐➲✉ ♥➔② ✈➔ ❣✐↔ t❤✉②➳t Q ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ❞➝♥ ✤➳♥ ① ❂ ❜✳ ❙✉② r❛✱ x ∈ B ✱ ❤❛② ❇ ❂ ❍✳ ❱➟② Q ✈➔ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ ♥â ❧➔ ♥❤â♠ ỡ t tỵ Qp õ ỡ t tỵ ❚➼♥❤ ❝❤➜t ✶✷✳ ∩{qQp |q ∈ P, q = p} = {0} ✈ỵ✐ P ❧➔ t➟♣ ❝→❝ sè ♥❣✉②➯♥ tè✳ ❈❤ù♥❣ ♠✐♥❤ ❈❤♦ a/pn ∈ ∩{qQp |q ∈ P, q = p}✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❛ ❂ ✵✳ ✣➸ t❤ü❝ ❤✐➺♥ ✤✐➲✉ ♥➔② ❝➛♥ ❝❤➾ r❛ ❝â ✈æ ❤↕♥ sè ♥❣✉②➯♥ tè ❝❤✐❛ ❤➳t ❛✳ ❈➛♥ ❝❤ù♥❣ ♠✐♥❤ q⑤❛ ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ tè q = p✳ ❱ỵ✐ a/pn ∈ qQp t❤➻ a/pn = q(c/pm )✱ ✈ỵ✐ c/pm ∈ Qp ✳ ❑❤✐ ✤â✱ qcpn = apm ✱ s✉② r❛ q|apm ✳ ❚✉② ♥❤✐➯♥✱ q ✈➔ pm ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✭✈➻ ♣ ✈➔ q ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✮✱ ♥➯♥ q⑤❛✳ ❱➟② ∩{qQp |q ∈ P, q = p} = {0}✳ ❚➼♥❤ ❝❤➜t ✶✸✳ ❱ỵ✐ ♠å✐ sè ♥❣✉②➯♥ tè q = p t❤➻✿ ❛✮ Qp = (qQp ) + Z✳ ❜✮ q = qQp ∩ Z✳ ❝✮ Qp /qQp ∼ = Z(q)✳ ❈❤ù♥❣ ♠✐♥❤ ❛✮ ●✐↔ sû a/pn ∈ Qp ✱ ❞♦ ♣ ✈➔ q ❧➔ ❝→❝ sè ♥❣✉②➯♥ tè ❦❤→❝ ♥❤❛✉✱ ♥➯♥ pn ✈➔ q ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✱ ❞♦ õ tỗ t số t iq + jpn = 1✳ ◆❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥ ✈ỵ✐ a/pn ❞➝♥ ✤➳♥ iq(a/pn ) + ja = a/pn ✳ ❙✉② r❛✱ a/pn ∈ (qQp ) + Z✱ ❤❛② Qp ⊆ (qQp ) + Z✳ ◆❣÷đ❝ ❧↕✐ t❤❡♦ ❚➼♥❤ ❝❤➜t ✺ ❝â Z ⊆ Qp ✳ ❉♦ q ❧➔ sè ♥❣✉②➯♥ tè ❦❤→❝ ♣ ♥➯♥ qQp ⊆ Qp ✳ ❙✉② r❛✱ qQp + Z ⊆ Qp ✳ ❱➟② qQp + Z = Qp ✳ ❜✮ ●✐↔ sû k ∈ qQp ∩ Z✳ ❑❤✐ ✤â k = q(c/pm ) ✈ỵ✐ c/pm ∈ Qp ✳ ❙✉② r❛ pm k = qc✱ ♥➯♥ q|pm k ✳ ❚ø ✤✐➲✉ ♥➔② ✈➔ ❣✐↔ t❤✉②➳t q ✈➔ pm ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ❞➝♥ ✤➳♥ q⑤❦✳ ❙✉② r❛✱ k ∈ q ✈➔ qQp ∩ Z ⊆ q ✳ ❈❤✐➲✉ ♥❣÷đ❝ ❧↕✐ t❤➻ ❤✐➸♥ ♥❤✐➯♥✳ ❉♦ ✤â✱ q = qQp ∩ Z✳ ✸✾ ❈❍×❒◆● ✷✳ ◆❍➶▼ QP ❱⑨ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ ◆⑨❨ ✹✵ ❝✮ ❚ø ❛✮ ✈➔ ❜✮ ❞➝♥ ✤➳♥ Qp /qQp ∼ = = Z/ q ✳ ▼➦t ❦❤→❝✱ Z/ q ∼ = Z(q)✳ ❱➟② Qp /qQp ∼ Z(q)✳ ❚➼♥❤ ❝❤➜t ✶✹✳ Qp ❧➔ ♥❤â♠ ❝â t❤➦♥❣ ❞÷ ❤ú✉ ❤↕♥✳ ❈❤ù♥❣ ♠✐♥❤ ●✐↔ sû Γ = {qQp |q ∈ P, q = p}✳ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✶✷ t❤➻ ∩Γ = {0} ✈➔ tø ❚➼♥❤ ❝❤➜t ✶✸✱ ♠ö❝ ❝✮ Qp /qQp ❧➔ ❤ú✉ ❤↕♥✳ ❱➟② Qp ❝â t❤➦♥❣ ❞÷ ❤ú✉ ❤↕♥ ✭t❤❡♦ ỵ ộ õ tỹ sü ❝õ❛ ♥❤â♠ ❝→❝ sè ❤ú✉ t✛ Q ✈ỵ✐ ♣❤➨♣ t♦→♥ ❝ë♥❣ ❧➔ ♥❤â♠ rót ❣å♥✳ ❈❤ù♥❣ ♠✐♥❤ ●✐↔ sû ❍ ❧➔ ♠ët ♥❤â♠ ❝♦♥ ❦❤ỉ♥❣ rót ❣å♥ ❝õ❛ Q✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ Q ❂ ❍✳ ❉♦ ❍ ❧➔ ♥❤â♠ ❦❤ỉ♥❣ rót ❣å♥ ♥➯♥ ❍ ❝â ♠ët ♥❤â♠ ❝♦♥ ❝❤✐❛ ✤÷đ❝ ❉ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ Q ⊆ D✳ ●å✐ x/y ∈ Q✱ ♥❤➟♥ ①➨t y = 0✳ ❉♦ ❉ ❧➔ ♥❤â♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ♥➯♥ ❉ ❝â ♣❤➛♥ tû a/b = 0✱ ♥➯♥ a = r ay = õ tỗ t t ∈ D ♠➔ ✭❛②✮t ❂ ❛✴❜ ❞♦ t➼♥❤ ❝❤✐❛ ✤÷đ❝ ❝õ❛ ❉✳ ❱➻ ✈➟②✱ t ❂ ✭❛✴❜✮✭✶✴❛②✮❂✶✴❜②✳ ▼➦t ❦❤→❝✱ t ∈ D✱ ❞➝♥ ✤➳♥ (xb)t ∈ D✳ ❉♦ ✤â✱ (xb)(1/by) = x/y ∈ D ❤❛② Q ⊆ D✱ s✉② r❛ Q ❂ ❍✳ ❱➟② ♠é✐ ♥❤â♠ ❝♦♥ t❤ü❝ sü ❝õ❛ Q ❧➔ ♥❤â♠ rót ❣å♥✳ ❚➼♥❤ ❝❤➜t ✶✺✳ Qp ❧➔ ♥❤â♠ rót ❣å♥✳ ❈❤ù♥❣ ♠✐♥❤ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✶✹ t❤➻ Qp ❝â t❤➦♥❣ ❞÷ ❤ú✉ ❤↕♥ ✈➔ t❤❡♦ ✣à♥❤ ỵ s r Qp õ rút ❝❤ù♥❣ ♠✐♥❤ ❦❤→❝✱ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✹ t❤➻ ♠å✐ ♥❤â♠ ❝♦♥ t❤ü❝ sü ❝õ❛ Q ❧➔ ♥❤â♠ rót ❣å♥✱ ❞♦ ✤â Qp ❧➔ ♥❤â♠ rót ❣å♥✳ ❚➼♥❤ ❝❤➜t ✶✻✳ Qp ❦❤æ♥❣ ❧➔ ♥❤â♠ ❝♦❤♦♣❢✳ ❈❤ù♥❣ ♠✐♥❤ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✶✸ t❤➻ Qp /qQp ∼ = Z(q)✱ s✉② r❛ Qp ∼ = Z(q) ì qQp ợ q số tè ❦❤→❝ ♣✱ tr♦♥❣ ✤â Z(q) × qQp = Qp ✳ ❱➟② Qp ❦❤æ♥❣ ❧➔ ♥❤â♠ ❝♦❤♦♣❢✳ ❚➼♥❤ ❝❤➜t ✶✼✳ ❚❛ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ❛✮ Qp ∩ Qp = Z✳ ❜✮ Qp + Qp = Q✳ ❝✮ Qp ∩ Qq = Z ✭✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ tè q = p✮✳ ❈❤ù♥❣ ♠✐♥❤ ❛✮ ❱ỵ✐ ♣ ❧➔ sè ♥❣✉②➯♥ tố trữợ t õ Qp = {a/pn |a/pn Q} ✈➔ Qp = {a/b|a/b ∈ Q, (b, p) = 1}✳ ❚❤❡♦ ❝→❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❤➻ Z ⊆ Qp ✈➔ Z ⊆ Qp ✳ ❙✉② r❛✱ Z ⊆ Qp ∩ Qp ✳ ▼➦t ❦❤→❝✱ ❣✐↔ sû x ∈ Qp ∩ Qp ✱ t❤➻ ① ❝â ❞↕♥❣ x = a/pn ✈➔ x = a /b ✭✈ỵ✐ ❜✬ ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉ ✈ỵ✐ ♣✮✳ ❙✉② r❛ ab = pn a ⇒ b |a ✱ ✈➻ (b , pn ) = 1✳ ❉♦ ✤â✱ x ∈ Z✳ ❱➟② Qp ∩ Qp = Z✳ ✷✳✶✳ ◆❍➶▼ QP ❜✮ ❚❛ ❝â Qp + Qp ⊆ Q ✭❤✐➸♥ ♥❤✐➯♥ ❞♦ ✈ỵ✐ x ∈ Qp + Qp ⇒ x = a/pn + a /b ✈ỵ✐ ✭❜✬✱ q ỗ số t ữủ ♠ët ♣❤➛♥ tû t❤✉ë❝ Q✮✳ ▼➦t ❦❤→❝✱ ❞♦ Qp ✈➔ Qp ❧➛♥ ❧÷đt ❧➔ ❝→❝ ♥❤â♠ ❝♦♥ t❤ü❝ sü ❝õ❛ Q ♥➯♥ Qp + Qp ❧➔ ♠ët t➟♣ ❝❤ù❛ tr♦♥❣ Q ữủ ợ x Q t ❛✴❜ ✈➔ b = t❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✿ ◆➳✉ ✭❜✱ ♣✮ ❂ ✶ t❤➻ a/b ∈ Qp ⊆ Qp + Qp ✳ ◆➳✉ ♣⑤ ❜ t❤➻ b = pn u✱ ✈ỵ✐ (pn , u) = 1✳ ❑❤✐ õ tỗ t số s t s spn + tu = 1✳ ❉♦ ✤â✱ a/b = a/pn u = a(spn + tu)/pn u = as/u + at/pn ∈ Qp + Qp ✱ s✉② r❛ Q ⊆ Qp + Qp ❱➟② Qp + Qp = Q✳ ❝✮ ❚❛ ❝â Z ⊆ Qp ✈➔ Z ⊆ Qq t❤❡♦ ❚➼♥❤ ❝❤➜t ✺ ♥➯♥ Z ⊆ Qp ∩ Qq ✭✈ỵ✐ ♣✱ q ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉✮✳ ▼➦t ❦❤→❝✱ ✈ỵ✐ x ∈ Qp ∩ Qq t❤➻ x = a/pn ✈➔ x = b/q m ✳ ❙✉② r❛✱ aq m = b.pn ✳ ❱➻ ✭♣✱ q✮ ❂ ✶ ♥➯♥ (pn , q m ) = 1✳ ❉♦ ✤â✱ x = b/q m ∈ Z✳ ❱➟② Z = Qp ∩ Qq ✳ ❚➼♥❤ ❝❤➜t ✶✽✳ ❈❤♦ q ❧➔ sè ♥❣✉②➯♥ tè ❦❤→❝ ♣✳ ❑❤✐ ✤â✱ Hom(Qp , Qq ) ❧➔ t➛♠ t❤÷í♥❣✳ ❈❤ù♥❣ ♠✐♥❤ ●✐↔ sû f ∈ Hom(Qp , Qq )✳ ◆➳✉ f = t tỗ t ởt số ỳ t a = u/ps a = tỗ t b = f (a) = v/q t ✱ s❛♦ ❝❤♦ b = f (a) = 0✳ v u u f (1) ✳ ❙✉② r❛✱ = f (1) ps qt ps ✭✯✮✳ ữợ ợ t ✈✱ ❝❤✐❛ ❤❛✐ ✈➳ ❝õ❛ ✭✯✮ ❝❤♦ ❞ t❛ ❝â v1 u1 = s f (1) ⇒ v1 ps = u1 q t f (1)✳ t q p ▼➦t ❦❤→❝ ❛ ❂ ❛✳✶ ♥➯♥ b = f (a) = a.f (1) = ❉♦ (u1 , v1 ) = ✈➔ (ps , q t ) = 1✱ ♥➯♥ ps |f (1) t f (1) Qq tỗ t↕✐ a a a s❛♦ ❝❤♦ f (1) = r ✳ ❙✉② r❛✱ ps | r ✳ ❉♦ (ps , q r ) = 1✱ ♥➯♥ ps |a ✳ ❱➻ ♣ ❧➔ ♠ët sè r q q q s ♥❣✉②➯♥ tố t ý q p ổ ữợ ❝õ❛ ❛✬ ♥➯♥ ❛✬ ❂ ✵✱ s✉② r❛ ❢✭✶✮ ❂ ✵✳ ❚ø ✤â✱ t❛ ❝â f (n) = 0, ∀n ∈ Z ✈➔ f ( n ) = 0✱ ✈ỵ✐ ♣ ❧➔ ♠ët sè ♥❣✉②➯♥ tè✳ ❱➟② ❢ ❂ ✵✱ ❤❛② p p q Hom(Q , Q ) ❧➔ t tữớ t Q ởt ợ ♣❤➨♣ ♥❤➙♥ ❜➻♥❤ t❤÷í♥❣✳ ❑❤✐ ✤â✱ Qp ❧➔ ✈➔♥❤ ❝♦♥ ❝õ❛ Q✳ ❈❤ù♥❣ ♠✐♥❤ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✶ t❤➻ Qp ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ Q ✤è✐ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ Qp ❧➔ ♥û❛ ♥❤â♠ ✤è✐ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ tr♦♥❣ Q✳ ●å✐ a/pn ✈➔ c/pm ❧➔ ❤❛✐ ♣❤➛♥ tû t❤✉ë❝ a c ac Qp t❤➻ n m = m+n ∈ Qp ✳ p p p ✹✶ ❈❍×❒◆● ✷✳ ◆❍➶▼ QP ❱⑨ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ ◆⑨❨ ✹✷ ❱➟② Qp ❧➔ ✈➔♥❤ ❝♦♥ ❝õ❛ Q✳ ❍ì♥ ♥ú❛ ❞♦ t➼♥❤ ❣✐❛♦ ❤♦→♥ ❝õ❛ ♣❤➨♣ ♥❤➙♥ ❤❛✐ sè ❤ú✉ t➾ ♥➯♥ ♣❤➨♣ ♥❤➙♥ tr♦♥❣ Qp ❝â t➼♥❤ ❣✐❛♦ ❤♦→♥✱ ❝â ✤ì♥ ✈à✱ ✈ỵ✐ ♣❤➛♥ tû ✤ì♥ ✈à ❧➔ = 1/p0 ✳ ❱➟② Qp ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à tr♦♥❣ tr÷í♥❣ ❝→❝ sè ❤ú✉ t✛ Q ✈➔ ❦❤ỉ♥❣ ❝â ÷ỵ❝ ❝õ❛ ❦❤ỉ♥❣ ♥➯♥ Qp ❧➔ ♠✐➲♥ ♥❣✉②➯♥ ♥❤÷♥❣ ❦❤ỉ♥❣ trữớ ổ õ ợ tû ♥❣❤à❝❤ ✤↔♦✳ ▼➦t ❦❤→❝✱ ❞♦ Q ❧➔ tr÷í♥❣ ♥❣✉②➯♥ tè ♥➯♥ ♥â ❦❤ỉ♥❣ ❝❤ù❛ ❜➜t ❦ý tr÷í♥❣ ❝♦♥ t❤ü❝ sü ♥➔♦✳ ❇ê ✤➲ ✷✳✶✳✺✳ ❈❤♦ ❆ ❧➔ ♥❤â♠ ❛❜❡♥ ❦❤æ♥❣ ①♦➢♥ ✈➔ x ∈ A, x = 0✳ ✣➦t P (A, x) = {a|a ∈ A, ma ∈ x , m ∈ N, m = 0}✳ ❑❤✐ ✤â✿ ❛✮ P õ t tỵ x ∈ P (A, x)✳ ❝✮ P✭❆✱①✮ ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ ♥❤â♠ ❝→❝ sè ❤ú✉ t✛ Q✳ ✣➦❝ ❜✐➺t ❤➔♠ sè g : P (A, x) −→ Q ❧➔ ♣❤➨♣ ♥❤ó♥❣ g(a) = j/m ⇔ ma = jx, m = 0✳ ❈❤ù♥❣ ♠✐♥❤ ◆➳✉ ❛ ✈➔ ❜ t❤✉ë❝ ✈➔♦ P✭❆✱①✮ t❤➻ ♠❛ ❂ ❥① ✈➔ ♥❜ ❂ ❦① ✈ỵ✐ m, n = 0✳ ❱➻ t❤➳✱ mn = ✈➔ ♠♥✭❛ ✲ ❜✮ ❂ ♠♥❛ ✲ ♠♥❜ ❂ ♥❥① ✲ ♠❦① ❂ ✭♥❥ ✲ ♠❦✮①✱ ❞♦ ✤â a − b ∈ P (A, x)✳ ❉➝♥ ✤➳♥ P✭❆✱①✮ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ ự t t tỵ P t❛ ❣å✐ n ∈ N, n = ✈➔ ❣å✐ a ∈ nA ∩ P (A, x)✳ ❑❤✐ ✤â ❛ ❂ ♥② ✈➔ ♠❛ ❂ ❥① ✈ỵ✐ y ∈ A✱ ♠ ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ ♠ët sè ♥❣✉②➯♥ ❥✳ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ♠♥② ❂ ❥①✳ ❚✉② ♥❤✐➯♥✱ mn = 0✱ ✈➻ m, n = 0✱ s✉② r❛✱ y ∈ P (A, x)✱ ❞♦ ✤â a ∈ nP (A, x) P õ t tỵ ❆ ✭s✉② r❛ ✭❛✮ ✈➔ ✭❜✮✮✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✭❝✮ trữợ t t tr ởt ●✐↔ sû ♠❛ ❂ ❥① ✈➔ ♥❛ ❂ ❦① ✈ỵ✐ m, n = t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❥✴♠ ❂ ❦✴♥✳ ❈â ❤❛✐ tr÷í♥❣ ❤đ♣ ①↔② r❛ ❛ ❂ ✵ ❤♦➦❝ a = 0✳ ◆➳✉ ❛ ❂ ✵ t❤➻ ♠❛ ❂ ♥❛ ❂ ✵✱ ❞♦ ✤â ❥① ❂ ❦① ❂ ✵✳ ✣✐➲✉ ♥➔② ✈➔ ❣✐↔ t❤✉②➳t ❆ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ✈➔ x = ❞➝♥ ✤➳♥ ❥ ❂ ❦ ❂ ✵✳ ❉♦ ✤â ❥✴♠ ❂ ❦✴♥✳ ◆➳✉ a = t❤➻ ♠❦❛ ❂ ❥❦① ❂ ❥♥❛✳ ✣✐➲✉ ♥➔② ✈➔ ❆ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ✈➔ ❣✐↔ t❤✉②➳t a = 0✱ s✉② r❛ ♠❦ ❂ ❥♥✱ ♥➯♥ ❥✴♠ ❂ ❦✴♥✳ ❱➟② ❣ ❧➔ ♠ët ❤➔♠ sè✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❣ ỗ sỷ a, b P (A, x) ♠➔ ♠❛ ❂ ❥① ✈➔ ♥❜ ❂ ❦① ✈ỵ✐ m, n = 0✳ ❑❤✐ ✤â✱ ♠♥✭❛ ✰ ❜✮ ❂ ♠♥❛ ✰ ♠♥❜ ❂ ✭♥❥ ✰ ♠❦✮①✳ ✣✐➲✉ ♥➔② ✈➔ mn = ❞➝♥ ✤➳♥ ❣✭❛ ✰ ❜✮ ❂ ✭♥❥ ✰ ♠❦✮✴♠♥ ❂ ✭❥✴♠✮ ✰ ✭❦✴♥✮ ❂ ❣✭❛✮ ✰ ❣✭❜✮✳ ❈✉è✐ ❝ị♥❣ ✤➸ ❦✐➸♠ tr❛ ❣ ❧➔ ✤ì♥ →♥❤✱ t❛ ❣✐↔ sû ❣✭❛✮ ❂ ✵ ✈ỵ✐ ♠❛ ❂ ❥①✳ ❑❤✐ ✤â ✵ ❂ ❥✴♠ ❞➝♥ ✤➳♥ ✵ ❂ ❥✳ ❙✉② r❛✱ ♠❛ ❂ ✵✳ ✣✐➲✉ ♥➔② ✈➔ ❣✐↔ t❤✉②➳t ❆ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✱ m = ♥➯♥ ❛ ❂ ✵✳ ❱➟② ❣ ❧➔ ✤ì♥ →♥❤✳ ❚➼♥❤ ❝❤➜t ✷✵✳ ❈❤♦ ❆ ❧➔ ♥❤â♠ ❛❜❡♥ ✈ỵ✐ ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ❛✮ ❆ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ❜✮ ❆ ❧➔ ♥❤â♠ ♣✲❝❤✐❛ ✤÷đ❝✳ ❝✮ ❆ ❧➔ ♠ð rë♥❣ ❝õ❛ ♠ët ♥❤â♠ ①②❝❧✐❝ ✈æ ❤↕♥ ❜ð✐ ♠ët ♣✲♥❤â♠✳ ❑❤✐ ✤â✱ A ∼ = Qp ✳ ✷✳✶✳ ◆❍➶▼ QP ❈❤ù♥❣ ♠✐♥❤ ❱ỵ✐ x ∈ A, x = ✤➦t P (A, x) = {a|a ∈ A, ma ∈ x , m ∈ N, m = 0}✳ ❉♦ ❆ ❧➔ ♥❤â♠ ❛❜❡♥ ❦❤æ♥❣ ①♦➢♥ ♥➯♥ →♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✶✳✺ t❤➻ P✭❆✱ ①✮ ❧➔ ♥❤â♠ t tỵ P õ t ♥❤ó♥❣ ✈➔♦ ♥❤â♠ ❝ë♥❣ ❝→❝ sè ❤ú✉ t✛ q✉❛ ✤ì♥ ❝➜✉ g : P (A, x) → Q t❤ä❛ g(a) = j/m ⇔ ma = jx✳ ❚❛ ❝â P✭❆✱ ①✮ ❂ ❆✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ P✭❆✱ ①✮✳ ❱➻ P✭❆✱ ①✮ ♥❤ó♥❣ ✈➔♦ ♥❤â♠ ✭Q✱ ✰✮ ♥➯♥ ♥❤â♠ ♥➔② ❝â t❤➸ ✤÷đ❝ ①❡♠ ♥❤÷ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ Q✳ ❉♦ ♥❤â♠ ✭Q✱ ✰ ✮ ✈➔ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ ♥â õ ỡ t tỵ õ ỡ t tỵ õ tỗ t ởt ỡ g : A → Q t❤ä❛ g(a) = j/m ⇔ ma = jx✳ ❉♦ ❆ ❧➔ ♥❤â♠ ♣✲❝❤✐❛ ✤÷đ❝ ✈➔ a, x A tỗ t yn , ym A✱ s❛♦ ❝❤♦ a = pn yn ✈➔ x = pm ym t tỗ t pl s ❝❤♦ ym = pl yn ✱ ♥➯♥ g(a) = j/m = l+m−n ∈ Qp ✳ p p ∼ ❉♦ ✤â✱ Img Q ỵ t❤ù ♥❤➜t Img = A ⊆ Qp ✱ ✈ỵ✐ Kerg = {0} ✭❞♦ ❣ ❧➔ ✤ì♥ ❝➜✉✮✳ ◆❣÷đ❝ ❧↕✐✱ ❞♦ ❆ ❧➔ ♠ët ♠ð rë♥❣ ❝õ❛ ♠ët ♥❤â♠ ①②❝❧✐❝ ✈æ ❤↕♥ ❜ð✐ ♠ët ♣ ✲ ♥❤â♠ ♥➯♥ ❣å✐ ❑ ❧➔ ♠ët ♥❤â♠ ①②❝❧✐❝ ✈æ ❤↕♥ ✈➔ ▲ ❧➔ ♠ët ♣✲♥❤â♠ t❤➻ t❛ ❝â ♠ët ❞➣② ❦❤ỵ♣ f g ♥❣➢♥ −→ K −→ A −→ L −→ 0✱ tr♦♥❣ ✤â ❢ ❧➔ ✤ì♥ ❝➜✉✱ ❣ ❧➔ t♦➔♥ ❝➜✉✳ ❑❤✐ ✤â✱ K∼ = Z ✈➔ A/K ∼ = L ✈ỵ✐ ▲ ❧➔ ♠ët ♣ ✲ ♥❤â♠ ❛❜❡♥ ✈æ ❤↕♥ ♥➯♥ ▲ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♥❤â♠ ♣✲Pr✉❢❢❡r Z(p∞ )✳ ▼➦t ❦❤→❝✱ Z(p∞ ) = Qp /Z✳ ❙✉② r❛✱ A ∼ = Qp ✳ ❚➼♥❤ ❝❤➜t ✷✶✳ Qp ∼ = G({Xi |i ∈ N, i = 0}|(Xi+1 )p Xi−1 = 1, ∀i)✳ ❈❤ù♥❣ ♠✐♥❤ ✣➦t G = G({Xi |i ∈ N, i = 0}|(Xi+1 )p Xi−1 = 1, ∀i)✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤✐➲✉ s❛✉✿ ● ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ● ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ Q ✈➔ ● ❧➔ ♥❤â♠ ❝♦♥ ✤ì♥ t tỵ ứ õ s r G = Qp ✳ ✯ ● ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ●å✐ ❛ ❧➔ ♠ët ♣❤➛♥ tû t❤✉ë❝ ●✱ s✉② r❛ a = X1e1 X2e2 Xnen , ei ∈ Z, ∀i : 1, n✳ ◆➳✉ p an = t❤➻ (X1e1 X2e2 Xnen )n = 1✳ ❉♦ Xi = Xi+1 ✈ỵ✐ i ∈ N, i = ♥➯♥ q✉② ♥↕♣ t❤❡♦ ♥ n−2 n−1 e1 pn−1 +e2 pn−2 + +en−1 p+en n t❛ ❝â✱ a = Xn ✳ ◆➳✉ a = t❤➻ (Xne1 p +e2 p + +en−1 p+en )n = n−1 n−2 1✳ ❙✉② r❛ Xne1 p +e2 p + +en−1 p+en = 1✱ ❤❛② ❛ ❂ ✶✳ ❱➟② ● ❧➔ ♥❤â♠ ❦❤ỉ♥❣ ①♦➢♥✳ ✯ ● ❧➔ ♥❤â♠ ❛❜❡♥ ❚❤➟t ✈➟② ✈ỵ✐ ❤❛✐ ♣❤➛♥ tû Xi ✈➔ Xj ❜➜t ❦ý t❤✉ë❝ t➟♣ s✐♥❤ ❝õ❛ ●✱ ❣✐↔ sû ✐ ❁ ❥✱ j−i (j−i+1) (j−i) t❤➻ Xi Xj = Xjp Xj = Xjp = Xj Xjp = Xj Xi ✳ ❱➟② ● ❧➔ ♥❤â♠ ❛❜❡♥ ❞♦ ❜➜t ❦ý ❤❛✐ ♣❤➛♥ tû s✐♥❤ ♥➔♦ ❝õ❛ õ t ữủ ợ ● ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ ♥❤â♠ ❝ë♥❣ ❝→❝ sè ❤ú✉ t✛ ✭Q✱ ✰ ✮✳ ✹✸ ❈❍×❒◆● ✷✳ ◆❍➶▼ QP ❱⑨ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ ◆⑨❨ ✹✹ ❉♦ ● ❧➔ ♥❤â♠ ❛❜❡♥ ❦❤ỉ♥❣ ①♦➢♥ ♥➯♥ →♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✶✳✺ t❤➻ ✈ỵ✐ x ∈ G, x = 0✱ ✤➦t P (G, x) = {a|a ∈ G, ma ∈ x , m ∈ Z, m = 0}✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ t q s P õ t tỵ ❝õ❛ ●✱ x ∈ P (G, x)✱ P✭●✱ ①✮ ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ ♥❤â♠ ✭Q✱ ✰ ✮ ❜ð✐ ✤ì♥ ❝➜✉ g : P (G, x) → Q t❤ä❛ g(a) = j/m ✈ỵ✐ ♠❛ ❂ ❥①✱ a ∈ P (G, x)✳ ❉♦ ♥❤â♠ ✭Q✱ ✰✮ ✈➔ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ ♥â ỡ t tỵ P õ ỡ t tỵ õ t ỗ g : G → Q t❤ä❛ ❞♦ ❣ ❧➔ ✤ì♥ ❝➜✉ r {0} ỵ t❤ù ♥❤➜t ❝â G ∼ = Img ✳ ▼➦t ❦❤→❝✱ Img = {j/m|ma = jx, m = 0, a, x ∈ G, x = 0}} ✈➔ ma ∈ x ✭t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ P✭●✱①✮✮✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ x ∈ G s✉② r❛ x = xe11 xe22 xenn ✳ ●å✐ t➟♣ s✐♥❤ ❝õ❛ x ❧➔ S = {x1 , x2 , , xn }✳ ❱➻ ma ∈ x ♥➯♥ ma = xk11 xk22 xknn ✱ ✈ỵ✐ ki , ei ∈ Z, ∀i : 1, n✳ ❙✉② r❛✱ j/m = a/x = xk11 xk22 xknn ⇒ j = e1 −k1 e2 −k2 = e1 e2 e −k e −k e 2 1 n e −k mx1 x2 xn xnn n xnen −kn x2 x2 mx1 x1 ❑❤✐ ✤â✱ j= (e −k )pn−1 +(e2 −k2 )pn−2 + +(en−1 −kn−1 )p+(en −kn ) xn 1 ❱➻ j/m ∈ Q✱ ♥➯♥ j, m ∈ Z ✈➔ m = 0✱ ♥➯♥ −k1 )p x(e n n−1 +(e −k )pn−2 + +(e 2 n−1 −kn−1 )p+(en −kn ) |1 ✣➦t M = (e1 − k1 )pn−1 + (e2 − k2 )pn−2 + + (en−1 − kn−1 )p + (en − kn ) t❤➻ xM n − = õ xn ữợ ❜➟❝ ▼ ❝õ❛ ♣❤➛♥ tû ✤ì♥ ✈à✳ ❱➻ ① ❜✐➸✉ ❞✐➵♥ n−1 n−2 ✤÷đ❝ q✉❛ xn ✱ tù❝ ❧➔ ♥➳✉ x = xα1 xα2 xαnn t❤➻ x = xαn1 p +α2 p + +αn ✱ s✉② r❛ ① ụ ữợ tỷ ✤ì♥ ✈à ✶ ✈ỵ✐ M = α1 pn−1 +α2 pn−2 + +αn ✳ ❱➟②✱ x ❧➔ ♥❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣✱ ❤❛② ♥❤â♠ s✐♥❤ ❜ð✐ t➟♣ ❙ ❧➔ ♥❤â♠ ①②❝❧✐❝✳ ●✐↔ sû ❝➜♣ ❝õ❛ ① ❧➔ ❦✱ ❦❤✐ ✤â✱ ❦ ⑤ ▼✬✱ s✉② r❛ k|pn l ✈ỵ✐ ❧ ❂ ❇❈◆◆ {αi |i : 1, n}✱ ♥➯♥ k|pn ✭❦ ❦❤æ♥❣ ❝❤✐❛ ❤➳t ữủ t ợ ❧➔ tê♥❣ ❝õ❛ ❝→❝ ❧ô② t❤ø❛ pn ✮✳ ❉♦ ma ∈ x , m ∈ Z✱ ♥➯♥ (ma)k = 1✳ r ữợ pn ✳ ❉♦ ✤â✱ m = pi ✭❞♦ a ∈ x ✱ ♥➯♥ ❝➜♣ ❝õ❛ ❛ ❝ô♥❣ ❧➔ ❧ô② t❤ø❛ ❝õ❛ ♣✮✳ ❱➟② G ∼ = Img ⊆ Qp ✳ ▼➦t ❦❤→❝✱ ❞♦ ■♠❣ ✈➔ Qp ❧➔ ♥❤â♠ ✤ì♥ t❤✉➛♥ tó② ♥➯♥ G∼ = Qp ✳ ◆❤â♠ Qp ❧➔ ♠ët ✈➼ ❞ư ❝ư t❤➸ ✈➲ ❧ỵ♣ ♥❤â♠ ❛❜❡♥ ✈ỉ ❤↕♥✱ ✈ỵ✐ ❝→❝ t➼♥❤ ❝❤➜t ♥❤÷✿ ◆❤â♠ ❦❤ỉ♥❣ ①♦➢♥❀ ◆❤â♠ ♣✲❝❤✐❛ ✤÷đ❝ ✈➔ ❦❤ỉ♥❣ ❝â ♥❤â♠ ❝♦♥ q✲❝❤✐❛ ✤÷đ❝ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ✭✈ỵ✐ q = p ✮❀ ◆❤â♠ ♥➔② ❝❤ù❛ ♥❤â♠ ❝→❝ số ợ t Z ữ ❧➔ ♠ët ♥❤â♠ ❝♦♥❀ ◆❤â♠ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ✈➔ ❦❤ỉ♥❣ t❤♦↔ ✤✐➲✉ ❦✐➺♥ ❆❈❈ ✈➔ ❉❈❈❀ ◆❤â♠ ❤♦♣❢ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ ♥❤â♠ ❝♦❤♦♣❢❀ ◆❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❝â ❞➔♥ ❝→❝ ♥❤â♠ ❝♦♥ ❧➔ ❞➔♥ ♣❤➙♥ ♣❤è✐❀ ◆❤â♠ ✤ì♥ t tỵ õ t ữ é ế QP ỳ ỗ tớ õ rút ❣å♥✳ ▼➦t ❦❤→❝✱ ❞♦ Qp ❦❤æ♥❣ ❝â ♥❤â♠ ❝♦♥ q✲❝❤✐❛ ữủ ổ t tữớ ợ q = p Qp ❦❤æ♥❣ ❝â ♥❤â♠ ❝♦♥ tè✐ ✤↕✐ ✭♠❛①✐♠❛❧ s✉❜❣r♦✉♣✮✱ t❤❡♦ ♥❣❤➽❛ ▼ ❧➔ ♥❤â♠ ❝♦♥ tè✐ ✤↕✐ ❝õ❛ ● ♥➳✉ ▼ ❧➔ ♠ët ♥❤â♠ ❝♦♥ t❤ü❝ sü ❝õ❛ ● ✈➔ ♥➳✉ ❝â ♠ët ♥❤â♠ ❍ t❤ä❛ M ≤ H ≤ G t❤➻ ❍ ❂ ▼ ❤♦➦❝ ❍ ❂ ●✳ ◆❣♦➔✐ r❛✱ ♥❤â♠ Qp ❝ơ♥❣ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ♠ët ❝→❝❤ t÷í♥❣ ♠✐♥❤ tổ q t s ỗ t tự ①➨t✿ ◆➳✉ ①➨t tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣ ❂ ✷ ✈➔ ♥ t❤✉ë❝ t➟♣ ❤đ♣ sè tü ♥❤✐➯♥ t❤➻ t❛ ✤÷đ❝ ♥❤â♠ ❝→❝ sè ❤ú✉ t✛ ♥❤à ♥❣✉②➯♥ ✈ỵ✐ ♣❤➨♣ t♦→♥ ❝ë♥❣✳ ❚ø ♥❤â♠ Qp t❛ ❝â t❤➸ ①➙② ❞ü♥❣ ✤÷đ❝ ♥❤â♠ t❤÷ì♥❣ ❝õ❛ ♥❤â♠ ♥➔②✱ ♥❤â♠ ♣✲ Pr✉❢❡r✱ Z(p∞ ) = Qp /Z✳ ✣➙② ❧➔ ♠ët ✈➼ ❞ö ✈➲ ♥❤â♠ ✈æ ❤↕♥ ♠➔ ♠å✐ ♥❤â♠ ❝♦♥ t❤ü❝ sü ❝õ❛ ♥â ✤➲✉ ❤ú✉ ❤↕♥✱ ♥❤â♠ ♥➔② ❝ô♥❣ ❝â ✈❛✐ trá q✉❛♥ trồ ố ợ ợ õ ữủ r❛✱ ✤➸ ❤✐➸✉ rã ❤ì♥ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥❤â♠ ❛❜❡♥ ✈ỉ ❤↕♥ ❝ơ♥❣ ♥❤÷ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥❤â♠ Qp ✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ s➩ tr➻♥❤ ❜➔② t❤➯♠ ♠ð rë♥❣ ❝õ❛ ♥❤â♠ ♥➔② tr♦♥❣ ♥❤â♠ ✭Q × Q❀ ✰ ✮✳ ✷✳✷ ▼ð rë♥❣ ❝õ❛ ♥❤â♠ Qp ●✐↔ sû ♣✱ q✱ r ❧➔ ❝→❝ sè ♥❣✉②➯♥ tè rí✐ ♥❤❛✉✳ ●å✐ B = Qp × {0}✱ C = {0} × Qq ✈➔ D = 1/r, 1/r ✳ ✣➦t ❆ ❂ ❇ ✰ ❈ ✰ ❉✳ ❑❤✐ ✤â ❆✱ ❇✱ ❈ ❧➔ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ ♥❤â♠ Q × Q ✈ỵ✐ ♣❤➨♣ t♦→♥ ❝ë♥❣✳ c2 c1 ❚❤➟t ✈➟②✱ A = ∅✳ ❱ỵ✐ (x, y) ∈ A ✈➔ (g, h) ∈ A t❤➻ (x, y) = ( i , 0) + (0, j ) + p q 1 b1 b2 1 c3 ( , ) ✈➔ (g, h) = ( t , 0) + (0, k ) + b3 ( , )✱ ✈ỵ✐ i, j, k, t ∈ Z✳ ❑❤✐ ✤â✱ (x, y) − r r p q r r c b1 c2 b 1 c p t − b1 p i c q k − b2 q j (g, h) = ( i − t , 0)+(0, j − k )+(c3 −b3 )( , ) = ( , 0)+(0, )+ p p q q r r pi+t q j+k 1 (c3 − b3 )( , ) ∈ A✳ r r ◆❤â♠ ❆ ❧➔ ♠ð rë♥❣ ❝õ❛ ♥❤â♠ Qp ✈➔ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ❚➼♥❤ ❝❤➜t ✷✷✳ ❚❛ ❝â ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ♣❤➛♥ tû t❤✉ë❝ ♥❤â♠ ❆ ✈ỵ✐ ♣❤➛♥ tû t❤✉ë❝ ♥❤â♠ Qp ♥❤÷ s❛✉✿ ❛✮ ◆➳✉ (x, 0) ∈ A t❤➻ x ∈ Qp ✳ ❜✮ ◆➳✉ (0, y) ∈ A t❤➻ y ∈ Qq ✳ ❈❤ù♥❣ ♠✐♥❤ ❉♦ (x, 0) ∈ A ♥➯♥ (x, 0) = ( c2 1 c1 , 0) + (0, j ) + c3 ( , ) i p q r r ❉♦ ✤â✱ = (c2 /q j ) + (c3 /r) ⇒ c3 q j = −c2 r✱ ❤❛② r|c3 q j ❞➝♥ ✤➳♥ r|c3 ✭❞♦ ✭q✱ r✮ ❂ ✶✮✳ c1 ❙✉② r❛✱ c3 = rt✱ ✈ỵ✐ t ❧➔ sè ♥❣✉②➯♥✳ ❱➟② x = i + t ∈ Qp ✱ ✈➻ Z ⊆ Qp ✳ p ❈❍×❒◆● ✷✳ ◆❍➶▼ QP ❱⑨ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ ◆⑨❨ ✹✻ ❚÷ì♥❣ tü t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ♥➳✉ (0, y) ∈ A t❤➻ y ∈ Qq ✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ◆❤â♠ ● ✤÷đ❝ ❣å✐ ❧➔ ❝â sè ♠ơ ♥ ✭❡①♣♦♥❡♥t ♥✮ ✈ỵ✐ ♥ ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥➳✉ ♥ t❤ä❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ❛✮ xn = 1, ∀x ∈ G✳ ❜✮ ◆➳✉ ♠ ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ xm = 1, ∀x ∈ G t❤➻ ♥⑤♠✳ ❚➼♥❤ ❝❤➜t ✷✸✳ ❚❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ❛✮ ❜✮ ❝✮ ❞✮ ❆ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ❆✴❇ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ❆✴❈ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ◆➳✉ sè ♠ô ✭❡①♣♦♥❡♥t✮ ❝õ❛ ❆✴✭❇✰❈✮ ❜➡♥❣ r t❤➻ ❆✴✭❇✰❈✮ ❧➔ ♥❤â♠ ①♦➢♥✳ ❈❤ù♥❣ ♠✐♥❤ ❛✮ ●å✐ ✭①✱ ②✮ ❧➔ ♣❤➛♥ tû ❝õ❛ ❆ ❝â ❝➜♣ ♥✳ ❑❤✐ ✤â✱ ♥① ❂ ♥② ❂ ✵✱ ❞♦ x, y ∈ Q, n = ✈➔ Q ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ♥➯♥ ❞➝♥ ✤➳♥ ① ❂ ✵✱ ② ❂ ✵✱ ❤❛② ✭①✱ ②✮ ❂ ✭✵✱ ✵✮✳ ❱➟② ❆ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ❜✮ ●✐↔ sû ❇ ✰ ✭①✱ ②✮ ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ ❆✴❇ ❝â ❝➜♣ ❧➔ ♥✳ ❑❤✐ ✤â✱ (nx, ny) ∈ B t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❇ ❞➝♥ ✤➳♥ ♥② ❂ ✵ ❞♦ ✤â ② ❂ ✵✱ ✈➻ Q ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✷✷ t❛ ❝â ♥➳✉ (x, 0) ∈ A t❤➻ x ∈ Qp ✳ ❉♦ ✤â✱ (x, 0) ∈ B s✉② r❛✱ ❇ ✰ ✭①✱ ②✮ ❂ ✵✱ s✉② r❛ ❆✴❇ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ❝✮ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ❝â ❆✴❈ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ ❞✮ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r(x, y) ∈ B + C, ∀(x, y) ∈ A✳ ❚❤➟t ✈➟②✱ ❞♦ (x, y) ∈ A ♥➯♥ (x, y) = c2 1 c1 , + 0, j + c3 , i p q r r c1 ) + c3 ∈ Qp ✱ ✈➻ Z ⊆ Qp ✳ ❚÷ì♥❣ tü✱ ry ∈ Qq ✳ ❱➟② (rx, ry) = pi (rx, 0) + (0, ry) ∈ B + C ✳ ❙✉② r❛✱ ❆✴✭❇ ✰ ❈✮ ❧➔ ♥❤â♠ ①♦➢♥✳ ❉♦ ✤â✱ rx = r( ❇ê ✤➲ ✷✳✷✳✷✳ ●å✐ ♣ ❧➔ ♠ët sè ♥❣✉②➯♥ tè✱ ● ❧➔ ♥❤â♠ ❛❜❡♥ ♣✲❝❤✐❛ ✤÷đ❝ ✈➔ ❍ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ ●✳ ❑❤✐ ✤â✱ ●✴❍ ❝ô♥❣ ❧➔ ♥❤â♠ ♣✲❝❤✐❛ ✤÷đ❝✳ ❈❤ù♥❣ ♠✐♥❤ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ♣✭●✴❍✮ ❂ ●✴❍✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû H + x ∈ G/H t❤➻ x ∈ G✱ s✉② r❛ ① ❂ ♣② ✈ỵ✐ ② ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ ●✱ ❞♦ ● ❧➔ ♥❤â♠ ♣✲❝❤✐❛ ✤÷đ❝✳ ❑❤✐ ✤â✱ H + x = H + py = p(H + y) ∈ pG/H ✳ ❱➟② ♣✭●✴❍✮ ❂ ●✴❍✳ ❇ê ✤➲ ✷✳✷✳✸✳ ◆➳✉ ❆ ❧➔ ♠ët ♥❤â♠ ❛❜❡♥ ✈➔ ♥ ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ ❦❤✐ ✤â ❝↔ ❆❬♥❪ ✈➔ ♥❆ ❧➔ ❝→❝ ♥❤â♠ ❝♦♥ ❜➜t ❜✐➳♥ ❝õ❛ ❆ ✈➔ A/A[n] ∼ = nA✳ ❈❤ù♥❣ ♠✐♥❤ ✷✳✷✳ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ QP ●å✐ d : A → A ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ❞✭①✮ ❂ ♥①✳ ❑❤✐ ✤â✱ ❞✭① ✰ ②✮ ❂ ♥✭① ✰②✮ ❂ ♥① ✰ ♥② r ởt ỗ ỵ tự t t A/kerd ∼ = Imd✳ ❚❛ ❝â✱ x ∈ kerd ❦❤✐ ✈➔ tữỡ ữỡ ợ ✵✱ ❞♦ ✤â kerd = A[n]✳ ▼➦t ❦❤→❝✱ x ∈ Imd ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ① ❂ ❞✭②✮ ✈ỵ✐ ♠ët ❣✐→ trà ② ♥➔♦ ✤â ✤✐➲✉ ♥➔② ❝❤➾ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ① ❂ ♥② ✈ỵ✐ ♠ët ❣✐→ trà ② ♥➔♦ ✤â✱ ✈➟② ■♠❞ ❂ ♥❆✳ ❆❬♥❪ ✈➔ õ ỗ tớ t ởt ỗ s r A/A[n] ∼ = nA✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❆❬♥❪ ❧➔ ♥❤â♠ ❝♦♥ ❜➜t ❜✐➳♥ ❝õ❛ ❆✱ ❣å✐ x ∈ A[n] ✈➔ f : A A ởt ỗ ✤â✱ ♥① ❂ ✵✱ s✉② r❛ ♥❢✭①✮ ❂ ❢✭♥①✮ ❂ ❢✭✵✮ ❂ ✵✳ ❱➟② f (x) ∈ A[n]✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ♥❆ ❧➔ ♥❤â♠ ❝♦♥ ❜➜t ❜✐➳♥ ❤♦➔♥ t♦➔♥✱ ❣å✐ x ∈ nA ✈➔ f : A → A ❧➔ ởt ỗ õ ợ trà ② ♥➔♦ ✤â✱ s✉② r❛ f (x) = f (ny) = nf (y) ∈ nA✳ ❚➼♥❤ ❝❤➜t ✷✹✳ ❚❛ ❝â ❝→❝ ❦➳t q✉↔ s❛✉✿ ❛✮ B = ∩{pn A|n ∈ N} ✈➔ C = ∩{q n A|n ∈ N}✳ ❜✮ ❇ ❧➔ ❈ ❧➔ ❝→❝ ♥❤â♠ ❝♦♥ ❜➜t ❜✐➳♥ ❤♦➔♥ t♦➔♥ ❝õ❛ ❆ ✭❢✉❧❧② ✐♥✈❛r✐❛♥t s✉❜❣r♦✉♣ ♦❢ ❆✮✳ ❈❤ù♥❣ ỵ W = {pn A|n N }✳ ◆➳✉ (x, 0) ∈ B ✱ t❤➻ ✈ỵ✐ ♠ët số tỹ tỗ t y Qp s ❝❤♦ x = pn y ✱ ❞♦ Qp ❧➔ ♥❤â♠ ♣✲❝❤✐❛ ✤÷đ❝✳ ❉♦ ✤â✱ (x, 0) = (pn y, 0) = pn (y, 0) ∈ pn A✳ ❙✉② r❛✱ (x, 0) W B W rữợ ❦❤✐ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐✳ ✣➦t H = B+ (1/r, 0) ✱ K = C+ (0, 1/r) ✈➔ ❋ ❂ ❍ ✰ ❑✳ ◆❤➟♥ ①➨t r➡♥❣✱ ❍✱ ❑ ✈➔ ❋ ❦❤æ♥❣ ❧➔ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ ❆✳ ❚✉② ♥❤✐➯♥✱ ❝❤ó♥❣ ❧➔ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ ♥❤â♠ ✭Q × Q✱ ✰ ✮✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ❍♦♠ ✭❲✱ ❑✮ ❧➔ t➛♠ t❤÷í♥❣✳ ●✐↔ sû f ∈ Hom(W, K)✳ ●å✐ d : K → K ❧➔ ❤➔♠ sè t❤ä❛ ❞✭①✱②✮ ❂ r r r ự ữủ ỗ ❝➜✉✳ ◆➳✉ (x, y) ∈ K t❤➻ (x, y) = (0, a/q m ) + i(0, 1/r)✱ ✈ỵ✐ ❣✐→ trà ❛✱ ♠ ✈➔ ✐ ♥➔♦ ✤â✳ ❉♦ ✤â✱ d(x, y) = r(x, y) = (0, ar/q m ) + (0, i)✳ ❱➻ Z ⊆ Qp ✈➔ (0, i) ∈ C ✱ ♥➯♥ d(x, y) ∈ C ✳ ❙✉② r❛✱ ■♠❞ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ ❈✱ ❞♦ ✤â d ◦ f : W → C ✳ ❑➳t ❧✉➟♥ ❲ ❧➔ ♣✲❝❤✐❛ ✤÷đ❝ ❜ð✐ ♥➳✉ t ∈ W t❤➻ t ∈ pA✱ s✉② r❛✱ t ❂ ♣s ✈ỵ✐ s ∈ A✳ ❚✉② ♥❤✐➯♥✱ s ♣❤↔✐ t❤✉ë❝ ❲ ❞♦ ✈ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ ♥✱ t ∈ pn+1 A ❞➝♥ ✤➳♥✱ t = pn+1 a✱ ✈ỵ✐ a ∈ A✱ ❞♦ ✤â ps = pn+1 a ⇒ s = pn a✳ ❆ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ t❤❡♦ ❚➼♥❤ ❝❤➜t ✷✸✳ ❉♦ ✤â✱ s ∈ W ✈➔ t ❂ ♣s s✉② r❛ ❲ ❧➔ ♣✲❝❤✐❛ ✤÷đ❝✱ ♥➯♥ Imd ◦ f ❧➔ ♣✲❝❤✐❛ ✤÷đ❝✳ ❚✉② ♥❤✐➯♥✱ C ∼ = Qq ✈➔ Qq ❦❤æ♥❣ ❝â ♥❤â♠ ❝♦♥ ữủ ổ t tữớ ợ p = q ❱➻ t❤➳ Imd ◦ f ❧➔ t➛♠ t❤÷í♥❣✳ ❙✉② r❛ ❢ ❂ ✵✱ ✈ỵ✐ (x, y) ∈ W ✳ ❉♦ ✤â✱ d ◦ f (x, y) = 0✱ ✤✐➲✉ ♥➔② s✉② r❛ r❢✭①✱ ②✮ ❂ ✵✱ ♥➯♥ ❢✭①✱②✮ ❂ ✵ ❞♦ ❈ ❧➔ ❦❤æ♥❣ ①♦➢♥✳ ❱➟② ❢ ❂ ✵✱ ❤❛② ❍♦♠ ✭❲✱ ❑✮ ❧➔ t➛♠ t❤÷í♥❣✳ ◆❤➟♥ t❤➜② t➜t ❝↔ ❝→❝ t♦↕ ✤ë t❤ù ❤❛✐ ❝õ❛ ❝→❝ ♣❤➛♥ tû t❤✉ë❝ ❍ ❧➔ ❜➡♥❣ ✵✱ ✈➻ ❝❤ó♥❣ ❧➔ t♦↕ ✤ë t❤ù ♥❤➜t ❝→❝ ♣❤➛♥ tû ❝õ❛ ❑✳ ❉♦ ✤â H ∩ K = {(0, 0)}✱ s✉② r❛ F = H ⊕ K k : F K ỗ ❝➜✉ ♥❤ó♥❣✱ ❞♦ (1/r, 1/r) = (1/r, 0) + (0, 1/r) ✈➔ D ⊆ F ✳ ▼➦t ❦❤→❝✱ B, C ⊆ F ✈➻ B ⊆ H ✈➔ C ⊆ K ✳ ❙✉② r❛✱ A = B + C + D ⊆ F ✳ ●å✐ λ ❧➔ ❤↕♥ ❝❤➳ ❝õ❛ πk tr➯♥ ❲ ✭❞♦ W ⊆ F ✈➻ W ⊆ A✮✳ ❉♦ ✤â✱ λ ∈ Hom(W, K) ♥➯♥ λ = 0✱ ✈➻ Hom(W, K) = {0}✳ ✹✼ ❈❍×❒◆● ✷✳ ◆❍➶▼ QP ❱⑨ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ ◆⑨❨ ✹✽ ✣➸ ❝❤ù♥❣ ♠✐♥❤ W ⊆ B ✳ ❚❛ ❣å✐ (x, y) ∈ W ✳ ❑❤✐ ✤â✱ λ(x, y) = 0✳ ❱➻ (x, y) ∈ A ✈➔ ② ❂ ✵ ♥➯♥ (x, y) ∈ B ✱ ❤❛② W ⊆ B ✳ ❱➟② B = ∩{pn A|n ∈ N}✳ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ✤÷đ❝✱ C = ∩{q n A|n ∈ N}✳ ❜✮ ❙✉② ✤÷đ❝ trü❝ t✐➳♣ tø ✭❛✮✱ ❇ê ✤➲ ✷✳✷✳✸ ✈➔ ♥❤➟♥ ①➨t ♣❤➛♥ ❣✐❛♦ ❝õ❛ ❝→❝ ♥❤â♠ ❝♦♥ ❜➜t ❜✐➳♥ ❤♦➔♥ t♦➔♥ ❧➔ ♠ët ♥❤â♠ ❝♦♥ ❜➜t ❜✐➳♥✳ ❇ê ✤➲ ✷✳✷✳✹✳ ◆➳✉ G = AB ✈ỵ✐ A ∩ B = 1✱ ❆ ✈➔ ❇ ❧➔ ❝→❝ ♥❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ❝õ❛ ●✱ ❣✐↔ sû ❋ ❧➔ ♠ët ♥❤â♠ ❝♦♥ ❜➜t ❜✐➳♥ ❝õ❛ ● t❤➻ F = (F ∩ A)(F ∩ B)✳ ❈❤ù♥❣ ♠✐♥❤ ❉♦ ❆ ❧➔ ♥❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ❝õ❛ ●✱ t❤➻ F ∩ A ❧➔ ♥❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ❝õ❛ ❋✳ ❚÷ì♥❣ tü✱ F ∩ B ❧➔ ♥❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ❝õ❛ ❋✳ ❉♦ ❆✱ ❇ ❧➔ rí✐ ♥❤❛✉✱ ♥➯♥ F ∩ A ✈➔ F ∩ B ❧➔ ❝→❝ ♥❤â♠ rí✐ ♥❤❛✉✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ F = (F ∩ A)(F ∩ B)✳ ❚❛ ❝â✱ πA : G → A ✈➔ B : G B ỗ ✤â✱ πA [F ] ⊆ F ✈➔ πB [F ] ⊆ F ✱ ❞♦ t➼♥❤ ❜➜t ❜✐➳♥ ❝õ❛ ❋✳ ●å✐ x ∈ F t❤➻ x ∈ G ♥➯♥ ① ❂ ❛❜ ✈ỵ✐ a ∈ A, b ∈ B ✳ ❉♦ ✤â✱ a = πA (x)✳ ❚✉② ♥❤✐➯♥✱ πA (F ) ⊆ F ❞➝♥ ✤➳♥ a ∈ F ✳ ❙✉② r❛✱ a ∈ F ∩ A✱ t÷ì♥❣ tü b ∈ F ∩ B ✳ ▼➦t ❦❤→❝ ❞♦ ① ❂ ❛❜ ♥➯♥ x ∈ (F ∩ A)(F ∩ B)✱ ❤❛② F ⊆ (F ∩ A)(F ∩ B)✱ ❝❤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❤✐➸♥ ♥❤✐➯♥✳ ❱➟②✱ F = (F ∩ A)(F ∩ B)✳ ❚➼♥❤ ❝❤➜t ✷✺✳ ❆ ❧➔ ♥❤â♠ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝✳ ❈❤ù♥❣ ♠✐♥❤ ●✐↔ sû A = M L t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ▼ ❤♦➦❝ ▲ ❧➔ t➛♠ t❤÷í♥❣✳ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✷✹ ✈➔ ❇ê ✤➲ ✷✳✷✳✹ t❤➻ B = (M ∩ B)(L ∩ B) ✈➔ C = (M ∩ C)(L ∩ C)✳ ❚✉② ♥❤✐➯♥✱ ❝↔ ❇ ✈➔ ❈ ✤➲✉ ❧➔ ♥❤â♠ ❦❤æ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ✭✈➻ B ∼ = Qp ✱ C ∼ = Qq tr♦♥❣ ✤â✱ Qp ✈➔ Qq ❧➔ ❝→❝ ♥❤â♠ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝✮✳ ❉♦ ✤â✱ ❤♦➦❝ B ⊆ M ❤♦➦❝ B ⊆ L✱ t÷ì♥❣ tü ❤♦➦❝ C ⊆ M ❤♦➦❝ C ⊆ L ❞♦ t➼♥❤ ❦❤æ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ❝õ❛ ❇ ✈➔ ❈✳ ❚r♦♥❣ ✹ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❤➻ t❛ s➩ ❝❤➾ r❛ ✷ tr÷í♥❣ ❤đ♣ ❧➔ ❦❤ỉ♥❣ t❤➸ ①↔② r❛✳ ❚❤➟t ✈➟②✱ tr÷í♥❣ ❤đ♣ ✶✱ ❣✐↔ sû B ⊆ M ✈➔ C ⊆ L✳ ❑❤✐ ✤â✱ B + C ⊆ B + L✳ ❚ø ✤✐➲✉ ♥➔② ✈➔ ❆✴✭❇✰❈✮ ❧➔ ♥❤â♠ ①♦➢♥ ❞➝♥ ✤➳♥ ❆✴✭❇ ✰ ▲✮ ❧➔ ♥❤â♠ ①♦➢♥✳ ❍ì♥ t❤➳✱ B ⊆ M ∩ (B + L)✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ ❝â t ∈ M ∩ (B + L)✱ ✈ỵ✐ t ❧➔ ❝➦♣ s➢♣ t❤ù tü✱ t❤➻ t ❂ ❜ ✰ ❡ ✈ỵ✐ b ∈ B, e ∈ L✳ ❉♦ ✤â✱ e = t − b ∈ M ∩ L✱ s✉② r❛ ❡ ❂ ✵✱ ♥➯♥ t = b ∈ B ✈➔ M ∩(B +L) = B ✳ ❱➻ t❤➳✱ A/(B +L) = (M +L)/(B +L) = (M +(B +L))/(B +L) ∼ = M/(M ∩ (B + L)) = M/B ✳ ❙✉② r❛✱ ▼✴❇ ❧➔ ♥❤â♠ ①♦➢♥✳ ❚✉② ♥❤✐➯♥✱ ▼✴❇ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ ❆✴❇ t❤❡♦ ❚➼♥❤ ❝❤➜t ✷✸ t❤➻ ❆✴❇ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✱ ❞➝♥ ✤➳♥ ▼ ❂ ❇✳ ❚÷ì♥❣ tü ▲ ❂ ❈✳ ❉♦ ✤â✱ A B + C ổ ỵ trữớ ủ ♥➔② ❦❤ỉ♥❣ ①↔② r❛✳ Ð tr÷í♥❣ ❤đ♣ ✷✱ ♥➳✉ B ⊆ L ✈➔ C ⊆ M ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤đ♣ tr➯♥ t❛ ❝ơ♥❣ s✉② r❛ ✤✐➲✉ ✈ỉ ỵ rữớ ủ B, C M t B + C ⊆ M ✳ ❚ø ✤✐➲✉ ♥➔② ✈➔ ❆✴✭❇ ✰ ❈✮ ❧➔ ♥❤â♠ ①♦➢♥ ❞➝♥ ✤➳♥ ❆✴▼ ❧➔ ♥❤â♠ ①♦➢♥✳ ❉♦ A/M ∼ = L✱ s✉② r❛✱ ▲ ❧➔ ♥❤â♠ ①♦➢♥✳ ❚✉② ♥❤✐➯♥✱ ▲ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ❜ð✐ ✈➻ ▲ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ ❆✳ ❱➟② ▲ ♣❤↔✐ ❧➔ ♥❤â♠ t➛♠ t❤÷í♥❣✳ ✷✳✷✳ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ QP ❚r÷í♥❣ ❤đ♣ ❝✉è✐ ❝ị♥❣✱ ♥➳✉ B, C ⊆ L t❤➻ t÷ì♥❣ tü tr÷í♥❣ ❤đ♣ ✸ t❛ ❝ơ♥❣ s✉② r❛ ✤÷đ❝ ▼ ❧➔ ♥❤â♠ t➛♠ t❤÷í♥❣✳ ❱➟② ❆ ❧➔ ♥❤â♠ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝✳ ❚➼♥❤ ❝❤➜t ✷✻✳ ◆❤â♠ ❆ ❦❤ỉ♥❣ t❤➸ ♥❤ó♥❣ ✈➔♦ ♥❤â♠ ❝→❝ sè ❤ú✉ t✛ Q ✈ỵ✐ ♣❤➨♣ t♦→♥ ❝ë♥❣✳ ❈❤ù♥❣ ♠✐♥❤ ❉♦ ❇ ✰ ❈ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ ❆✱ ♥➯♥ ♥➳✉ ❆ ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ Q t❤➻ ❇ ✰ ❈ ❝ơ♥❣ ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ Q✳ ❚✉② ♥❤✐➯♥ B ∩ C = ∅ ✈➔ B + C = B ⊕ C ✳ ❉♦ ✤â ❇ ✰ ❈ ❧➔ ♥❤â♠ ♣❤➙♥ t➼❝❤ ✤÷đ❝✱ ♥❤÷♥❣ Q ✈➔ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ ♥â ❧➔ ♥❤â♠ ❦❤æ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝✳ ❱➟② ❆ ❦❤ỉ♥❣ t❤➸ ♥❤ó♥❣ ✈➔♦ Q✳ ❚➼♥❤ ❝❤➜t ✷✼✳ ❆ ❧➔ ♥❤â♠ rót ❣å♥✳ ❈❤ù♥❣ ♠✐♥❤ ●✐↔ sû ❍ ❧➔ ♥❤â♠ ❝♦♥ ❝❤✐❛ ✤÷đ❝ ❝õ❛ ❆✱ t❛ s ự õ t tữớ ợ sè tü ♥❤✐➯♥ ♥✱ H = pn H ✳ ❉♦ t➼♥❤ ❝❤✐❛ ✤÷đ❝ ❝õ❛ ❍ t❛ ❝â H ⊆ pn A✳ ❙✉② r❛ H ⊆ B ✭t❤❡♦ ❚➼♥❤ ❝❤➜t ✷✹✮✳ ❚÷ì♥❣ tü ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ H ⊆ C ✳ ❉♦ ✤â✱ H ⊆ B ∩ C = {0}✳ ❱➟② ❆ ❧➔ ♥❤â♠ rót ❣å♥✳ ◆❤➟♥ ①➨t✿ ◆❤â♠ Qp ❧➔ ♥❤â♠ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ✈➔ ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ Q✳ ❚✉② ♥❤✐➯♥✱ ♥❤â♠ ❆ ❧➔ ♠ët ♠ð rë♥❣ ❝õ❛ ♥❤â♠ Qp ❧➔ ♠ët ✈➼ ❞ư ✈➲ ❝→❝ ♥❤â♠ ❛❜❡♥ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ❦❤ỉ♥❣ ①♦➢♥ ✈➔ ❦❤ỉ♥❣ t❤➸ ♥❤ó♥❣ ✈➔♦ ♥❤â♠ ✭Q✱ ✰ ✮✳ ▼ët ♥❤â♠ ❛❜❡♥ ❦❤ỉ♥❣ ①♦➢♥ ✤÷đ❝ ❣å✐ ❧➔ ❝â ❤↕♥❣ ♥ ♥➳✉ ♥â ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ t➼❝❤ trü❝ t✐➳♣ ❝õ❛ ♥ t❤➔♥❤ ♣❤➛♥ ✭❝♦♣✐❡s✮ ❝õ❛ Q✱ ♥❤÷♥❣ ♥â ❦❤ỉ♥❣ t❤➸ ♥❤ó♥❣ ✈➔♦ t➼❝❤ trü❝ t✐➳♣ ❝õ❛ ♥ ✲✶ t❤➔♥❤ ♣❤➛♥ ❝õ❛ Q✳ ❉♦ ❆ ❧➔ ♥❤â♠ ❝â ❤↕♥❣ ❧➔ ✷✱ ❆ ❦❤ỉ♥❣ t❤➸ ♥❤ó♥❣ ✈➔♦ ♥❤â♠ ✭Q✱ ✰ ✮✳ ◆❤â♠ ❆ ❝á♥ ❧➔ ♠ët ✈➼ ❞ö ✈➲ ♥❤â♠ ❝♦♥ ❝õ❛ ♥❤â♠ ❛❜❡♥ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ❦❤ỉ♥❣ ♥❤➜t t❤✐➳t ❧➔ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ✭♥❤÷ ❚➼♥❤ ❝❤➜t ✷✻✱ t❛ ❝â ❇ ✰ ❈ ❧➔ ♥❤â♠ ❝♦♥ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ❝õ❛ ❆✮✳ ❚➼♥❤ ❝❤➜t ✷✽✳ ◆❤â♠ ❆ ❦❤ỉ♥❣ ❧➔ ♥❤â♠ ✤ì♥ t❤✉➛♥ tó②✳ ❈❤ù♥❣ ♠✐♥❤ ◆❤➟♥ t❤➜②✱ B ≤ A✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❇ ❧➔ õ t tú ổ t tữớ ợ (x, y) ∈ nA ∩ B t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ (x, y) ∈ nB ✳ ❚❤➟t ✈➟②✱ ♥➳✉ (x, y) ∈ nA ∩ B t❤➻ (x, y) ∈ nA ✈➔ (x, y) ∈ B ✳ ❚❛ ❝â✱ (x, y) = n[(c1 /pi , 0) + (0, c2 /q j ) + c3 (1/r, 1/r)] ✈➔ (x, y) ∈ B ✳ ❙✉② r❛✱ (x, y) = n(c1 /pi + c3 /r, c2 /q j + c3 /r) ỗ tớ x Qp , y = 0✳ ❉♦ ✤â✱ c2 /q j = −c3 /r ✈➔ x = n(c1 /pi + c3 /r)✳ ❱➻ (r, q j ) = ♥➯♥ r|c3 ✳ ❙✉② r❛✱ x = n(c1 /pi + c3 /r) ∈ Qp ✱ ❤❛② (x, y) ∈ nB ✱ ♥➯♥ ❇ ❧➔ ♥❤â♠ ❝♦♥ t❤✉➛♥ tó② ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ❆✳ ❱➟② ❆ ❦❤ỉ♥❣ ❧➔ ♥❤â♠ ✤ì♥ t❤✉➛♥ tó②✳ ✹✾ ❈❍×❒◆● ✷✳ ◆❍➶▼ QP ❱⑨ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ ◆⑨❨ ✺✵ ❚➼♥❤ ❝❤➜t ✷✾✳ ❆ ❧➔ ♥❤â♠ ❤♦♣❢✳ ❈❤ù♥❣ ♠✐♥❤ ●✐↔ sû ❝â ♠ët ♥❤â♠ ❝♦♥ ❑ t❤➟t sü ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ❆ ♠➔ A/K ∼ = A✳ ❑❤✐ ✤â✱ ❆✴❑ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ✭❞♦ ❆ ❧➔ ♥❤â♠ ❦❤ỉ♥❣ ①♦➢♥✮✳ ❑ ❝â t❤➸ ①❡♠ ♥❤÷ ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ Q × Q✳ ▼➦t ❦❤→❝✱ Q × Q ❧➔ ♥❤â♠ ❦❤ỉ♥❣ ①♦➢♥✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ (x, y) ∈ Q × Q✱ ❣✐↔ sû ✭①✱ ②✮ ❝â ❝➜♣ ♥ ❦❤✐ ✤â ✭♥①✱ ♥②✮ ❂ ✭✵✱ ✵✮✳ ❉♦ x ∈ Q ✈➔ y ∈ Q ✈➔ Q ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ ♥➯♥ ① ❂ ✵✱ ② ❂ ✵✳ ❙✉② r❛✱ ✭①✱ ②✮ ❂ ✭✵✱ ✵✮✳ ❱➟② Q × Q ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✱ ♥➯♥ ❑ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥✳ m k m k , ) ∈ K ✈➔ = 0, = 0✳ n l n l ❉♦ ✤â✱ m = 0, k = ✈➔ K + (x, y) ∈ (Q × Q)/K ✳ ▼➦t ❦❤→❝ ❞♦ x, y ∈ Q ♥➯♥ ① ❂ ♠✬✴♥✬ ✈➔ ② ❂ ❦✬✴❧✬ ✈ỵ✐ ❝→❝ sè ♥❣✉②➯♥ ♠✬❀ ♥✬❀ ❦✬❀ ❧✬ t❤ä❛ n = 0, l = 0✳ ❙✉② r❛✱ ▼➦t ❦❤→❝✱ (Q×Q)/K ❧➔ ♥❤â♠ ①♦➢♥✳ ❚❤➟t ✈➟②✱ ❣å✐ ( m k m k m k , , , + (x, y) = + n l n l n l = mn + m n kl + k l , nn ll ❉♦ ✤â✱ K + (x, y) ❝â ❝➜♣ ❤ú✉ ❤↕♥✱ ✈➻ (n , l )(K + (x, y)) = K + (n x, l y) ∈ K ✱ ✈ỵ✐ n = 0, l = 0✳ ❙✉② r❛✱ (Q × Q)/K ❧➔ ♥❤â♠ ①♦➢♥ ♥➯♥ ❆✴❑ ❧➔ ♥❤â♠ ①♦➢♥✳ ❙✉② r❛✱ ❆ ✈ø❛ ❧➔ ♥❤â♠ ①♦➢♥ ✈ø❛ ❧➔ ♥❤â♠ ❦❤æ♥❣ ①♦➢♥ s✉② r❛ ❆ ❧➔ ♥❤â♠ t➛♠ t❤÷í♥❣ ✭♠➙✉ t❤✉➝♥✮✳ ❱➟② ❑ ♣❤↔✐ ❧➔ ♥❤â♠ t➛♠ t❤÷í♥❣ ❤❛② ❆ ❧➔ ♥❤â♠ ❤♦♣❢✳ ❚➼♥❤ ❝❤➜t ✸✵✳ ◆❤â♠ ❆ ❦❤ỉ♥❣ ❧➔ ♥❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣ tø ✤â s✉② r❛ ❞➔♥ ❝→❝ ♥❤â♠ ❝♦♥ ▲✭❆✮ ❦❤æ♥❣ ❧➔ ❞➔♥ ♣❤➙♥ ♣❤è✐✳ ❈❤ù♥❣ ♠✐♥❤ ●å✐ u = (x, y) ∈ A ✈➔ v = (g, h) ∈ A✳ ❑❤✐ ✤â✱ (x, y) = ( c2 1 c1 , 0) + (0, j ) + c3 ( , ) i p q r r b1 b2 1 , 0) + (0, r ) + b3 ( , )✳ ◆➳✉ ❆ ❧➔ ♥❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣ t❤➻ ♥❤â♠ t p q r r s✐♥❤ ❜ð✐ {u, v} ❧➔ ♥❤â♠ ①②❝❧✐❝✳ ●✐↔ sû {u, v} = α t❤➻ α ♣❤↔✐ ❝â ❞↕♥❣ α = ( k , 0) + p 1 m n (0, l ) + ( , )✱ s✉② r❛ u = α ✈➔ v = α ✱ ✈ỵ✐ ♠✱ ♥ ❧➔ ❝→❝ sè ♥❣✉②➯♥✳ q r r ✈➔ (g, h) = ( ❚ø ✤â✱ t❛ ❝â ❝→❝ ❤➺ t❤ù❝ s❛✉✿ ( c1 c2 1 1 1 , 0) + (0, j ) + c3 ( , ) = (( k , 0) + (0, l ) + ( , ))m (1) i p q r r p q r r ( b1 b2 1 1 1 , 0) + (0, r ) + b3 ( , ) = (( k , 0) + (0, l ) + ( , ))n (2) t p q r r p q r r ✷✳✷✳ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ QP ✺✶ c1 c2 1 1 , 0)✱ b = (0, j )✱ c = c3 ( , )✱ g = ( k , 0)✱ h = (0, l ) ✈➔ w = ( , )✳ i p q r r p q r r ❚ø ✭✶✮ →♣ tr tự t ỗ t t ♠➝✉ sè ❧✉ÿ t❤ø❛ ❝õ❛ ❝→❝ sè ♥❣✉②➯♥ tè ♣✱ q✱ r t❛ ❝â✿ ✣➦t a = ( ✳ a + b + c = (g + h + w)m = g m + hm + wm + m−1 j m−j j w ✳ ❙✉② r❛ j=1 Cm−1 h a = b = c = m−1 i s i−s s m−i i Cm−1 Ci h w = i=1 s=0 g m−1 i=1 gm hm wm − i s=0 m−1 j=1 i Cis g m−i hi−s ws + Cm−1 j Cm−1 hm−j wj ❉♦ ♣✱ q✱ r ❧➔ ❝→❝ sè ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈➔ ð ❞á♥❣ ❝✉è✐ ❝ò♥❣ ❝õ❛ ❤➺ tr➯♥ t❤➻ ✈➳ tr→✐ ❝â ❝❤ù❛ t❤➔♥❤ ♣❤➛♥ ❧✉ÿ t❤ø❛ ❝õ❛ sè ♥❣✉②➯♥ tè ♣✱ ♥❤÷♥❣ ✈➳ ♣❤↔✐ ❦❤ỉ♥❣ ❝❤ù❛ t❤➔♥❤ ♣❤➛♥ ❝â ❧ô② t❤ø❛ ❝õ❛ sè ♥❣✉②➯♥ tè ♣ ✭♠➙✉ t❤✉➝♥✮✳ ❱➟② ✭❆✱ ✰ ✮ ❦❤ỉ♥❣ ❧➔ ♥❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣✳ ❚❤❡♦ t→❝ ❣✐↔ ❖r❡ ∅②st❡✐♥✳ ❙tr✉❝t✉r❡s ❛♥❞ ❣r♦✉♣ t❤❡♦r②✳ ■■✳ ❉✉❦❡ ▼❛t❤✳ ❏✳ ✹ ✭✶✾✸✽✮✱ ♥♦✳ ✷✱ ✷✹✼✕✷✻✾✳ t❤➻ ♠ët ♥❤â♠ ❧➔ ♥❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❞➔♥ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ ♥â ❧➔ ❞➔♥ ♣❤➙♥ ♣❤è✐✳ ❉♦ ❆ ❦❤ỉ♥❣ ❧➔ ♥❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣ ♥➯♥ ❞➔♥ ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ ❆ ❦❤æ♥❣ ❧➔ ❞➔♥ ♣❤➙♥ ♣❤è✐✳ ◆❣♦➔✐ r❛✱ ♥❣÷í✐ t❛ ❝á♥ t➻♠ ✤÷đ❝ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ♥❤â♠ ❆ ✈ỵ✐ ♥❤â♠ ♣✲Pr✉❢❢❡r q✉❛ ❝→❝ ❜✐➸✉ t❤ù❝ s❛✉✿ A/(C + D) ∼ = Z(p∞ ) ✈➔ A/(B + D) ∼ = Z(q ∞ )✳ ❈❍×❒◆● ✷✳ ◆❍➶▼ QP ❱⑨ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ ◆⑨❨ ✺✷ ❑➌❚ ▲❯❾◆ ▲ỵ♣ ❝→❝ ♥❤â♠ ❛❜❡♥ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤♦❛ ❤å❝ ✈➔ ❦ÿ t❤✉➟t✳ ❍✐➺♥ ♥❛②✱ ♠➦❝ ũ ợ õ ỳ ữủ ự ♠ët ❝→❝❤ ❤♦➔♥ ❝❤➾♥❤✱ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❧ỵ♣ ❝→❝ ♥❤â♠ ❛❜❡♥ ✈ỉ ❤↕♥ ✈➝♥ ❧➔ ❧➽♥❤ ✈ü❝ ✤❛♥❣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉✳ ❚r♦♥❣ ♣❤↕♠ ✈✐ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ t➟♣ tr✉♥❣ ❣✐ỵ✐ t❤✐➺✉ ✈➲ ♥❤â♠ Qp ✈ỵ✐ ♠ët số t t trữ ợ õ ✈æ ❤↕♥ ✈➔ tr➻♥❤ ❜➔② ♠ð rë♥❣ ❝õ❛ ♥❤â♠ ♥➔② tr♦♥❣ ♥❤â♠ ✭Q × Q✱ ✰✮✳ ❚ø ✈✐➺❝ s♦ s→♥❤ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥❤â♠ ❝ë♥❣ ❝→❝ sè ❤ú✉ t✛ ✭Q✱ ✰✮✱ ♥❤â♠ ❝♦♥ Q ✈➔ ♠ð rë♥❣ ❝õ❛ ♥â t❛ rót r❛ ✤÷đ❝ ♠ët sè ♥❤➟♥ ①➨t ✈➲ t➼♥❤ ✤â♥❣ ❝õ❛ ❝→❝ ❦❤→✐ ♥✐➺♠ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ♣❤➛♥ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❱➼ ❞ư ♥❤÷✿ ❑❤→✐ ♥✐➺♠ ♥❤â♠ ổ õ ợ ợ õ ữ ♥❤â♠ ❝♦♥ ❝õ❛ ♠ët ♥❤â♠ ❝❤✐❛ ✤÷đ❝ t❤➻ ❝❤÷❛ ❤➥♥ ❧➔ ♥❤â♠ ❝❤✐❛ ✤÷đ❝✱ ✈➼ ❞ư ♥❤â♠ ✭Z✱ ✰✮ ❦❤ỉ♥❣ ❧➔ ♥❤â♠ ❝❤✐❛ ✤÷đ❝ ♠➦❝ ❞ị ♥â ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ ♠ët ♥❤â♠ ❝❤✐❛ ✤÷đ❝ ✭Q✱ ✰✮✳ ❈→❝ ♥❤â♠ ❝♦♥ t❤ü❝ sü ❝õ❛ ♥❤â♠ ✭Q✱ ✰✮ ❧➔ ♥❤â♠ rót ❣å♥ ✈➔ ❝â t❤➦♥❣ ❞÷ ❤ú✉ ❤↕♥✱ ♥❤÷♥❣ ❜↔♥ t❤➙♥ ♥❤â♠ ❝ë♥❣ ❝→❝ sè ❤ú✉ t✛ ❧↕✐ ❦❤ỉ♥❣ ❧➔ ♥❤â♠ rót ❣å♥ ✈➔ ❦❤ỉ♥❣ ❝â t❤➦♥❣ ❞÷ ❤ú✉ ❤↕♥✳ ◆❣♦➔✐ r❛✱ ♥❤â♠ ✭Q✱ ✰✮ ✈ø❛ ❧➔ ♥❤â♠ ❤♦♣❢ ✈➔ ♥❤â♠ ❝♦❤♦♣❢ ữ t t õ ổ õ ố ợ ❧ỵ♣ ❝→❝ ♥❤â♠ ❝♦♥ ✈➼ ❞ư ♥❤â♠ Qp ✈➔ ♥❤â♠ ✭Z✱ ✰✮ ❦❤æ♥❣ ❧➔ ♥❤â♠ ❝♦❤♦♣❢✳ ❑❤→✐ ♥✐➺♠ ♥❤â♠ ❦❤æ♥❣ t ữủ ụ ổ õ ố ợ ợ ♥❤â♠ ❝♦♥ ❝ư t❤➸ ♥❤÷ ♥❤â♠ ❆ ❧➔ ♥❤â♠ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝✱ ♥❤÷♥❣ ♥❤â♠ ❝♦♥ t❤ü❝ sü ❇ ✰ ❈ ❝õ❛ ♥â ❧↕✐ ❧➔ ♥❤â♠ ♣❤➙♥ t➼❝❤ ✤÷đ❝✳ ◆❣♦➔✐ r❛✱ ♠ð rë♥❣ ❝õ❛ ♠ët ♥❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣ ❝❤÷❛ ❤➥♥ ❧➔ ♥❤â♠ ①②❝❧✐❝ ✤à❛ ♣❤÷ì♥❣ ✈➼ ❞ư ♥❤÷ ♥❤â♠ ❆✳ p ❚✉② ♥❤✐➯♥✱ ❞♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❝❤÷❛ t❤➸ tr➻♥❤ ❜➔② ✤÷đ❝ ♥❤✐➲✉ t➼♥❤ ❝❤➜t ❝õ❛ ❧ỵ♣ ❝→❝ ♥❤â♠ ❛❜❡♥ ✈ỉ ❤↕♥ ✈➔ ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ❧ỵ♣ ♥❤â♠ ♥➔② tr♦♥❣ ✤â ❝â ♥❤â♠ Qp ✈➔ ❝→❝ ♥❤â♠ ❝â ❧✐➯♥ q✉❛♥✳ ❱➻ t❤➳✱ ❝❤ó♥❣ tỉ✐ ữợ ự t t ❧➔✿ ✲ ❚➼♥❤ ❝❤➜t ✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ ♠ët sè ♥❤â♠ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ♥❤â♠ Qp ✈➔ ❝→❝ ♠ð rë♥❣ ❝õ❛ ♥❤â♠ ♥➔②✳ ✲ ▲ỵ♣ ❝→❝ ♥❤â♠ ❛❜❡♥ ✈ỉ ❤↕♥ t❤♦↔ ✤✐➲✉ ❦✐➺♥ ❆❈❈ ✈➔ ❉❈❈✳ ✲ ▲ỵ♣ ❝→❝ ♥❤â♠ ❛❜❡♥ ❤é♥ ❤đ♣✳ ✲ ❍↕♥❣ ❝õ❛ ♥❤â♠ ❛❜❡♥ ✈ỉ ❤↕♥ ❦❤æ♥❣ ①♦➢♥✳ ✷✳✷✳ ▼Ð ❘❐◆● ❈Õ❆ ◆❍➶▼ QP ✺✸ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❚✐➳♥❣ ❱✐➺t ✶✳ ❇ò✐ ❳✉➙♥ ❍↔✐✱ ❚rà♥❤ ❚❤❛♥❤ ✣➧♦ ✭✷✵✵✷✮✱ ✣↕✐ sè ❤✐➺♥ ✤↕✐✱ ◆❳❇ ✣↕✐ ố ố ỗ ✷✳ ❉❡r❡❝t ❏✳❙✳ ❘♦❜✐♥s♦♥ ✭✶✾✾✻✮✱ ❆ ❝♦✉rs❡ ♦♥ ❚❤❡♦r② ♦❢ ●r♦✉♣s✱ ❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣✱ ◆❡✇❨♦r❦✳ ✸✳ ❏♦s❡♣❤ ❏✳ ❘♦t♠❛♥ ✭✶✾✾✺✮✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ❚❤❡♦r② ♦❢ ●r♦✉♣s✱ ✹t❤ ❡❞✐t✐♦♥✱ ❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣✳ ✹✳ ▼✐❝❤❛❡❧ ❲❡✐♥st❡✐♥ ✭✷✵✵✼✮✱ ❊①❛♠♣❧❡s ♦❢ ❣r♦✉♣s✱ P♦❧②❣♦♥❛❧ P✉❜❧✐s❤✐♥❣ ❍♦✉s❡✳ ... r❛✱ a/pn ∈ (qQp ) + Z✱ ❤❛② Qp ⊆ (qQp ) + Z✳ ◆❣÷đ❝ ❧↕✐ t❤❡♦ ❚➼♥❤ ❝❤➜t ✺ ❝â Z ⊆ Qp ✳ ❉♦ q ❧➔ sè ♥❣✉②➯♥ tè ❦❤→❝ ♣ ♥➯♥ qQp ⊆ Qp ✳ ❙✉② r❛✱ qQp + Z ⊆ Qp ✳ ❱➟② qQp + Z = Qp ✳ ❜✮ ●✐↔ sû k ∈ qQp ∩ Z✳ ❑❤✐... x ∈ Z✳ ❱➟② Qp ∩ Qp = Z✳ ✷✳✶✳ ◆❍➶▼ QP ❜✮ ❚❛ ❝â Qp + Qp ⊆ Q ✭❤✐➸♥ ♥❤✐➯♥ ❞♦ ✈ỵ✐ x ∈ Qp + Qp ⇒ x = a/pn + a /b ợ q ỗ số ✈➳ ♣❤↔✐ t❛ ❧↕✐ ✤÷đ❝ ♠ët ♣❤➛♥ tû t❤✉ë❝ Q✮✳ ▼➦t ❦❤→❝✱ ❞♦ Qp ✈➔ Qp ❧➛♥ ❧÷đt... u = as/u + at/pn ∈ Qp + Qp ✱ s✉② r❛ Q ⊆ Qp + Qp ❱➟② Qp + Qp = Q✳ ❝✮ ❚❛ ❝â Z ⊆ Qp ✈➔ Z ⊆ Qq t❤❡♦ ❚➼♥❤ ❝❤➜t ✺ ♥➯♥ Z ⊆ Qp ∩ Qq ✭✈ỵ✐ ♣✱ q ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉✮✳ ▼➦t ❦❤→❝✱ ✈ỵ✐ x ∈ Qp ∩ Qq t❤➻ x = a/pn