Các số đại số và ứng dụng vào giải phương trình diophantine

TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN - NGUYỄN THỊ NGỌC HOA CÁC SỐ ĐẠI SỐ VÀ ỨNG DỤNG VÀO GIẢI PHƯƠNG TRÌNH DIOPHANTINE KHỐ LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Đại số Hà Nội, 2019 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN - NGUYỄN THỊ NGỌC HOA CÁC SỐ ĐẠI SỐ VÀ ỨNG DỤNG VÀO GIẢI PHƯƠNG TRÌNH DIOPHANTINE KHOÁ LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Đại số Người hướng dẫn khoa học TS Lê Quý Thường Hà Nội, 2019 ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✷ ▲í✐ ♥â✐ ✤➛✉ ✹ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✻ ✶✳✶ ❱➔♥❤ ✈➔ tr÷í♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ❱➔♥❤ ♥❤➙♥ tû ❤â❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ▼æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷ ❙ì ❧÷đ❝ ✈➲ sè ♥❣✉②➯♥ ✤↕✐ sè ✶✷ ✷✳✶ ❙è ♥❣✉②➯♥ ✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷ ❇❛♦ ✤â♥❣ ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸ ▼ð rë♥❣ ✤↕✐ sè ❝õ❛ ♠ët tr÷í♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✹ ▲✐➯♥ ❤đ♣ ❝õ❛ ♠ët ♣❤➛♥ tû tr➯♥ tr÷í♥❣ ✷✳✺ ❈→❝ ❧✐➯♥ ❤ñ♣ ❝õ❛ sè ♥❣✉②➯♥ ✤↕✐ sè ✷✳✻ K ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ❈→❝ sè ♥❣✉②➯♥ ✤↕✐ sè tr➯♥ ♠ët tr÷í♥❣ t♦➔♥ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ✸✸ ✸✳✶ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ x2 − 2y = ±1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ x2 − my = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✸ ▼ët sè tr÷í♥❣ ❤đ♣ ✈ỉ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ y − x3 = k ✸✳✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ y − x3 = k ✹✸ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❑➳t ❧✉➟♥ ✹✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✼ ỡ rữợ tr ❝õ❛ ❦❤â❛ ❧✉➟♥✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ tợ t ổ trữớ ữ P❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝æ tr♦♥❣ tê ❜ë ♠ỉ♥ ✣↕✐ sè ❝ơ♥❣ ♥❤÷ ❝→❝ t❤➛② ❝ỉ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ tr✐ t❤ù❝ qỵ t t ủ ❤♦➔♥ t❤➔♥❤ tèt ♥❤✐➺♠ ✈ö ❦❤â❛ ❤å❝ ✈➔ ❦❤â❛ ❧✉➟♥✳ ✣➦❝ ❜✐➺t✱ ❡♠ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ỏ t ỡ s s tợ ỵ ữớ ữớ trỹ t ữợ t t ❣✐ó♣ ✤ï ✤➸ ❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❉♦ t❤í✐ ❣✐❛♥✱ ♥➠♥❣ ❧ü❝ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❜↔♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✳ ❱➻ ✈➟②✱ ❡♠ r➜t ♠♦♥❣ ữủ ỳ ỵ õ ỵ qỵ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ◆❣å❝ ❍♦❛ ✸ ▲í✐ ♥â✐ ✤➛✉ P❤÷ì♥❣ tr➻♥❤ ♥❣❤✐➺♠ ♥❣✉②➯♥ ✭♣❤÷ì♥❣ tr➻♥❤ ❣✐↔✐ ✤÷đ❝ tr➯♥ t➟♣ ❤ñ♣ ❝→❝ sè ♥❣✉②➯♥✮ ❧➔ ♠ët ❜ë ♣❤➟♥ q✉❛♥ trå♥❣ ❝õ❛ t♦→♥ ❤å❝✳ ▲à❝❤ sû ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❧➔ ♠ët ❝✉ë❝ ❤➔♥❤ tr➻♥❤ ❧✐ ❦➻ ✈➔ ✤➛② ❤➜♣ ❞➝♥ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ❛✐ ②➯✉ t❤➼❝❤ t ữớ ỏ ợ ❜➔✐ t♦→♥ ❋❡r♠❛t✱ ❣✐ỵ✐ t♦→♥ ❤å❝ ♣❤↔✐ ♠➜t ❜❛ t❤➳ ❦➾ ✤➸ ✤÷❛ r❛ ❧í✐ ❣✐↔✐ trå♥ ✈➭♥ ❝❤♦ ❜➔✐ t♦→♥ ♥➔②✳ ❚✉② ♥❤✐➯♥ ❜➔✐ t♦→♥ ❋❡r♠❛t ✤â ❝❤➾ ❧➔ ♠ët ♣❤➛♥ tr♦♥❣ ❝❤✉é✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❉✐♦♣❤❛♥t✐♥❡✳ ✭❉✐♦♣❤❛♥t✐♥❡ ❧➔ ♠ët ♥❤➔ t♦→♥ ❤å❝ ♥❣÷í✐ ❍② ▲↕♣✱ ỉ♥❣ sè♥❣ ð ❆❧❡①❛♥❞r✐❛ ✈➔♦ t❤➳ ❦➾ t❤ù ■■■✱ ♥❣÷í✐ ❝❤✉②➯♥ ự ữỡ tr ự trữợ ởt ❝ỉ♥❣ tr➻♥❤ ✈➽ ✤↕✐ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ❉✐♦♣❤❛♥t✐♥❡✱ ♠ët ❝➙✉ ❤ä✐ ❧✉ỉ♥ ❧✉ỉ♥ ✤÷đ❝ ✤➦t r❛✿ ✏❈â t❤✉➟t t♦→♥ ❝❤✉♥❣ ♥➔♦ ✤➸ ❣✐↔✐ t➜t ❝↔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡❄✑ ◗✉↔ t❤ü❝ ✤➙② ❧➔ ♠ët ✤✐➲✉ ❦❤â ❦❤➠♥✳ ố ợ ởt số ợ ữỡ tr t t❛ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ ❜➡♥❣ ✈✐➺❝ →♣ ❞ư♥❣ sè ✤↕✐ sè✳ ❱➟② ❝❤ó♥❣ t❛ ❝â t❤➸ sû ❞ư♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ♥➔♦ ❝õ❛ sè ✤↕✐ sè ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ✈➔ ❝→❝❤ ❣✐↔✐ ♠ët sè ❧ỵ♣ ❉✐♦♣❤❛♥t✐♥❡ ➜② ♥❤÷ t❤➳ ♥➔♦❄ ✣➸ ❣✐↔✐ q✉②➳t ❝➙✉ ❤ä✐ ♥➔② ❝❤ó♥❣ tæ✐ ✤➣ ❧ü❛ ❝❤å♥ ❝❤õ ✤➲ ✏ ❈→❝ sè ✤↕✐ sè ✈➔ ù♥❣ ❞ư♥❣ ✈➔♦ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✑✳ ◆ë✐ õ ỗ ữỡ ữỡ ✏❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✑✳ ❈❤÷ì♥❣ ♥➔② ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ✈➔♥❤✱ tr÷í♥❣ ✈➔ ♠ỉ✤✉♥✱ ❧➔♠ ♥➲♥ t↔♥❣ ❝❤✉➞♥ ❜à ❝❤♦ ♥ë✐ ❞✉♥❣ ❝õ❛ ❝→❝ ❝❤÷ì♥❣ t✐➳♣ t❤❡♦✳ ✹ ✺ ❈❤÷ì♥❣ ✷✿ ✏❙ì ❧÷đ❝ ✈➲ sè ♥❣✉②➯♥ ✤↕✐ sè✑✳ ❈❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ♥ë✐ ❞✉♥❣ ✈➲ ❝→❝ sè ✤↕✐ sè✳ ✣➙② ❧➔ ❝→❝ ❦✐➳♥ t❤ù❝ q✉❛♥ trå♥❣ ❣✐ó♣ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ ❝õ❛ ❝❤÷ì♥❣ s❛✉✳ ❈❤÷ì♥❣ ✸✿ ✏❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✑✳ ✣➙② ❧➔ ❝❤÷ì♥❣ trå♥❣ t➙♠ ❝õ❛ ❦❤â❛ ❧✉➟♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✈➟♥ ❞ư♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ✤➣ ❝❤✉➞♥ ữỡ trữợ ự sỹ tỗ t ởt số ợ ữỡ tr ❉✐♦♣❤❛♥t✐♥❡✳ ❉♦ t❤í✐ ❣✐❛♥✱ ♥➠♥❣ ❧ü❝ ✈➔ ✤✐➲✉ ❦✐➺♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t sõt qỵ t ổ õ õ õ ỵ qỵ õ ❝õ❛ ❝❤ó♥❣ tỉ✐ ❝â t❤➸ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚r➙♥ trå♥❣✳ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ✤➣ ♥❤➢❝ ❧↕✐ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ✈➔♥❤✱ ✐✤➯❛♥✱ ♠✐➲♥ ♥❣✉②➯♥ ✈➔ tr÷í♥❣✱ ✈➔♥❤ ♥❤➙♥ tû ❤â❛ ✈➔ ♠ỉ✤✉♥✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ❧➔ ❬✶❪✳ ✶✳✶ ❱➔♥❤ ✈➔ tr÷í♥❣ ▼ỉt ✈➔♥❤ ❧➔ ♠ët t➟♣ ❤đ♣ R= ữủ tr t ổ P❤➨♣ ❝ë♥❣✿ + : R × R → R✱ (x, y) → x + y ✱ • P❤➨♣ ♥❤➙♥✿ · : R × R → R✱ (x, y) → x · y = xy ✱ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙②✿ ✭❛✮ (R, +) ❧➔ ♠æt ♥❤â♠ ❛❜❡❧❀ ✭❜✮ P❤➨♣ ♥❤➙♥ ❝â t➼♥❤ ❝❤➜t ❦➳t ❤ñ♣❀ ✭❝✮ P❤➨♣ ♥❤➙♥ ♣❤➙♥ ♣❤è✐ ✈➲ ❤❛✐ ♣❤➼❛ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣✿ (x + y)z = xz + yz, ✈ỵ✐ ♠å✐ x, y, z t❤✉ë❝ R✳ ✻ z(x + y) = zx + zy ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ✶✳ ❱➔♥❤ ❣å✐ ❧➔ R ❣✐❛♦ ❤♦→♥ ✤÷đ❝ ❣å✐ ❧➔ ❝â ✤ì♥ ✈à ✼ ♥➳✉ ♣❤➨♣ ♥❤➙♥ ❣✐❛♦ ❤♦→♥✳ ❱➔♥❤ R ✤÷đ❝ ♥➳✉ ♣❤➨♣ õ õ ỡ tự tỗ t ♠ët ♣❤➛♥ tû 1∈R · x = x · = x ổ õ ữợ ổ s ❝❤♦ ♥➳✉ x ✈ỵ✐ ♠å✐ xy = t❤✉ë❝ R✳ ❱➔♥❤ x=0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ R ✤÷đ❝ ❣å✐ y = 0✳ ❤♦➦❝ ▼ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ ❝â ✤ì♥ ✈à ổ õ ữợ ổ ữủ ởt ♠✐➲♥ ♥❣✉②➯♥✳ ▼ët ♠✐➲♥ ♥❣✉②➯♥ ♠➔ ♠å✐ ♣❤➛♥ tû ❦❤→❝ ❦❤ỉ♥❣ ✤➲✉ ❦❤↔ ♥❣❤à❝❤ ✤÷đ❝ ❣å✐ ❧➔ ♠ët tr÷í♥❣✳ ❚➟♣ ❤đ♣ ❝→❝ sè ♥❣✉②➯♥ Z ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥ t❤ỉ♥❣ t❤÷í♥❣ ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥✳ ❈→❝ t➟♣ ❤đ♣ Q✱ R ✈➔ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥ C tổ tữớ trữớ ủ Z/n ợ ỗ ữ số [x] + [y] := [x + y]✱ [x][y] := [xy] ♣❤➨♣ t♦→♥ ✈à ❝â ❤ú✉ ❤↕♥ ♣❤➛♥ tû✳ ◆â ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ❝→❝ sè ♥❣✉②➯♥ ♠♦❞✉❧♦ n✳ ❱➔♥❤ ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Z/n ❧➔ ♠✐➲♥ ♥❣✉②➯♥ ❦❤✐ ✈➔ ❝❤➾ õ ởt trữớ n ũ ợ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ Z/n ❚➟♣ ❤đ♣ ❝→❝ ✤❛ t❤ù❝ n n ❧➔ ♠ët sè ♥❣✉②➯♥ tè✳ ❍ì♥ ♥ú❛✱ ❜✐➳♥ ✈ỵ✐ ❤➺ sè tr➯♥ ♠ët ✈➔♥❤ R ❝ò♥❣ ợ tự tổ tữớ ❧➟♣ t❤➔♥❤ ♠ët ✈➔♥❤✱ ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ✤❛ t❤ù❝ ♥❣✉②➯♥ t❤➻ ▼ët n ❜✐➳♥ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ R[x1 , , xn ] ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ✐✤➯❛♥ tr→✐ ❝õ❛ ✈➔♥❤ R ❤➜♣ t❤ö ✤è✐ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ tø ❜➯♥ tr→✐✱ tù❝ ❧➔ a ∈ I✳ ▼ët ✐✤➯❛♥ ♣❤↔✐ ❝õ❛ ✈➔♥❤ R ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ tø ❜➯♥ ♣❤↔✐✱ tù❝ ❧➔ ❝õ❛ R✳ R ❧➔ ♠ët ♠✐➲♥ ∈ I ❧➔ ♠ët ✈➔♥❤ ❝♦♥ ar ∈ I ◆➳✉ R R ✤â♥❣ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❧➔ ♠ët ✈➔♥❤ ❝♦♥ ✈ỵ✐ ♠å✐ ✈ø❛ ❧➔ ♠ët ✐✤➯❛♥ tr→✐ ✈ø❛ ❧➔ ♠ët ✐✤➯❛♥ ♣❤↔✐ ❝õ❛ ✐✤➯❛♥ ✭❤❛✐ ♣❤➼❛✮ ◆➳✉ ❝ô♥❣ ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥✳ ✈➔♥❤ ❝♦♥ ❝õ❛ ♠ët ✈➔♥❤ R ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥✳ ▼ët R[x1 , , xn ]✳ I ✈ỵ✐ ♠å✐ ❝â t➼♥❤ r ∈R ✈➔ ❝â t➼♥❤ ❤➜♣ t❤ö ✤è✐ r∈R R I ✈➔ a ∈ I✳ ◆➳✉ I t❤➻ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ t❤➻ ❝→❝ ❦❤→✐ ♥✐➺♠ ✐✤➯❛♥ tr→✐ ✈➔ ✐✤➯❛♥ ♣❤↔✐ trò♥❣ ♥❤❛✉✳ ❈❤♦ S ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ ✈➔♥❤ R✳ ▼ët ✐✤➯❛♥ I ✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ s✐♥❤ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ✶✳ ❜ð✐ S ✱ I = (S)✱ ❦➼ ❤✐➺✉ ✤÷ì♥❣✱ (S) S = {a}✱ a ∈ R✱ ✐✤➯❛♥ ❝❤➼♥❤ ❝õ❛ ữủ ởt ợ n Z R ❧➔ ✐✤➯❛♥ ♥❤ä ♥❤➜t ❝❤ù❛ t❤➻ t❛ ✈✐➳t R S✳ ◆➳✉ t❤❛② ✈➻ ✈✐➳t ❧➔ ♠ët ♠✐➲♥ ❝❤➼♥❤✱ ♠å✐ ✐✤➯❛♥ tự õ tr S ỗ ({a})✱ ✈➔ ❣å✐ K[X]✱ K[X] ✈ỵ✐ K ✤➲✉ ❝â ❞↕♥❣ Z ✤➲✉ ❝â ❞↕♥❣ ❧➔ ♠ët tr÷í♥❣✱ ❝ơ♥❣ ❧➔ ♠ët (f (X))✱ ✐✤➯❛♥ s✐♥❤ ❜ð✐ ♠ët K[X]✳ ♥❣✉②➯♥ tè ♥➳✉ M R ▼ët ✐✤➯❛♥ t↕✐ ♠ët ✐✤➯❛♥ I ❝õ❛ (a) (n) = nZ✱ ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✈➔ ❝â ✤ì♥ ✈à✳ ❑❤✐ ✤â ♠ët ✐✤➯❛♥ ✤÷đ❝ ❣å✐ ❧➔ y ∈ P✳ (a) ❝❤ù❛ ▼ët ❝→❝❤ t÷ì♥❣ ▼ët ♠✐➲♥ ♥❣✉②➯♥ ♠➔ ♠å✐ ✐✤➯❛♥ ✤➲✉ ❧➔ ✐✤➯❛♥ ❧➔ ♠ët sè tü ♥❤✐➯♥✳ ❱➔♥❤ ●✐↔ sû R S✳ ♠✐➲♥ ❝❤➼♥❤✳ ♠✐➲♥ ❝❤➼♥❤✱ ♠å✐ ✐✤➯❛♥ ❝õ❛ R I ♥➳✉ ❧➔ ❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ❝õ❛ ♠ët ♣❤➛♥ tû ❧➔ ♠ët ✽ ❝õ❛ P =R ✈➔ ✤÷đ❝ ❣å✐ ❧➔ xy ∈ P ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝ü❝ ✤↕✐ ♥➳✉ M =R P xP ổ tỗ R s I = M ✱ I = R ✈➔ M ⊆ I R ú ỵ r ỹ ❧➔ ♥❣✉②➯♥ tè✳ ✣✐➲✉ ♥❣÷đ❝ ❧↕✐ ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ ✤ó♥❣✱ ữ õ ú ợ R Z K[X] ợ K ❧➔ ♠ët tr÷í♥❣✳ ▼➺♥❤ ✤➲ ✶✳✶✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✈➔ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ❑❤✐ ✤â ✭❛✮ I ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ R/I ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥❀ ✭❜✮ I ❧➔ ✐✤➯❛♥ ❝ü❝ ✤↕✐ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ R/I ❧➔ ♠ët tr÷í♥❣✳ ❱➼ ❞ư✱ ✤❛ t❤ù❝ (X + 1) ❜ð✐ ✈➻ K tr♦♥❣ R[X] ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ t❤ü❝ ♥➯♥ ✐✤➯❛♥ ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ ✈➔♥❤ R/(X + 1) ◆➳✉ X2 + ✤➥♥❣ ❝➜✉ ✈ỵ✐ C✱ R[X]✳ ■✤➯❛♥ ♥➔② ❝ơ♥❣ ❝ü❝ ✤↕✐ ❧➔ ♠ët tr÷í♥❣✳ ❧➔ ♠ët tr÷í♥❣ ❤ú✉ ❤↕♥✱ ♥â ❝â t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ s❛✉ ✤➙②✳ ▼➺♥❤ ✤➲ ✶✳✷✳ ❈❤♦ K ởt trữớ ỳ õ tỗ t ởt số ♥❣✉②➯♥ tè p ✈➔ ♠ët sè tü ♥❤✐➯♥ n s❛♦ ❝❤♦ |K| = pn ✳ ❇➔✐ t♦→♥ ①➙② ❞ü♥❣ ♠ët tr÷í♥❣ ❝â pn ♣❤➛♥ tû ❝â t❤➸ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ♥❤í ▼➺♥❤ ✤➲ ✶✳✶✳ ❚❤➟t ✈➟②✱ ①➨t ✈➔♥❤ ✤❛ t❤ù❝ (Z/p)[X] ✈➔ ①➙② ❞ü♥❣ ♠ët ✤❛ ❈❤÷ì♥❣ ✸✳ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ●✐↔ sû ♠å✐ √ σ : Q( 2) → C x ∈ Q✳ ❚❤➟t ✈➟②✱ ✈✐➳t ✸✹ ❧➔ ♠ët ✤ì♥ ❝➜✉ ❜➜t ❦➻✳ ❑❤✐ ✤â x ữợ tố ũ õ t ❝❤å♥ x= p q ✱ ✈ỵ✐ p, q q = 0✱ p > 0✳ σ(x) = x ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❉♦ x ❧➔ tê♥❣ ❝õ❛ σ(x) = pσ( 1q ) = pq = x ✭✈➻ qσ( 1q ) = σ(1) = 1✮✳ √ √ P√2 (X) = X − ∈ Q( 2) ❧➔ ✤❛ t❤ù❝ tè✐ t✐➸✉ ❝õ❛ ♥➯♥ p sè ❤↕♥❣ ✈ỵ✐ q ♥➯♥ ❱➻ P√2 (σ(X)) = σ(X)2 − 2σ(1) (✈➻ σ(1) = 1) = σ(X ) − σ(2) (✈➻ 2σ(1) = σ(1 + 1) = σ(2)) = σ(X − 2) = σ(P√2 (X)) √ √ √ P√2 (σ( 2)) = σ(P√2 ( 2)) = σ(0) = 0✱ ❤❛② σ( 2) ❧➔ ♠ët √ √ √ √ 2✱ t❛ ❝â ♥❣❤✐➺♠ ❝õ❛ P√2 (X)✳ ❉♦ ✤â σ( 2) = ± ✳ ◆➳✉ σ( 2) = √ √ √ √ √ σ(x+y 2) = σ(x)+σ(y) = x+y 2✱ tù❝ ❧➔ σ = σ1 ✳ ◆➳✉ σ( 2) = − √ √ √ t❤➻ σ(x+y 2) = σ(x)+σ(y)(− 2) = x−y 2✱ tù❝ ❧➔ σ = σ2 ✳ ❱➟② σ1 = Id √ ✈➔ σ2 ❧➔ t➜t ❝↔ ❝→❝ ✤ì♥ ❝➜✉ tr÷í♥❣ tø Q( 2) ✈➔♦ C✳ ◆❤÷ ✈➟② ✣à♥❤ ỵ s õ trỏ q trồ tr ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♥❣❤✐➺♠ ♥❣✉②➯♥ x2 − 2y = ỵ tỷ Z + Z√2 ❝â ❞↕♥❣ ±(1 + √2)n✱ ✈ỵ✐ n ∈ Z✳ √ U (Z + Z 2) ❧➔ ♥❤â♠ ỗ tt tỷ Z + Z rữợ t t ự r ổ tỗ t U (Z + Z 2) s❛♦ ❝❤♦ ❈❤ù♥❣ ♠✐♥❤✳ ❑➼ ❤✐➺✉ 1 t tr t õ tỗ t n ∈ N>0 η √ n √ −n s❛♦ ❝❤♦ = (1 + 2) s✉② r❛ η = (1 + 2) ✳ η √ √ ◆➳✉ η = t❤➻ t❛ ❝â t❤➸ ✈✐➳t η = (1 + 2) ✳ ❱➟② ♠å✐ η ∈ U (Z + Z 2) √ n ✈➔ η > t❤➻ η ❝â ❞↕♥❣✿ η = (1 + 2) , n ∈ Z✳ ❉➵ t❤➜② η(1 + √ η(1 + ❈❤÷ì♥❣ ✸✳ ◆➳✉ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ √ η ∈ U (Z + Z 2) −1 < η < ✈➔ t❤➻ ✸✻ −1 η √ √ ∈ U (Z + Z 2) ✈➔ −1 η √ > 1✳ n ∈ N>0 s❛♦ ❝❤♦ −1 2)n tù❝ ❧➔ η = −(1 + 2)−n ✳ η = (1 + √ √ −1 ∈ U ( Z + Z 2) ✈➔ < −1 ◆➳✉ η ∈ U (Z + Z 2) ✈➔ η < −1 t❤➻ η η < 1✳ √ r ữủ r tỗ t n N>0 s❛♦ ❝❤♦ 2)−n ✱ tù❝ ❧➔ η = (1 + √ η = −(1 + 2)n ✳ √ n √ ❱➟② ✈ỵ✐ ♠å✐ η ∈ U (Z + Z 2)✱ tỗ t n Z s = (1+ 2) r tỗ t ỵ ▼å✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ x2 − 2y2 = ❝â ❞↕♥❣✿ m m−1 k ± (x, y) = 2k C2m ,± k=0 2k+1 2k C2m , m ∈ N k=0 ❜✮ ▼å✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ x2 − 2y = −1 ❝â ❞↕♥❣✿ m ± (x, y) = ❈❤ù♥❣ ♠✐♥❤✳ m−1 k k=0 ❚❛ ự ỵ Pữỡ ự þ √ √ x + y ∈ U (Z + Z 2)✳ √ x2 − 2y = t❤➻ √ ±(1 + 2)n ✱ ✈ỵ✐ n ∈ Z✳ (x, y) ❧➔ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ √ x + y ❝â ❞↕♥❣ √ n ❝õ❛ ±(1 + 2) ✱ ❣♦♠ ❉♦ ✤â ❑❤❛✐ tr✐➸♥ ♥❤à t❤ù❝ ◆❡✇t♦♥ ✈➔ ❤➺ sè ❝õ❛ 2k+1 2k C2m+1 , m ∈ N k=0 ❜✮ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü✳ ◆➳✉ x✮ 2k C2m+1 ,± ✭t❤✉ ✤÷đ❝ y ✮✳ (x, y) = n = 2m✱ m ∈ N✱ ◆➳✉ m ❤➺ sè tü ❞♦ ✭t❤✉ ✤÷đ❝ m−1 ± k 2k C2m ,± k=0 2k+1 2k C2m , m ∈ N k=0 ❚❤û ❧↕✐ t❤➻ ✤➙② ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ n = 2m + 1✱ m ∈ N✱ t❤➻ x2 − 2y = 1✳ ◆➳✉ t❤➻ m m−1 k 2k C2m < k=0 ❜ð✐ ✈➻ ♥â ✤÷đ❝ s✉② r❛ tø k 2k+1 C2m+1 2, k=0 √ ( − 1)2m+1 > 0✳ m−1 (x, y) = √ ± m−1 k=0 ❈❤➼♥❤ ✈➻ ✤✐➲✉ ♥➔② ♠➔ ❝→❝ ❝➦♣ k 2k C2m+1 ,± 2k+1 2k C2m+1 k=0 ❈❤÷ì♥❣ ✸✳ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✸✼ x2 − 2y = 1✳ ❇➡♥❣ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü t❛ t❤➙② t➜t ❝↔ ❝→❝ tr÷í♥❣ ❤ñ♣ m∈N ✈➔ n = −2m + 1✱ m ∈ N✱ ♥❣❤✐➺♠ ❝õ❛ x2 − 2y = 1✳ x2 − 2y = ❝→❝ ❣✐→ trà (x, y) n = −2m✱ t➻♠ ✤÷đ❝ ✤➲✉ ❦❤ỉ♥❣ ❧➔ ❱➟② ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ❝â ❞↕♥❣ ♥❤÷ tr♦♥❣ ♣❤➛♥ ỵ m = ✱ t❛ ❝â✿ x = C60 + 2C62 + 4C64 + 8C66 = + 30 + 60 + = 99, y = C61 + 2C63 + 4C65 = + 40 + 24 = 70 ❱➟② (±99, ±70) ❧➔ ♠ët sè ♥❣❤✐➺♠ ❝õ❛ x2 − 2y = 1✳ ✸✳✷ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ x2 − my2 = m N>0 m ổ õ ữợ ữỡ ỵ sỷ m N>0 ổ õ ữợ ữỡ õ tỗ t (x, y) ∈ Z × Z \ {(±1, 0)} s❛♦ ❝❤♦ x2 − my = 1✳ ❈❤ù♥❣ ♠✐♥❤✳ ❝❤♦✿ ●✐↔ sû √ < |x − y m| < ❝õ❛ ❝→❝ ❦❤♦↔♥❣ ♥û❛ ♠ð ✤➦t N ∈ N>0 ✳ N ✈➔ 0 xt − yt m✳ ❦❤♦↔♥❣ ❈❤÷ì♥❣ ✸✳ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ❑❤✐ ➜②✱ t❛ ❝â y s − yt > √ (xs − xt ) − (ys − yt ) m < ✸✽ N ✈➔ ys − yt = s − t = 0✳ ◆➳✉ t❤➻ ✤➦t   x = xs − xt  y = ys − yt , ✈➔ ♥➳✉ ys − yt < t❤➻ ✤➦t   x = xt − xs  y = yt − ys √ < |x − y m| < N✱ y > ✈➔ y = |ys − yt | N s ự 2 tỗ t↕✐ ✈ỉ sè (x, y) ∈ (Z × Z) \ (Z × 0) s❛♦ ❝❤♦ < |x − my | < + m✳ √ ●✐↔ sû N1 N>0 tỗ t x1 , y1 Z s❛♦ ❝❤♦ < |x1 − y1 m| < ✈➔ N1 < y1 N ✳ ▲➜② N2 ∈ N>0 ✈➔ N2 > |x1 −y11 √m| > N1 ✳ ự tr tỗ t x2 , y2 ∈ N s❛♦ ❝❤♦ < |x2 − y2 m| < N2 ✳ N2 ✈➔ < y2 ❑❤✐ ✤â ❚÷ì♥❣ tü✱ t❛ t❤✉ ✤÷đ❝ ♠ët ❞➣② ❝→❝ sè ♥❣✉②➯♥ Nk > 0✱ xk ✱ yk ✭k ∈ N>0 ) s❛♦ ❝❤♦ √ √ 1 < |xk−1 − yk−1 m| < < |xk − yk m| < Nk Nk−1 √ 1 ✈➔ < yk < Nk ✳ ❙✉② r❛ < |xk − yk m| < Nk yk , ✈ỵ✐ ♠å✐ k t❤✉ë❝ N>0 ✳ √ √ √ ▼➦t ❦❤→❝✱ xk + yk m = xk − yk m + 2yk m ♥➯♥ t❛ ❝â √ |xk + yk m| √ √ √ |xk − yk m| + 2yk m < + 2yk m y ❙✉② r❛ √ √ √ √ √ < |x2k − yk2 m| = |(xk − yk m)(xk + yk m)| < + m + m yk √ ◗✉❛♥ s→t ❣✐ú❛ ✈➔ + m ❝❤➾ ❝â ❤ú✉ ❤↕♥ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ tr♦♥❣ ❦❤✐ ❝â ✈ỉ sè (x, y) ∈ (Z × Z) \ (Z × {0}) s❛♦ ❝❤♦ |x2 − my | số ữỡ õ tỗ t↕✐ τ ∈ Z, |τ | ∈ (0, + m) ∩ N s❛♦ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❈❤÷ì♥❣ ✸✳ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ❉✐♦♣❤❛♥t✐♥❡ x2 − my = τ ♥❣✉②➯♥ ❞÷ì♥❣✳ ●å✐ T ∈Z ❝â ✈æ sè ♥❣❤✐➺♠✱ ♥â✐ r✐➯♥❣ ♥â ❝â ✈æ sè ♥❣❤✐➺♠ s❛♦ ❝❤♦ |T | = min{|τ | | x2 − my = τ ❝â ✈ỉ sè ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣} s ự ữợ ợ t x2 − my = T ✳ ❝õ❛ T˜, y˜ ∈ Z (T, y) =✶ ✈ỵ✐ ✈ỉ sè ♥❣❤✐➺♠ s❛♦ ❝❤♦ (x, y) T = T˜d, y = y˜d ❝õ❛ ✈➔ x2 − my = T ✳ (T˜, y˜) = 1✳ (x, y) (T, y) = d > ❚❤➟t sỷ ữợ ợ t ố ợ ổ số ữỡ t r tỗ t ♣❤÷ì♥❣ tr➻♥❤ x2 − md2 y˜2 = T˜d✳ ●å✐ p ởt ữợ tố d d > ♥➯♥ p ❚❛ ❝â 2✳ T p2 ❝â ✈æ sè x2 ❝❤✐❛ ❤➳t ❝❤♦ p✱ s✉② r❛ x ❝❤✐❛ ❤➳t ❝❤♦ p✱ ♥➯♥ x2 −my = ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❤➟t ✈➟②✱ ♥➳✉ (x0 , y0 ) ❧➔ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ x2 − my = T ❝❤✐❛ ❤➳t ❝❤♦ ❝❤♦ (y0 , T ) = d✱ d s❛♦ ❝❤♦ p✱ d > p ✈➔ ( xp0 , yp0 ) ❧➔ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ x2 − my = |T |✳ ♥➯♥ ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ♥❤ä ♥❤➜t ❝õ❛ (T, y) = ✤è✐ ✈ỵ✐ ✈ỉ sè ♥❣❤✐➺♠ ♠ët ♥❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ (x, y) x2 − my = T ❝õ❛ t❤➻ x0 ❝❤✐❛ ❤➳t |T | T p2 ✳ ❉♦ p < |T | ❙✉② r ữợ ợ t x2 my = T ✳ s❛♦ ❝❤♦ (T, Y ) = 1✳ ●å✐ (X, Y ) r tỗ t u, v ∈ Z s❛♦ ❝❤♦ T v¯ − Y u¯ = 1✳ ✣➦t u = min{¯ u ∈ (0, |T |) | T v¯ − Y u¯ = 1}✳ ❙✉② r❛   uY ≡ (mod T )  0 < u < |T | ữ ợ ÷ỵ❝ ❝❤✉♥❣ ❧ỵ♥ ♥❤➜t ❝õ❛ ❝❤♦ uY ≡ (mod T )✳ x2 − my = T ♠➣♥ T u21 s❛♦ ❝❤♦ T (X, Y ) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ x2 my = T y tỗ t↕✐ ❞✉② ♥❤➜t u ∈ (0, |T |) ∩ N s❛♦ ❚❛ ❣å✐ (X1 , Y1 ), (X2 , Y2 ) tũ ỵ (T, Y1 ) = (T, Y2 ) = 1✳ ●å✐ u1 , u2 ∈ (0, |T |) ∩ N t❤ä❛ u1 Y1 ≡ (mod T ), u2 Y2 ≡ (mod T )✳ ✈➔ t❤ä❛ ♠➣♥ X22 u22 − mY22 u22 = T u22 ✳ ▲➜② ♠♦❞✉❧♦ T ❙✉② r❛ X12 u21 − mY12 u21 = ❤❛✐ ✈➳ t❛ t❤✉ ✤÷đ❝ (X1 u1 )2 − m ≡ (mod T ) ❈❤÷ì♥❣ ✸✳ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ✹✵ ✈➔ (X2 u2 )2 − m ≡ (mod T ) ❙✉② r❛ (X1 u1 )2 − (X2 u2 )2 ≡ (mod T ) ❚÷ì♥❣ ✤÷ì♥❣ (X1 u1 − X2 u2 )(X1 u1 + X2 u2 ) ≡ (mod T ) ❱➟② X1 u1 ≡ X2 u2 (modT ) ❤♦➦❝ ❉➵ ❞➔♥❣ ♥❤➟♥ t❤➜② ♥➳✉ t❤➻ (±x, ±y) X1 u1 ≡ X2 u2 (modT )✳ (x, y) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❝ô♥❣ ✤➲✉ ❧➔ ♥❣❤✐➺♠ ❝õ❛ t❤➸ ❝❤å♥ ❤❛✐ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ X1 u1 ≡ X2 u2 (mod ❚❛ ❝❤ù♥❣ ♠✐♥❤ x2 − my = T ✳ (X1 , Y1 ) ❉♦ ✤â✱ t❛ ❝â (X2 , Y2 ) ữ tr ợ = X+Y m Q+Q m ✳ √ X1 +Y1 √m T )✳ ❚❛ ❝â X +Y2 m X, Y ∈ Z✳ x2 − my = T ✈➔ ❚❤➟t ✈➟②✱ t❛ ❝â u1 u2 (X2 Y1 − X1 Y2 ) = (u1 Y1 )(u2 X2 ) − (u2 Y2 )(u1 X1 ) ≡ (u2 X2 − u1 X1 ) (modT ) ≡ (mod T ) ❙✉② r❛ u1 u2 (X2 Y1 − X1 Y2 ) ❝❤✐❛ ❤➳t ❝❤♦ T ✳ ◆➳✉ T = ±1 t❤➻ X, Y ∈ Z✳ ◆➳✉ |T | > t❛ ❝â u1 Y1 ≡ (mod T ), u2 Y2 ≡ (mod T )✳ ❝❤✐❛ t t ữợ Y Z tù❝ ❧➔ ❚÷ì♥❣ tü✱ t❛ ❝â T✳ ❉➝♥ ✤➳♥ X ∈ Z✳ ❙✉② r❛ X2 Y1 − X1 Y2 u1 u2 ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ T ❙✉② r❛ √ √ √ (X1 + Y1 m) = (X2 − Y2 m)(X + Y m) ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ tr♦♥❣ ✈➔♥❤ √ Z + Z m✳ ◆➳✉ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ t❤❛② t❤➻ ✈❛✐ trá ❝õ❛ Y s➩ ✤÷đ❝ t❤❛② ❜➡♥❣ Y1 −Y ✱ ❜➡♥❣ −Y1 ✈➔ t❤❛② Y2 ❜➡♥❣ −Y2 ❞♦ ✤â √ √ √ (X1 − Y1 m) = (X2 + Y2 m)(X − Y m) ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ tr♦♥❣ ❉♦ ✤â tr➻♥❤ √ Z + Z m✳ ❙✉② r❛ T = X12 − mY12 = T (X − mY )✳ X − mY = 1✳ ❱➟② (X, Y ) ❧➔ ♠ët ♥❣❤✐➺♠ ❦❤→❝ (±1, 0) ❝õ❛ ♣❤÷ì♥❣ x2 − my = 1✳ ❈❤÷ì♥❣ ✸✳ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ✹✶ ✸✳✸ ▼ët sè tr÷í♥❣ ❤đ♣ ✈ỉ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t y2 x3 = k ỵ M, N ❧➔ ❝→❝ sè ♥❣✉②➯♥ s❛♦ ❝❤♦ M ≡ (mod 4) ✈➔ N ≡ (mod 4)✳ ●✐↔ sû N ❝â t❤➯♠ t➼♥❤ ❝❤➜t ♥➳✉ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè s❛♦ ❝❤♦ p | N t❤➻ p ≡ (mod 4)✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ y − x3 = M − N ✈æ ♥❣❤✐➺♠✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤↔♥ ❝❤ù♥❣✳ ●✐↔ sû ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ −1 (mod 4) ◆➳✉ y (x, y) ❧➫ t❤➻ ♥➯♥ y − x3 = M − N ✳ ❑❤✐ ✤â✱ ✈➻ M − N ≡ y ≡ x3 − (mod 4)✳ y ≡ (mod 4)✳ ❱➟② ♥➳✉ ◆➳✉ x y ❝❤➤♥ t❤➻ ❝❤➤♥ ❤♦➦❝ y ≡ (mod 4)✳ x ≡ (mod 4) t❤➻ ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ♥❣✉②➯♥ ♥➯♥ s✉② r❛ x ≡ (mod 4)✱ y − x3 = M N tữỡ ữỡ ợ y2 + N = x2 − xM + M ≡ (mod 4) tỗ t số t (x + M )(x2 − xM + M )✳ ♠ët sè ♥❣✉②➯♥ tè ❧➫ r❛ (x, y) ∈ Z ì Z p ữợ y + N ≡ (mod p)✳ tø ✤â ❉♦ õ tỗ t p ữ t x2 − xM + M = ( −1 p ) = (1) tọ ữỡ tr t ỵ M, N z2 p p−1 z∈Z ✈➔ ❞➝♥ ✤➳♥ p ≡ (mod 4)✳ s❛♦ ❝❤♦ y ❙✉② z ≡ −1 (mod p)✱ = = −1✱ ♠➙✉ t❤✉➝♥✦ ✣✐➲✉ ♥➔② ❝❤ù♥❣ y − x3 = M − N ✈æ ♥❣❤✐➺♠✳ ❧➔ ❝→❝ sè ♥❣✉②➯♥ s❛♦ ❝❤♦ M ≡ (mod 4) ✈➔ N ≡ (mod 2)✳ ●✐↔ sû N ❝â t❤➯♠ t➼♥❤ ❝❤➜t ♥➳✉ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè s❛♦ ❝❤♦ p | N t❤➻ p ≡ (mod 4)✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ y − x3 = M − N ✈æ ♥❣❤✐➺♠✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤↔♥ ❝❤ù♥❣✳ ●✐↔ sû ♠ët ♥❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ✈➻ (x, y) ∈ Z × Z ❧➔ y −x3 = M −N ✳ ❙✉② r❛ y −x3 ≡ −1 ( mod 4) N ≡ ( mod 4)✱ tù❝ ❧➔ y ≡ x3 − ( mod 4)✳ ❚❛ ❝â y ≡ 0, ( mod 4) ❈❤÷ì♥❣ ✸✳ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ x3 − ≡ 0, 2, (mod 4)✳ ✈➔ sè ❝❤➤♥ ✈➔ x ≡ (mod 4)✳ ❱➟② ✹✷ y ≡ x3 − (mod 4) ▼➦t ❦❤→❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ y − x3 = M − N y ❧➔ t÷ì♥❣ ✤÷ì♥❣ y + N = x3 + M = (x + M )(x2 − M x + M ) ≡ (mod 4), tr♦♥❣ ✤â x + M ≡ x2 − M x + M ≡ ( mod 4)✳ ❉♦ ✤â✱ x2 − M x + M ♠ët sè ❧➫✳ ●✐↔ sû i ✱ pi = pj ♠å✐ r➡♥❣ x2 −M x+M = pα1 · · · pαl l ♥➳✉ i = j✳ ◆➳✉ ✈ỵ✐ pi ≡ (mod4) ❧➔ pi ❧➔ ❝→❝ sè ♥❣✉②➯♥ tè ❧➫ ✈ỵ✐ ✈ỵ✐ ♠å✐ i t❤➻ t❛ s✉② r❛ ✤÷đ❝ x2 − M x + M ≡ 1α1 · · · 1αl ≡ ( mod 4) t õ tỗ t p ∈ {p1 , , pl } s❛♦ ❝❤♦ p ≡ ( mod 4)✳ ❚❛ ❝â y + N ≡ ( mod p)✳ ❙✉② r tỗ t zZ z + (mod p) s t ợ ữỡ tr➻♥❤ −1 p = (−1) y − x3 = M − N p−1 −1 p ❙✉② r❛ = −1 ✈➻ = z2 p p ≡ (mod 4)✳ = 1✱ ❱➟② ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥✳ ✣à♥❤ ỵ M, N Z ợ M 4, (mod 8) ✈➔ N ≡ 1(mod 2)✳ ●✐↔ sû N ❝â t❤➯♠ t➼♥❤ ❝❤➜t ♥➳✉ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè s❛♦ ❝❤♦ p | N t❤➻ p ≡ ±1 (mod 8)✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ y − x3 = M + 2N ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤↔♥ ❝❤ù♥❣✳ ●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ y − x3 = M + 2N ❝õ❛ M ❚❛ ❝â ✈➔ N t❛ ❝â (x, y) t❤✉ë❝ Z × Z✳ ❚❤❡♦ ❣✐↔ t❤✐➳t M + 2N ≡ (mod 4)✳ y ≡ 0, (mod 4) x ≡ (mod 4)✳ ✈➔ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ ✈➔ ❙✉② r❛ y ≡ x3 + (mod 4)✳ x3 + ≡ 1, 2, (mod 4)✳ ▼➦t ❦❤→❝ y − x3 = M + 2N ◆❤÷ ✈➟② y số tữỡ ữỡ ợ y 2N = x3 + M = (x + M )(x2 − M x + M ) ❉♦ x ≡ (mod4) ◆➳✉ ✈➔ pi ♥➯♥ s✉② r❛ x ≡ (mod 8) x ≡ 3, (mod 8)✳ t❤➻ t❛ s✉② r❛ ✤÷đ❝ r➡♥❣ x2 − M x + M ≡ ±3 (mod8)✳ ❱✐➳t x + M ≡ ±1 (mod 8) x2 − M x + M = pα1 · · · pαl l ✱ ❧➔ ❝→❝ sè ♥❣✉②➯♥ tè ❧➫ ♣❤➙♥ ❜✐➺t✳ ◆➳✉ ✈ỵ✐ ♠å✐ x2 − M x + M ≡ (mod 8)✱ i✱ pi ≡ (mod8)✱ ♠➙✉ t❤✉➝♥✦ tỗ t t p {p1 , , pl } ❈❤÷ì♥❣ ✸✳ s❛♦ ❝❤♦ z ∈Z ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ p ≡ ±3 (mod 8)✳ s❛♦ ❝❤♦ z ≡ (mod p)✳ p ✤à♥❤ ♥❣❤➽❛ = (−1) x ≡ (mod 8) ◆➳✉ ❙✉② r❛ p2 −1 y − 2N ≡ (mod p)✱ ❉♦ ✤â p = z p = tỗ t t❤✉➝♥ ✈ỵ✐ = −1✳ t❤➻ ✹✸ x + M ≡ ±3 (mod 8) ▲➟♣ ❧✉➟♥ t÷ì♥❣ tü ♥❤÷ tr➯♥✱ ❝ơ♥❣ ❞➝♥ ✤➳♥ ♠➙✉ t❤✉➝♥✳ ❱➟② ♣❤÷ì♥❣ tr➻♥❤ y − x3 = M + 2N ❦❤æ♥❣ ❝â ỵ M, N Z ✈ỵ✐ M ≡ (mod 8) ✈➔ N ≡ (mod 2)✳ ●✐↔ sû N ❝â t❤➯♠ t➼♥❤ ❝❤➜t ♥➳✉ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè s❛♦ ❝❤♦ p | N t❤➻ p ≡ 1, (mod 8)✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ y − x3 = M − N ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤ù♥❣ ♠✐♥❤ tữỡ tỹ ỵ ữỡ tr t y2 − x3 = k m ❈❤♦ ●å✐ OK ❧➔ ởt số ổ õ ữợ ữỡ ✈➔♥❤ t➜t ❝↔ ❝→❝ sè ♥❣✉②➯♥ ✤↕✐ sè tr♦♥❣ m ≡ (mod 4) √ OK = Z + Z( m)✳ ♥➳✉ ▼ët t➟♣ A⊆K t❤➻ OK = Z + ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦✐➺♥ s❛✉ ✤➙② t❤ä❛ ♠➣♥ ✭❛✮ ♥➳✉ x ∈ OK t❤➻ αx ∈ A✱ t❤➻ ●å✐ OK ✭❝❤➾ ❝➛♥ ❧➜② y ❜➡♥❣ m ≡ (mod 4) α + β ∈ A✱ y = 0✱ y OK ỵ r t tổ tữớ tự ỵ tự OK , A tỗ t↕✐ K✳ √ 1+ −3 Z( )✱ ✈➔ ♥➳✉ s❛♦ ❝❤♦ OK ♥➳✉ ❜❛ ✤✐➲✉ ✭❜✮ ♥➳✉ α ∈ A✱ yA ⊆ OK ✳ ❈❤ó ✤➲✉ ❧➔ ✐✤➯❛♥ ♣❤➙♥ ✤è✐ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ✐✤➯❛♥✿ bi | ∈ A, bi ∈ B, |S| < ∞ i∈S t❤➻ tr♦♥❣ ♣❤➛♥ ✭❝✮ ❝õ❛ ✤à♥❤ ♥❣❤➽❛✮✳ I(K) ❧➔ ♠ët ♥❤â♠ ❝→❝ ✐✤➯❛♥ ♣❤➙♥ t❤ù❝ ❝õ❛ OK A·B = √ K = Q( m)✳ ❈❤÷ì♥❣ ✸✳ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ✹✹ ✭❉➵ ❦✐➸♠ tr❛ I(K) ♥❤â♠ ❝♦♥ ❝õ❛ I(K) ỗ tt tự tr♦♥❣ I(K)✳ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ♥â✐ tr➯♥ ❧➔ ♠ët ♥❤â♠✳✮ ●å✐ P (K) ❧➔ ■✤➯❛♥ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ ❧➔ ♠ët ✐✤➯❛♥ ♣❤➙♥ t❤ù❝ s✐♥❤ ❜ð✐ ♠ët ♣❤➛♥ tû ❝õ❛ K✳ tr÷í♥❣ ❑❤✐ ✤â ♥❣÷í✐ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝♦♥ ❝❤✉➞♥ t➢❝ ❝õ❛ ♥❤â♠ ❣å✐ ❧➔ P (K) ❧➔ ♠ët ♥❤â♠ I(K)✳ ◆❤â♠ t❤÷ì♥❣ H(K) := I(K)/P (K) ✤÷đ❝ õ ợ K ỵ ❤✐➺✉ ✳ ◆❤â♠ H(K) ❧➔ ♠ët ♥❤â♠ ❤ú✉ ❤↕♥✳ ✭❬✸❪✮ h(K) := |H(K)| ỵ k < ởtsố ổ õ ữợ ữỡ s k ≡ 2, (mod 4)) ✈➔ h(Q k) ≡ (mod 3) tỗ t a Z s❛♦ ❝❤♦ k = 1−3a2 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♣❤❛♥t✐♥❡ y − x3 = k ❝â t➟♣ ♥❣❤✐➺♠ ❧➔ (x, y) = (4a2 − 1, ±(3a − 8a3 ))✳ tỗ t a Z ợ k = −1 − 3a2 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ y − x3 = k ❝â t➟♣ ♥❣❤✐➺♠ ❧➔ (x, y) = (4a2 + 1, ±(3a + 8a2 ))✳ • ✭❝✮ ◆➳✉ k = ±1−3a2 t❤➻ ✈ỵ✐ ♠å✐ a ∈ Z t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ y − x3 = k ✈æ ♥❣❤✐➺♠✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✭❝✮ ❜➡♥❣ ♣❤↔♥ ❝❤ù♥❣✳ ●✐↔ sû ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ỵ õ x (mod 2) y ≡ 0, (mod 4) ❙✉② r❛ ♥➳✉ (x, k) = 1✳ ●✐↔ sû x py12 = p2 x3 + k1 ✱ ❣✐↔ t❤✐➳t ❉♦ k s✉② r❛ k ✈➔ t❤➻ ✈➔ lx + 2mk = 1✳ x ✈➔ k ✳ ❙✉② p | k1 ✱ r❛ ❞➝♥ ✤➳♥ k t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t x3 + ≡ 0, (mod 4) x ≡ (mod 2)✳ ♥➳✉ ❚❛ s➩ ❝❤ù♥❣ 1✳ ●å✐ p ❧➔ sè p | y ✳ ❱✐➳t x = px1 ✱ k = pk1 p2 | k ✳ t❛ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ (x, k) = 1✳ ♥➯♥ (x, 2k) = 1✳ ❱➟② tỗ t l, m Z K = Q( k)✳ ❑❤✐ ✤â OK = Z + Z k ✱ ✈➻ t❤❡♦ (x, k) = ✣➦t ✈ỵ✐ ❝â ởt ữợ ợ ỡ ổ õ ữợ ữỡ ✈➟② x ≡ (mod 2) s❛♦ ❝❤♦ y = x3 + k tố ữợ ❝õ❛ ❝â y − x3 = k (x, y) Z ì Z ữỡ ố số P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ❣✐↔ t❤✐➳t k ≡ 2, (mod 4)✳ (y − √ k) tr♦♥❣ OK ✹✺ ❚❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ❝→❝ ✐✤➯❛♥ (y + √ k) ✈➔ ♥❣✉②➯♥ tố ũ sỷ ữủ õ tỗ t P s P ữợ ❝õ❛ (y + k) ✈➔ (y − k)✱ tù❝ √ √ √ √ √ ❧➔ y + k, y − k ∈ P ✳ ❙✉② r❛ k = (y + k) − (y − k) ∈ P ❞➝♥ ✤➳♥ √ √ √ √ 2k = k(2 k) ∈ P ✳ ◆❤÷♥❣ (y + k)(y − k) = (y − k) = (x3 ) s✉② r❛ ✐✤➯❛♥ ♥❣✉②➯♥ tè x3 ∈ P ✳ ❉♦ P ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥➯♥ x ∈ P ✳ ❙✉② r❛ = lx + m(2k) ∈ P ✱ P = OK ✱ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ♥❣✉②➯♥ tè ❝õ❛ P ✳ √ k) ✈➔ (y − k) ❧➔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈➔ √ √ (y + k)(y − k) = (y − k) = (x3 ) = (x)3 ♥❣❤➽❛ ❧➔ ❦➨♦ t❤❡♦ ❱➟② (y + √ ởt t q tr số tỗ t↕✐ ♠ët ✐✤➯❛♥ ❝❤♦ (x + √ k) = A3 ❝ô♥❣ ❧➔ ♠ët A3 A ❝õ❛ OK s❛♦ h(K) ≡ (mod3) ♥➯♥ A √ ✐✤➯❛♥ ❝❤➼♥❤✳ ●✐↔ sû A = (a + b k) ✈ỵ✐ a, b ∈ Z s✉② r❛ √ √ √ (y + k) = A3 = (a + b k)3 = ((a + b k)3 ) s✉② r❛ ❧➔ ✐✤➯❛♥ ❝❤➼♥❤✳ ❱➻ √ k = (a + b k), ∈ OK ✳ ◆➳✉ ❦❤æ♥❣ ❦❤↔ ♥❣❤à❝❤ t❤➻ √ √ √ √ (y + k) = ( (a + b k)3 ) ((a + b k)3 ) = (y + k) y+ ❚ø ✤â √ ✣✐➲✉ ♠➙✉ t❤✉➝♥ ♥➔② ❝❤ù♥❣ tä k ≡ 2, (mod4) OK ✱ tù❝ ❧➔ ∈ U (OK )✳ ❉♦ U (OK ) = {±1}✳ ❱➟② √ √ √ = ±1✳ ▲➜② ❧✐➯♥ ❤ñ♣ ❝õ❛ y+ k = (a+b k) t❛ ❝â y− k = (a−b k)3 ✳ ✈➔ k < tr t ỵ √ ❙✉② r❛ √ √ √ k)(y − k) = (a + b k)3 (a − b k)3 √ √ = ((a + b k)(a − b k))3 x3 = y − k =(y + √ = (a2 − b2 k)3 tù❝ ❧➔ x = a2 − kb2 ✳ ▼➦t ❦❤→❝ √ √ √ √ 2y = (y + k) + (y − k) = ((a + b k)3 + (a − b k)3 ) ❈❤÷ì♥❣ ✸✳ ❙è ✤↕✐ sè ✈➔ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ✹✻ √ √ √ √ √ k = (y + k) − (y − k) = ((a + b k)3 − (a − b k)3 ) ❉♦ ✤â   y = (a3 + 3kab2 ) ✭✸✳✶✮  1 = (3a2 b + kb3 ) ❙✉② r❛ b = ±1✱ k = − 3a2 ✳ ◆➳✉ ❦➨♦ t❤❡♦ b=− b = ± t❤➻ ✳ ◆➳✉ b = t❤➻ = −(3a2 + k) = 3a2 + k s✉② r❛ s✉② r❛ k = −1 − 3a2 ✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ tt ữủ ự ự ỵ ✭❝✮✱ ♥➳✉ ❤♦➦❝ k = −1 − 3a2 ✈ỵ✐ a ∈ Z✳ y − x3 = k ❝â ♥❣❤✐➺♠ t❤➻ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ k = − 3a2 y = ±(3a − 8a3 ) ✈➔ k = − 3a2 t❤➻ = b✱ x = 4a2 − 1✳ ❞♦ ◆➳✉ k = −1 − 3a2 t❤➻ = −b✱ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ y = ±(3a + 8a3 ) ✈➔ x = 4a2 + t ỵ M = 2✱ N = 3✱ t❛ ❝â k = M −N = −1✱ ♣❤÷ì♥❣ tr➻♥❤ y −x3 = −1 ✈ỉ ♥❣❤✐➺♠✳ ❍✐➸♥ ♥❤✐➯♥ ♣❤÷ì♥❣ tr➻♥❤ y −x3 = ❝â ✈æ sè ♥❣❤✐➺♠ ♥❣✉②➯♥✳ ❈â ♠ët ❧ỵ♣ ❝→❝ ❣✐→ trà y − x3 = k k > t↕✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ ✭①❡♠ ❬✸✱ ✣à♥❤ ❧② ✶✹✳✷✳✹❪✮✳ ❑➳t ❧✉➟♥ ❳✉②➯♥ s✉èt ❦❤♦→ ❧✉➟♥ ✏ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ ❈→❝ sè ✤↕✐ sè ✈➔ ù♥❣ ❞ư♥❣ ✈➔♦ ❣✐↔✐ ♣❤÷ì♥❣ ✑✱ ❝❤ó♥❣ tỉ✐ t➟♣ tr✉♥❣ ự sỹ tỗ t ữỡ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❝ư t❤➸✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ♠➔ ❝❤ó♥❣ tỉ✐ ✤↕t ✤÷đ❝ tr♦♥❣ ❦❤♦→ ❧✉➟♥ ♥➔② ❧➔✿ ✭✶✮ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❦❤↔ ♥❣❤à❝❤ ❝õ❛ ❞↕♥❣ ±(1 + √ √ Z+Z ❝â 2)n , n ∈ Z✳ ✭✷✮ Ù♥❣ ❞ö♥❣ sè ✤↕✐ sè t➻♠ ✤÷đ❝ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ x2 − 2y = ✈➔ x2 − 2y = −1✳ ✭✸✮ ❈❤ù♥❣ ♠✐♥❤ sü tỗ t t x2 my = ợ m (1, 0) ữỡ tr số tỹ ổ õ ữợ ữỡ ự sỹ tỗ t ữỡ tr t y = x3 + k ✳ ✭✺✮ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡ y − x3 = k tr♦♥❣ ♥❤✐➲✉ tr÷í♥❣ ❤đ♣✳ ✭✻✮ ❳➙② ❞ü♥❣ ✤÷đ❝ ❝→❝ ✈➼ ❞ư ♠✐♥❤ t q ỵ õ tr♦♥❣ ❦❤♦→ ❧✉➟♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ sû ❞ü♥❣ ❦✐➳♥ t❤ù❝ ✈➲ sè ✤↕✐ sè ✤➸ ❣✐↔✐ q✉②➳t ✤÷đ❝ ♠ët số trữớ ủ ự sỹ tỗ t t ởt số ợ ữỡ tr t ứ ✤➙② ❧➔♠ ♥➲♥ t↔♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❧ỵ♣ ❜➔✐ t♦→♥ ❦❤→❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥❣❤✐➺♠ ♥❣✉②➯♥✳ ✹✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆❣✉②➵♥ ❍ú✉ ❱✐➺t ❍÷♥❣✱ ✣↕✐ sè ✤↕✐ ❝÷ì♥❣✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ư❝✱ ✶✾✾✾✳ ❬✷❪ ◆❣ỉ ❱✐➺t ❚r✉♥❣✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ✷✵✵✻✳ ❬✸❪ ❙❛❜❛♥ ❆❧❛❝❛ ✈➔ ❑❡♥♥❡t❤ ❲✐❧❧✐❛♠s✱ ■♥tr♦❞✉❝t♦r② ❆❧❣❡❜r❛✐❝ ♥✉♠❜❡r t❤❡♦r②✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✷✵✵✸✳ ✹✽ ...TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN - NGUYỄN THỊ NGỌC HOA CÁC SỐ ĐẠI SỐ VÀ ỨNG DỤNG VÀO GIẢI PHƯƠNG TRÌNH DIOPHANTINE KHỐ LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Đại số Người... ❈→❝ ủ số số ỵ ◆➳✉ α ∈ C ❧➔ ♠ët sè ♥❣✉②➯♥ ✤↕✐ sè ✭tù❝ ❧➔✱ α ❧➔ ♠ët ♣❤➛♥ tû ♥❣✉②➯♥ tr➯♥ Z✮ t❤➻ ❝→❝ ❧✐➯♥ ❤đ♣ ❝õ❛ ♥â tr➯♥ Q ❝ơ♥❣ ❧➔ ❝→❝ sè ♥❣✉②➯♥ ✤↕✐ sè✳ ❈❤ù♥❣ ♠✐♥❤✳ t❤ù❝ ❳➨t α ởt số số õ... √1 ∈C ❦❤ỉ♥❣ ❧➔ sè ♥❣✉②➯♥ ✤↕✐ sè✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû √ ✶✷ ❧➔ ♠ët sè ♥❣✉②➯♥ ❈❤÷ì♥❣ ✷✳ ❙ì ❧÷đ❝ ✈➲ số số số õ tỗ t tù❝ ❧➔ a0 , , an−1 ∈ Z n √ ✶✸ + an−1 √ n−1 s❛♦ ❝❤♦ + · · · + a1 √ + a0 = 0, √