TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* VŨ THỊ NGỌC ANH MƠĐUN TRÊN MIỀN CHÍNH KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Đại số HÀ NỘI – 2018 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TỐN ************* VŨ THỊ NGỌC ANH MƠĐUN TRÊN MIỀN CHÍNH KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chun ngành: Đại số Người hướng dẫn khoa học TS LÊ QUÝ THƯỜNG HÀ NỘI – 2018 ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▲í✐ ♥â✐ ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✸ ✺ ✶✳✶ ▼✐➲♥ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ▼æ✤✉♥ ✈➔ ♠æ✤✉♥ ❝♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ỹ ỗ ởt ổ tỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ▼æ✤✉♥ tr➯♥ ♠✐➲♥ ❝❤➼♥❤ ✶✺ ✸ ▼æ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣ ✷✻ ❑➳t ❧✉➟♥ ✸✻ ✷✳✶ ❈➜✉ tró❝ ❝õ❛ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ♠✐➲♥ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❈→❝ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ❊✉❝❧✐❞ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✶ ❈➜✉ tró❝ K[x]✲♠ỉ✤✉♥ tr➯♥ ♠ët K ✲❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ✳ ✳ ✳ ✳ ✷✻ ✸✳✷ ❚➼♥❤ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠ët ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸ Ù♥❣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ t➻♠ ✤❛ t❤ù❝ tè✐ t✐➸✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỡ rữợ tr ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❦❤â❛ ❧✉➟♥✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ tỵ✐ ❝→❝ t❤➛② ❝ỉ ❦❤♦❛ ❚♦→♥✱ tr÷í♥❣ ✣↕✐ ❍å❝ ❙÷ P❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ tê ❜ë ♠ỉ♥ ✣↕✐ sè ❝ơ♥❣ ♥❤÷ ❝→❝ t❤➛② ❝æ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ t ỳ tr tự qỵ t t❤✉➟♥ ❧đ✐ ✤➸ ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ♥❤✐➺♠ ✈ư ❦❤â❛ ❤å❝ ✈➔ ❦❤â❛ ❧✉➟♥✳ ✣➦❝ ❜✐➺t✱ ❡♠ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ 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✳ ❱➻ rt ữủ ỳ ỵ õ þ q✉þ ❜→✉ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ❱ô ❚❤à õ ỵ t ỵ tt ổ ữủ ự tứ t õ tr ỵ tt ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ♥❣➔♥❤ t♦→♥ ❤å❝✳ ❱✐➺❝ t➻♠ ❤✐➸✉ t tr ỳ t q q trồ ỵ tt ổ ỵ tt ổ tr ởt ữợ ự õ t t tớ sỹ tr ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ✣↕✐ sè ❤✐➺♥ ✤↕✐✳ õ r ỵ tt ổ ỳ s õ ù♥❣ ❞ö♥❣ ❤➳t sù❝ ❤ú✉ ➼❝❤ tr♦♥❣ ✣↕✐ sè t✉②➳♥ t➼♥❤✱ ❝❤➥♥❣ ❤↕♥ ♥â ❝❤♦ ♠ët ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ ❞↕♥❣ ❝❤✉➞♥ ❤ú✉ t✛ ✈➔ ❞↕♥❣ ❝❤✉➞♥ ❏♦r❞❛♥ ❝õ❛ ♠ët ♠❛ tr ổ ợ số tr ởt trữớ ❏♦r❞❛♥ ❝õ❛ ♠ët ♠❛ tr➟♥ ✈✉æ♥❣ A ❧➔ ♠ët ♠❛ tr ổ tữỡ ữỡ õ ợ tr ✤÷í♥❣ ❝❤➨♦ ♥❤➜t✱ ♥❤÷♥❣ ✤➸ t❤✉ ✤÷đ❝ ♥â✱ ②➯✉ ❝➛✉ t✐➯♥ q✉②➳t ❧➔ tr÷í♥❣ ♥➲♥ ♣❤↔✐ ❝❤ù❛ t➜t ❝↔ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ A✳ ❈❤ó♥❣ tỉ✐ s➩ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❞↕♥❣ ❝❤✉➞♥ ❤ú✉ t✛ ✈➻ ♥â ❦❤æ♥❣ ✤á✐ ❤ä✐ ✤✐➲✉ ❦✐➺♥ ✈➲ tr÷í♥❣ ♥➲♥ ✈➔ ✈➝♥ ✤↔♠ ❜↔♦ ✤÷đ❝ ②➯✉ ❝➛✉ t➻♠ ✤❛ t❤ù❝ tè✐ t✐➸✉ ❝õ❛ ♠❛ tr➟♥ ❝❤➾ ❞ò♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝➜♣✳ ❳✉➜t t tứ q st tr ũ ợ sỹ ữợ t t ỵ ữớ tổ t➔✐ ▼æ✤✉♥ tr➯♥ ♠✐➲♥ ❝❤➼♥❤ ✤➸ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❚➻♠ ❤✐➸✉ ❝➜✉ tró❝ ❝õ❛ ♠ët ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ♠ët ♠✐➲♥ ❝❤➼♥❤✱ ❝→❝ ✤➦❝ ✤✐➸♠ t❤ó ✈à ❝õ❛ ❝→❝ ♠ỉ✤✉♥ ♥➔② tr➯♥ ❝→❝ ♠✐➲♥ ❝❤➼♥❤✱ ✤➦❝ ❜✐➺t ❧➔ tr➯♥ ✈➔♥❤ ❊✉❝❧✐❞ tự K[x] ợ K ởt trữớ ❑❤â❛ ❧✉➟♥ tr➻♥❤ ❜➔② ✈✐➺❝ ✤å❝ ❤✐➸✉ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ tr♦♥❣ ❈❤÷ì♥❣ ✶✷ ❝õ❛ ❝✉è♥ s→❝❤ ❆❜str❛❝t ❆❧❣❡❜r❛ ❝õ❛ ❉✉♠♠✐t ✈➔ ❋♦♦t❡ ✭①❡♠ ❬✷❪✮✳ ❑✐➳♥ t❤ù❝ ✸ ✹ ❝❤✉➞♥ ❜à ❝õ❛ ❦❤â❛ ❧✉➟♥ ❝á♥ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝✉è♥ s→❝❤ ✣↕✐ sè ✤↕✐ ❝÷ì♥❣ ❝õ❛ ◆❣✉②➵♥ ❍ú✉ ❱✐➺t ❍÷♥❣ ✭①❡♠ ❬✶❪✮✳ ✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❑❤â❛ ❧✉➟♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ♠✐➲♥ ❝❤➼♥❤ ✈➔ ù♥❣ ❞ư♥❣✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ự t ỗ ỵ ❝➜✉ tró❝ ❝õ❛ ♠ët ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ♠ët ♠✐➲♥ ❝❤➼♥❤✳ ✭❜✮ ❚❤✉➟t t♦→♥ ❞ò♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝➜♣ tr➯♥ ♠❛ tr➟♥ ✤➸ t➼♥❤ ❝→❝ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠ët ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ♠ët ✈➔♥❤ ❊✉❝❧✐❞✳ ✭❝✮ ❉↕♥❣ ❝❤✉➞♥ ❤ú✉ t✛ ❝õ❛ ♠ët ♠❛ tr ổ ợ tỷ tr ởt trữớ ù♥❣ ❞ö♥❣ ✤➸ t➼♥❤ ✤❛ t❤ù❝ tè✐ t✐➸✉ ❝õ❛ ♠ët ♠❛ tr➟♥ ✈✉ỉ♥❣✳ ✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❚❤❛♠ ❦❤↔♦ t➔✐ ❧✐➺✉✱ ♣❤➙♥ t➼❝❤ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ❝❤ù♥❣ ♠✐♥❤ ỵ q trồ tỹ ộ t t ữợ ỵ tt ổ tr ❝❤➼♥❤ ✈➔♦ ❜➔✐ t♦→♥ t➻♠ ✤❛ t❤ù❝ tè✐ t✐➸✉ ❝õ❛ ♠❛ tr➟♥✳ ✺✳ ❈➜✉ tró❝ ✤➲ t➔✐ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✿ ❈❤÷ì♥❣ ♥➔② tâ♠ ❧÷đ❝ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ỡ tt ỗ ♥✐➺♠ ✈➔♥❤✱ ♠✐➲♥ ❝❤➼♥❤✱ ♠æ✤✉♥ tr➯♥ ♠ët ✈➔♥❤ ✈➔ ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ tr➯♥ ♠ỉ✤✉♥✳ ❈❤÷ì♥❣ ✷✿ ▼ỉ✤✉♥ tr➯♥ ♠✐➲♥ ❝❤➼♥❤✿ ❚r♦♥❣ t♦➔♥ ❜ë ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❧✉ỉ♥ ❣✐↔ sû R ❧➔ ♠ët ♠✐➲♥ ❝❤➼♥❤ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ởt ỵ q trồ trú ổ tr ♠✐➲♥ ❝❤➼♥❤ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠ët ♠✐➲♥ ❝❤➼♥❤✱ ❝ö t❤➸ ❧➔ ♠ët ✈➔♥❤ ❊✉❝❧✐❞✳ ❈❤÷ì♥❣ ✸✿ ▼ỉ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥✿ ❈❤ó♥❣ tỉ✐ →♣ ❞ư♥❣ ♠ët ❦➳t q✉↔ q✉❛♥ trå♥❣ tr♦♥❣ ❝❤÷ì♥❣ ✷ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❦❤â❛ ❧✉➟♥ ❱ơ ❚❤à ◆❣å❝ ❆♥❤ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② tâ♠ ❧÷đ❝ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ tt ỗ ♠✐➲♥ ❝❤➼♥❤✱ ♠æ✤✉♥ tr➯♥ ♠ët ✈➔♥❤ ✈➔ ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ tr➯♥ ♠ỉ✤✉♥✳ ✶✳✶ ▼✐➲♥ ❝❤➼♥❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❚❛ ❣å✐ ♠ët ✈➔♥❤ ♠é✐ t➟♣ ❤ñ♣ R = ∅ ũ ợ t ổ ỗ P ❝ë♥❣✿ + : R × R → R, (x, y) → x + y, · : R × R → R, (x, y) → x · y, ✭❜✮ P❤➨♣ ♥❤➙♥✿ t❤ä❛ ♠➣♥ ❜❛ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙②✿ ✭❘✶✮ R ❧➔ ♠ët ♥❤â♠ ❛❜❡❧ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣❀ ✭❘✷✮ P❤➨♣ ♥❤➙♥ ❝â t➼♥❤ ❝❤➜t ❦➳t ❤ñ♣❀ ✭❘✸✮ P❤➨♣ ♥❤➙♥ ♣❤➙♥ ♣❤è✐ ✈➲ ❤❛✐ ♣❤➼❛ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣✿ (x + y)z = xz + yz, ✈ỵ✐ ♠å✐ x, y, z ∈ R✳ ✺ z(x + y) = zx + zy, ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✻ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ▼ët ✈➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ♥➳✉ ♣❤➨♣ ♥❤➙♥ ❝õ❛ ♥â ❣✐❛♦ ❤♦→♥✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ❝â ✤ì♥ ✈à ♥➳✉ ♣❤➨♣ ♥❤➙♥ ❝õ❛ ♥â ❝â ỡ tự tỗ t ởt tỷ ∈ R s❛♦ ❝❤♦ · x = x · = x✱ ✈ỵ✐ ♠å✐ x ∈ R✳ ❱➔♥❤ R ữủ ổ õ ữợ ổ tỗ t↕✐ x, y ∈ R \ {0} s❛♦ ❝❤♦ xy = 0✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ♥➳✉ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ õ ỡ ổ õ ữợ ổ ❞ö ✶✳✹✳ ✭❛✮ ▼é✐ t➟♣ sè Z✱ Q✱ R✱ C ✤➲✉ ❧➟♣ t❤➔♥❤ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✤è✐ ✈ỵ✐ ❤❛✐ ♣❤➨♣ t♦→♥ ❝ë♥❣ ✈➔ ♥❤➙♥ ❝→❝ sè t❤ỉ♥❣ t❤÷í♥❣✳ ✭❜✮ ❚r➯♥ ♥❤â♠ ❝ë♥❣ Z/n ❝→❝ sè ♥❣✉②➯♥ ♠æ✤✉♥ n ✭n ≥ 1✮ t❛ tr❛♥❣ ❜à ♠ët ♣❤➨♣ ♥❤➙♥ ♥❤÷ s❛✉ [a][b] := [ab] ◆❤â♠ ❝ë♥❣ Z/n ❝ò♥❣ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ✤â ❧➟♣ t❤➔♥❤ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ ❝â ✤ì♥ ✈à ❧➔ ❬✶❪✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ▼ët tr÷í♥❣ ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ ❝â ✤ì♥ ✈à = s❛♦ ❝❤♦ ♠å✐ ♣❤➛♥ tû ❦❤→❝ ✵ ❝õ❛ ♥â ✤➲✉ ❦❤↔ ♥❣❤à❝❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ●✐↔ sû R ❧➔ ♠ët ✈➔♥❤✳ ❚➟♣ ❝♦♥ S ⊂ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ R ♥➳✉ S ❧➔ ♠ët ♥❤â♠ ❝♦♥ ❝õ❛ ♥❤â♠ ❝ë♥❣ R ✈➔ ❦❤➨♣ ❦➼♥ ✤è✐ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥✱ tù❝ ❧➔ ✈ỵ✐ ♠å✐ x, y ∈ S t❛ ❝â xy ∈ S ✳ ❑❤✐ ✤â✱ S ❝ô♥❣ ❧➔ ♠ët ✈➔♥❤ ✈ỵ✐ ❤❛✐ ♣❤➨♣ t♦→♥ ❧➔ ❤↕♥ ❝❤➳ ❝õ❛ ❝→❝ ♣❤➨♣ t♦→♥ t÷ì♥❣ ù♥❣ ❝õ❛ R ❧➯♥ S ✳ ◆➳✉ ✈➔♥❤ ❝♦♥ S ❧➔ ♠ët tr÷í♥❣ t❤➻ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ ✈➔♥❤ R✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ✭❛✮ ▼ët ✐✤➯❛♥ tr→✐ ❝õ❛ ✈➔♥❤ R ❧➔ ♠ët ✈➔♥❤ ❝♦♥ A ⊂ R ❝â t➼♥❤ ❝❤➜t ∈ A✱ ✈ỵ✐ ♠å✐ r ∈ R✱ ✈ỵ✐ ♠å✐ a ∈ A✳ ✭❜✮ ▼ët ✐✤➯❛♥ ♣❤↔✐ ❝õ❛ ✈➔♥❤ R ❧➔ ♠ët ✈➔♥❤ ❝♦♥ A ⊂ R ❝â t➼♥❤ ❝❤➜t ar ∈ A✱ ✈ỵ✐ ♠å✐ a ∈ A✱ ✈ỵ✐ ♠å✐ r ∈ R✳ ✭❝✮ ◆➳✉ ✈➔♥❤ ❝♦♥ A ⊂ R ✈ø❛ ❧➔ ✐✤➯❛♥ tr→✐✱ ✈ø❛ ❧➔ ✐✤➯❛♥ ♣❤↔✐ t❤➻ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✐✤➯❛♥ ✭❤❛✐ ♣❤➼❛✮ ❝õ❛ R✳ ✭❞✮ ❈❤♦ t➟♣ ❝♦♥ X ⊂ R✳ ■✤➯❛♥ ♥❤ä ♥❤➜t ❝õ❛ R ❝❤ù❛ X ✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ s✐♥❤ ❜ð✐ X ✳ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✼ ▼é✐ ✐✤➯❛♥ ❝õ❛ ♠ët ✈➔♥❤ R ❦❤→❝ ✵ ✈➔ ❦❤→❝ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✐✤➯❛♥ t❤➟t sü✳ ✣è✐ ✈ỵ✐ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ ❝→❝ ❦❤→✐ ♥✐➺♠ ✐✤➯❛♥ tr→✐✱ ✐✤➯❛♥ ♣❤↔✐✱ ✐✤➯❛♥ ❧➔ trò♥❣ ♥❤❛✉✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✽✳ ●✐↔ sû a✱ b ❧➔ ❝→❝ ♣❤➛♥ tû ❦❤→❝ ✵ tr♦♥❣ ♠ët ♠✐➲♥ ♥❣✉②➯♥ R✳ ❚❛ ♥â✐ ♣❤➛♥ tû d ∈ R✱ d = 0✱ ❧➔ ÷ỵ❝ ❝❤✉♥❣ ❧ỵ♥ ♥❤➜t ❝õ❛ a ✈➔ b ♥➳✉ d | a✱ d | b ✈➔ ✈ỵ✐ ♠å✐ c ∈ R s❛♦ ❝❤♦ c | a✱ c | b t❤➻ c | d✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ●✐↔ sû R ✈➔ R ❧➔ ❝→❝ ✈➔♥❤✳ ⑩♥❤ ①↕ ϕ : R → R ữủ ởt ỗ õ ❜↔♦ t♦➔♥ ❝↔ ❤❛✐ ♣❤➨♣ t♦→♥ ❝õ❛ ✈➔♥❤✱ tù❝ ❧➔ ♥➳✉ ✈ỵ✐ ♠å✐ x, y ∈ R✱ ϕ(x + y) = ϕ(x) + ϕ(y), ϕ(xy) = ϕ(x)ϕ(y) ✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳ ởt ỗ ỗ tớ ởt ỡ →♥❤ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤ì♥ ❝➜✉ ✈➔♥❤ ✭❤❛② ♠ët ú ởt ỗ ỗ tớ ❧➔ ♠ët t♦➔♥ →♥❤ ✤÷đ❝ ❣å✐ ❧➔ ♠ët t♦➔♥ ❝➜✉ ởt ỗ ỗ tớ ởt s♦♥❣ →♥❤ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤➥♥❣ ❝➜✉ ✈➔♥❤✳ ◆➳✉ ❝â ♠ët ✤➥♥❣ ❝➜✉ ✈➔♥❤ ϕ : R → R t❤➻ t❛ ♥â✐ ✈➔♥❤ R ✤➥♥❣ ❝➜✉ ✈ỵ✐ ✈➔♥❤ R ✱ ✈➔ ✈✐➳t R ∼= R ✳ ▼➺♥❤ ✤➲ ✶✳✶✶✳ ●✐↔ sû A ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ ◆â✐ r✐➯♥❣ A ❧➔ ♠ët ♥❤â♠ ❝♦♥ ❝õ❛ ♥❤â♠ ❛❜❡❧ R ✤è✐ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣✳ ❱➻ R ❛❜❡❧ ♥➯♥ A ❧➔ ❝❤✉➞♥ t➢❝✳ ❚➟♣ R/A ❧➔ ♠ët ♥❤â♠ ❛❜❡❧ ✤è✐ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣✿ (x + A) + (y + A) = (x + y) + A, ✈ỵ✐ x, y ∈ R✳ ❚❛ tr❛♥❣ ❜à ❝❤♦ R/A ♠ët ♣❤➨♣ ♥❤➙♥ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ (x + A) · (y + A) = (xy + A), ✈ỵ✐ x, y ∈ R✳ ❑❤✐ ✤â R/A ❧➔ ♠ët ✈➔♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✷✳ ❱➔♥❤ R/A ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ t❤÷ì♥❣ ❝õ❛ ✈➔♥❤ R t❤❡♦ ✐✤➯❛♥ A✳ ỵ sỷ : R R ởt ỗ õ tỗ t ♥❤➜t ♠ët ✤➥♥❣ ❝➜✉ ✈➔♥❤ ϕ : R/ ker ϕ → Imϕ s❛♦ ❝❤♦ ϕ = ϕ · π ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✽ ❈❤♦ ϕ ❧➔ ỗ õ r ởt ỗ ❝õ❛ ❝→❝ ♥❤â♠ ❝ë♥❣ R ✈➔ R ✳ ❚❤❡♦ ✣à♥❤ ỵ ỗ õ : R/ ker → Imϕ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ϕ(x + ker ϕ) = ϕ(x) ✭✈ỵ✐ ♠å✐ x ∈ R✮ ❧➔ ✤➥♥❣ ❝➜✉ ♥❤â♠ ❞✉② ♥❤➜t ❧➔♠ ❝❤♦ ϕ = ϕ.π ❚❛ ❝❤ù♥❣ ởt ỗ t ❜↔♦ t♦➔♥ ♣❤➨♣ ❝ë♥❣✳ ✣è✐ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ ϕ[(x + ker ϕ)(y + ker ϕ)] = ϕ(xy + ker ϕ) = ϕ(xy) = ϕ(x)ϕ(y) = ϕ(x + ker ϕ)ϕ(y + ker ϕ), ✈ỵ✐ ♠å✐ x, y R ỵ ữủ ự ●✐↔ sû R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ✭❛✮ ■✤➯❛♥ A ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ♥❣✉②➯♥ tè ♥➳✉ A = R ✈➔ ✈ỵ✐ ♠å✐ x, y ∈ R✱ tø xy ∈ A s✉② r❛ x ∈ A ❤♦➦❝ y ∈ A✳ ✭❜✮ ■✤➯❛♥ A ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ❝ü❝ ✤↕✐ ♥➳✉ A = R ✈➔ ổ tỗ t B A s B = A ✈➔ B = R✳ ◆â✐ ❝→❝❤ ❦❤→❝ A ❧➔ ❝ü❝ ✤↕✐ t❤❡♦ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠ tr♦♥❣ t➟♣ ❝→❝ ✐✤➯❛♥ t❤➟t sü ❝õ❛ R✳ ▼➺♥❤ ✤➲ ✶✳✶✺✳ ❚r♦♥❣ ♠ët ✈➔♥❤ R ❣✐❛♦ ❤♦→♥ ✈➔ ❝â ✤ì♥ ✈à✱ ♠å✐ ✐✤➯❛♥ ❝ü❝ ✤↕✐ ✤➲✉ ♥❣✉②➯♥ tè✳ ▼➺♥❤ ✤➲ ✶✳✶✻✳ ●✐↔ sû A ❧➔ ♠ët ✐✤➯❛♥ tr♦♥❣ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✈➔ ❝â ✤ì♥ ✈à R✳ ❑❤✐ ✤â✿ ✭❛✮ A ❧➔ ♥❣✉②➯♥ tè ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ R/A ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥✳ ✭❜✮ A ❧➔ ❝ü❝ ✤↕✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ R/A ❧➔ ♠ët tr÷í♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✼✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❦❤→❝ {0}✳ ✭❛✮ ▼ët ✐✤➯❛♥ I ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✐✤➯❛♥ ❝❤➼♥❤ ♥➳✉ ♥â ✤÷đ❝ s✐♥❤ ❜ð✐ ♠ët ♣❤➛♥ tû✳ ✭❜✮ ▼ët ♠✐➲♥ ❝❤➼♥❤ ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ tr♦♥❣ ✤â ♠å✐ ✐✤➯❛♥ ✤➲✉ ❧➔ ✐✤➯❛♥ ❝❤➼♥❤✳ ❱➼ ❞ö ✶✳✶✽✳ ❱➔♥❤ sè ♥❣✉②➯♥ Z✱ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët K[x] ợ số tở ởt trữớ K ❝→❝ ♠✐➲♥ ❝❤➼♥❤✳ ❈❤÷ì♥❣ ✷✳ ▼ỉ✤✉♥ tr➯♥ ♠✐➲♥ ❝❤➼♥❤ ✷✸ õ tr q tữỡ ự ợ ỡ sð x , , x ❝õ❛ R ✈➔ ❤➺ s✐♥❤ ♠ỵ✐ y , , y , , y , , y❧➔  1 j i k           a11 aj1 ai1 ak1 ❜✮ ❳➨t ❤➺ s✐♥❤ ♠ỵ✐ y , , y , , y             y1 yi yj − ayi yk           =               =     i a1n ajn ain akn j           − ayi , , yk i j i           t❛ ❝â a11 x1 + · · · + a1n xn ai1 x1 + · · · + ain xn aj1 x1 + · · · + ajn xn − a(ai1 x1 + + ain xn ) ak1 x1 + · · · + akn xn  a11 x1 + · · · + a1n xn    ai1 x1 + · · · + ain xn     (aj1 − aai1 )x1 + · · · + (ajn − aain )xn    ak1 x1 + · · · + akn xn ▼❛ tr q tữỡ ự ợ ỡ s y , , y , , y − ay , , y ❧➔  n n x1 , , xn k a11 ai1 aj1 − aai1 ak1 ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ a1n ain ajn − aain akn                       ✈➔ ❤➺ s✐♥❤ ♠ỵ✐ ❈❤÷ì♥❣ ✷✳ ▼ỉ✤✉♥ tr➯♥ ♠✐➲♥ ❝❤➼♥❤ ✷✹ ❇ê ✤➲ ✷✳✶✶✳ ✭❛✮ ❉ò♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝➜♣ ❤➔♥❣ ✈➔ ❝ët t❛ ❝â t❤➸ ✤÷❛ ♠❛ tr➟♥ q✉❛♥ ❤➺ A ✈➲ ❞↕♥❣✿  a1  a21 A =  ak1 a12 a22 ak2  a1n a2n  ,  akn tr♦♥❣ ✤â a1 ữợ ợ t tt tû aij ✱ ✈➔ a1 | aij ✈ỵ✐ ♠å✐ ≤ i ≤ k ✈➔ ≤ j ≤ n✳ ✭❜✮ ❉ò♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝➜♣ ❤➔♥❣ ✈➔ ❝ët t❛ ❝â t❤➸ ✤÷❛ ♠❛ tr➟♥ A ✈➲ ❞↕♥❣✿   a1    B=   0 b22 b32 bk2 b2n b3n bkn   , tr õ a1 ữợ ❧ỵ♥ ♥❤➜t ❝õ❛ t➜t ❝↔ aij ✱ ✈➔ a1 | bij ✈ỵ✐ ♠å✐ ≤ i ≤ k ✈➔ ≤ j ≤ n✳ ✭❝✮ ❉ò♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝➜♣ ❤➔♥❣ ✈➔ ❝ët t❛ ❝â t❤➸ ✤÷❛ ♠❛ tr➟♥ B ✈➲ ❞↕♥❣✿   a1 0  a   C= 0 c33 c3n   0 ck3 ckn    ,   tr♦♥❣ ✤â a2 ữợ ợ t tt bij ✱ a1 | a2 ✱ ✈➔ a2 | cij ✈ỵ✐ ♠å✐ ≤ i ≤ k ✈➔ ≤ j ≤ n✳ ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✭❛✮ ✈➔ ✭❜✮✱ þ ✭❝✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü✳ ❇➡♥❣ ❝→❝❤ ✤ê✐ ❝❤é ❝→❝ ❤➔♥❣ ❝❤♦ ♥❤❛✉✱ ❝→❝ ❝ët ❝❤♦ ♥❤❛✉ ✭♥➳✉ ❝➛♥✮ t❛ ❝â t❤➸ ❣✐↔ sû a = 0✳ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐❛ ❝â ❞÷ tr♦♥❣ ✈➔♥❤ ❊✉❝❧✐❞ t❛ ❝â a = b a + r ✱ ✈ỵ✐ b , r ∈ R✱ r = ❤♦➦❝ δ(r ) < δ(a )✱ ✈ỵ✐ ♠å✐ ≤ i ≤ k A ợ b rỗ ✈➔♦ ❤➔♥❣ i t❛ ✤÷❛ ❈❤ù♥❣ ♠✐♥❤✳ 11 i1 i1 11 i1 i1 i1 i1 i1 i1 11 ❈❤÷ì♥❣ ✷✳ ▼æ✤✉♥ tr➯♥ ♠✐➲♥ ❝❤➼♥❤ ✷✺ ♠❛ tr➟♥ A ✈➲ ♠❛ tr➟♥ ❝â ❞↕♥❣ s❛✉ ✤➙② tr♦♥❣ M (k × n, R)✿  a11  r21 A˜ =   rk1 c12 c22 ck2  c1n c2n    ckn ❈❤å♥ ♠ët ♣❤➛♥ tû λ = tr♦♥❣ ❝ët ✤➛✉ t✐➯♥ ❝õ❛ A˜ s❛♦ ❝❤♦ δ(λ) ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥❤ä ♥❤➜t✳ ❉ò♥❣ ♣❤➨♣ ✤ê✐ ❤➔♥❣ ✤➸ ✤÷❛ λ ❧➯♥ ✈à tr➼ ❤➔♥❣ ✶ ❝ët ✶✱ t÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐❛ ❝â ❞÷ tr♦♥❣ ✈➔♥❤ ❊✉❝❧✐❞ ❝õ❛ ❝→❝ ♣❤➛♥ tû ❝õ❛ ❝ët ✶ ❝❤♦ λ✳ ▲➦♣ ❧↕✐ q✉→ tr➻♥❤ s ỳ ữợ t t ữủ tỷ a ð ✈à tr➼ ❤➔♥❣ ✶ ❝ët ✶ ✈➔ a ữợ tt tỷ tr ởt t❤ù ♥❤➜t ❝õ❛ ♠❛ tr➟♥ t÷ì♥❣ ù♥❣✳ ▲➦♣ ❧↕✐ q✉→ tr➻♥❤ tr➯♥ ❝❤♦ ❤➔♥❣ ✶ ❝õ❛ ♠❛ tr➟♥ ✈ø❛ t❤✉ ữủ t t ữủ ởt tỷ a ữợ ởt õ ữợ ❝õ❛ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❦❤→❝✳ ❱➟② ❜➡♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝➜♣ ❝❤♦ ❤➔♥❣ ✈➔ ❝ët t❛ ✤➣ ✤÷❛ A ✈➲ ❞↕♥❣ A tr♦♥❣ ♣❤➛♥ ✭❛✮✳ ✣➦t a = a d ✱ d ∈ R✱ ✈ỵ✐ ≤ i ≤ k✳ ◆❤➙♥ ❤➔♥❣ ✶ ❝õ❛ A ✈ỵ✐ −d ✈➔ ❝ë♥❣ ✈➔♦ ❤➔♥❣ ✐ t❛ t❤✉ ✤÷đ❝ ♠❛ tr➟♥   1 1 i1 i1 i1 i1 a1    B =   a12 b22 b32 bk2 a1n b2n b3n bkn       ❚÷ì♥❣ tü✱ t❛ t❤ü❝ ❤✐➺♥ ✤è✐ ợ B t t ữủ tr B ❝â ❞↕♥❣ ♥❤÷ tr♦♥❣ ♣❤➛♥ ✭❜✮✳ ❈❤ù♥❣ ♠✐♥❤ ✣à♥❤ ỵ ✈➔ ❇ê ✤➲ ✷✳✶✶✱ t❛ ❝â t❤➸ ❞ò♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝➜♣ ❝❤♦ ❤➔♥❣ ✈➔ ❝ët ✤➸ ✤÷❛ ♠❛ tr➟♥ A ✈➲ ♠❛ tr➟♥ ❝â ❞↕♥❣ s❛✉ ✤➙② ✭❣✐↔ sû k ≤ n✮✿  a1  a2 C=  0 0 0 ak   ,  tr♦♥❣ ✤â a | a | · · · | a ✳ ❚❛ ❧➜② k ❧➔ ❝❤➾ sè ❧ỵ♥ ♥❤➜t ♠➔ a t❛ ❤♦➔♥ t❤➔♥❤ ♣❤➨♣ ❝❤ù♥❣ ỵ k k =0 ❑❤✐ ✤â ❈❤÷ì♥❣ ✸ ▼ỉ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ư♥❣ ✸✳✶ ❈➜✉ tró❝ K[x]✲♠ỉ✤✉♥ tr➯♥ ♠ët K ✲❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ❈❤♦ K ❧➔ ♠ët tr÷í♥❣✳ ❑❤✐ ✤â ✈➔♥❤ ✤❛ t❤ù❝ K[x] ❧➔ ♠ët ✈➔♥❤ ❊✉❝❧✐❞✳ ❈❤♦ V ❧➔ ♠ët K ✲❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì n ❝❤✐➲✉ ✈➔ T : V V ởt tỹ ỗ K t t rữợ t t ỹ ởt tró❝ K[x]✲♠ỉ✤✉♥ tr➯♥ V ✳ ❚❛ sû ❞ư♥❣ ♣❤➨♣ ❝ë♥❣ ✈➨❝tì tr➯♥ V ✈➔ ✤à♥❤ ♥❣❤➽❛ ♣❤➨♣ ♥❤➙♥ ♠ët ✈➨❝tì tr V ợ ởt ổ ữợ tr K[x] ữ s f (x) · v := f (T )(v), tr♦♥❣ ✤â ♥➳✉ f (x) = a x s f (T ) = as T s + · · · + a1 T + a0 Id s ✳ + · · · + a1 x + a0 ∈ K[x] ❇ê ✤➲ ✸✳✶✳ ❱ỵ✐ ❝→❝ ♣❤➨♣ t♦→♥ ✈ø❛ ✤à♥❤ ♥❣❤➽❛✱ V ❧➔ ♠ët K[x]✲♠æ✤✉♥ ①♦➢♥✳ ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ trü❝ t✐➳♣ ✤➸ t❤➜② V ❧➔ ♠ët K[x]✲♠æ✤✉♥✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ♥â ❧➔ ♠ët K[x]✲♠ỉ✤✉♥ ①♦➢♥✱ tù❝ ❧➔ ✈ỵ✐ ♠å✐ v ∈ V tỗ t f (x) K[x] s f (x) · v = 0✳ ❚❤➟t ✈➟②✱ ✤➦t f (x) := P (x)✱ ✤❛ t❤ù❝ ✤➦❝ tr÷♥❣ ❝õ❛ T ✳ ❚❤❡♦ ỵ t t õ ự T f (x) · v = f (T )(v) = PT (T )(v) = 0(v) = ❱➟② V ❧➔ ♠ët K[x]✲♠æ✤✉♥ ①♦➢♥✳ ✷✻ ▼æ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ữỡ ỵ tỗ t ❝→❝ ✤❛ t❤ù❝ ❧ã✐ a (x), a (x), , a (x) ∈ K[x] ✭❝→❝ ✤❛ t❤ù❝ ❝â ❤➺ sè ❝❛♦ ♥❤➜t ❜➡♥❣ ✶✮ s❛♦ ❝❤♦ a (x) | a (x) | · · · | a (x) ✈➔ ♠ët ✤➥♥❣ ❝➜✉ K[x]✲♠æ✤✉♥ ✭✸✳✶✮ V ∼ = K[x]/a (x) ⊕ K[x]/(a (x)) ⊕ · · · ⊕ K[x]/(a (x)) ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳ ❈→❝ ✤❛ t❤ù❝ a (x), a (x), , a (x) ð tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ T ✱ ❝❤ó♥❣ ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠❛ tr➟♥ ❝õ❛ T tr♦♥❣ ❜➜t ❦➻ ❝ì sð ♥➔♦ ❝õ❛ V ✳ ✣➦t V := K[x]/(a (x))✱ ✈ỵ✐ ≤ i ≤ m✳ ❇ê ✤➲ s❛✉ ✤➙② ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ m 1 m m i 2 m i ❇ê ✤➲ ✸✳✸✳ ❱ỵ✐ ♠å✐ ≤ i ≤ m✱ Vi ❧➔ ♠ët K ✲❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ K ✲❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì V ✳ ❍ì♥ ♥ú❛✱ Vi ❧➔ ❝→❝ K ✲❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❜➜t ❜✐➳♥ ✤è✐ ✈ỵ✐ T ✱ tù❝ ❧➔ T (Vi ) ⊂ Vi ✳ ❉♦ ✤â ♣❤➙♥ t➼❝❤ K[x]✲♠æ✤✉♥ tr♦♥❣ ✭✸✳✶✮ ❦➨♦ t❤❡♦ ♣❤➙♥ t➼❝❤ K ✲❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì V ∼ ✭✸✳✷✮ = V ⊕ V ⊕ ··· ⊕ V ❇ê ✤➲ ✸✳✸ ❝ô♥❣ s✉② r❛ r➡♥❣ t❛ ❝â t❤➸ t tỹ ỗ T : V V t tờ trỹ t tỹ ỗ T := T | ✱ ≤ i ≤ m✱ tù❝ ❧➔ T = T ⊕ T ⊕ ··· ⊕ T ✭✸✳✸✮ ❱ỵ✐ ♠é✐ ≤ i ≤ m✱ t❛ ①➨t tỹ ỗ T : V V ❦➼ ❤✐➺✉ ♠ët ♣❤➛♥ tû ❝õ❛ V = K[x]/(a (x)) ❧➔ p(x)✱ ❧ỵ♣ ❝õ❛ ✤❛ t❤ù❝ p(x) ∈ K[x]✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝➜✉ tró❝ K[x]✲♠ỉ✤✉♥ tr➯♥ V t❛ ❝â m i Vi m i i i i i Ti (p(x)) = xp(x) ●✐↔ sû a (x) = x + b x + · · · + b x + b ✳ ❑❤✐ ✤â 1, x, x , , x ❧➔ ♠ët ❝ì sð ❝õ❛ K ✲❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì V ✳ ❇ð✐ ✈➻ T (1) = x✱ T (x) = x ✱ ✳✳✳✱ T (x ) = x ✈➔ k i k−1 k−1 i i k−1 i i k Ti (xk−1 ) = −b0 − b1 x − · · · − bk−1 xk−1 , ♠❛ tr➟♥ ❝õ❛ T tr♦♥❣ ❝ì sð 1, x, x , , x ❧➔ i     Ci :=     0 0 0 0 0 k−1 −b0 −b1 −b2 −bk−2 −bk−1          k−1 ❈❤÷ì♥❣ ✸✳ ▼ỉ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣ ✷✽ ✣à♥❤ ♥❣❤➽❛ tr Ci ữủ tr ỗ ❤➔♥❤ ❝õ❛ ai(x)✳ ◆❤÷ ✈➟② t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ♠ët ❝ì sð ❝õ❛ V tø ❝ì sð ❝õ❛ ❝→❝ K ✲❦❤æ♥❣ ❣✐❛♥ ❝♦♥ V ✱ V ✱ ✳✳✳✱ V t❤❡♦ ❝→❝❤ tr➯♥✳ ❚r♦♥❣ ❝ì sð ♥➔②✱ ♠❛ tr➟♥ ❝õ❛ T = T ⊕ T ⊕ · · · ⊕ T ❧➔ C ⊕ C ⊕ · · · ⊕ C ✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✺✳ ▼❛ tr➟♥ C ⊕ C ⊕ · · · ⊕ C ✤÷đ❝ ❣å✐ ❧➔ ❞↕♥❣ ❝❤✉➞♥ ❤ú✉ t✛ ❝õ❛ T ✳ ◆➳✉ A ❧➔ ♠❛ tr➟♥ ❝õ❛ T tr♦♥❣ ♠ët ❝ì sð ♥➔♦ ✤â t❤➻ t❛ ❝ô♥❣ ♥â✐ ♠❛ tr➟♥ C ⊕ C ⊕ · · · ⊕ C ❧➔ ❞↕♥❣ ❝❤✉➞♥ ❤ú✉ t✛ ❝õ❛ A✳ ❉♦ ✤â ✤❛ t❤ù❝ ✤➦❝ tr÷♥❣ ❝õ❛ T ✈➔ ❝õ❛ ♠ët ♠❛ tr➟♥ A ❝õ❛ T tr♦♥❣ ♠ët ❝ì sð ♥➔♦ ✤â ❧➔ 1 m m 1 2 m m m PT (x) = PA (x) = PC1 (x)PC2 (x) · · · PCm (x) ❉➵ ❞➔♥❣ t➼♥❤ ✤÷đ❝ P Ci (x) = (x) ✈ỵ✐ ♠å✐ ≤ i ≤ m✳ ❱➻ ✈➟② PT (x) = PA (x) = a1 (x)a2 (x) · · · am (x) ▼➺♥❤ ✤➲ ✸✳✻✳ ❈❤♦ A ∈ M (n × n, K)✳ ✣❛ t❤ù❝ tè✐ t✐➸✉ mA(x) ❝õ❛ A ❜➡♥❣ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❧ỵ♥ ♥❤➜t am (x) ❝õ❛ A✳ ●✐↔ sû A ❧➔ ♠❛ tr ởt tỹ ỗ T : K K tr♦♥❣ ❝ì sð ❝❤➼♥❤ t➢❝ ❝õ❛ K ✳ ❑❤✐ ✤â✱ ♥❤÷ ♣❤➙♥ t➼❝❤ ð tr➯♥✱ T ❝↔♠ s✐♥❤ tr➯♥ K ♠ỉt ❝➜✉ tró❝ tü ♥❤✐➯♥ ❝õ❛ ♠ët K[x]✲♠ỉ✤✉♥✳ ❙û ❞ö♥❣ ♣❤➙♥ t➼❝❤ ✭✸✳✶✮ t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ n n n n Kn ∼ = K[x]/a1 (x) ⊕ K[x]/(a2 (x)) ⊕ · · · ⊕ K[x]/(am (x)), ✈ỵ✐ a (x), , a (x) ❧➔ ❝→❝ ♥❤➙♥ tû ❜➜t ❜✐➸♥ ❝õ❛ T ✭✈➔ ❝õ❛ A✮✳ ❈❤å♥ ♠ët ❝ì sð ❝õ❛ K ❜➡♥❣ ❝→❝❤ ❣ë♣ ❝→❝ ❝ì sð ❝õ❛ ❝→❝ K ✲❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ❝♦♥ K[x]/a (x)✱ ≤ i ≤ m✳ ◆➳✉ v = p(x) ❧➔ ♠ët ♣❤➛♥ tỷ tũ ỵ tr ởt ỡ s K ổ ❣✐❛♥ ✈➨❝tì K[x]/a (x) t❤➻ m n i i am (T )(v) = am (x) · p(x) = am (x)p(x) = tr♦♥❣ K[x]/a (x)✱ ❜ð✐ ✈➻ a (x) ❝❤✐❛ ❤➳t ❝❤♦ a (x)✳ ❱➟② a (T ) = 0✱ ✈➔ ❤➺ q✉↔ ❧➔ m (x) | a (T )✳ ▼➦t ❦❤→❝✱ ❧➜② ✈➨❝tì tr♦♥❣ K[x]/a (x)✳ ❉♦ m (T ) = ♥➯♥ i m A i m m m A mA (x) = mA (x) · = mA (T )(1) = 0(1) = tr♦♥❣ K[x]/a (x)✳ ❉♦ ✤â a (x) | m (x)✳ ❱➻ a t❤ù❝ ❧ã✐ ♥➯♥ m (x) = a (x)✳ m A m m A m (x) ✈➔ m (x) ❧➔ ❝→❝ ✤❛ A ❈❤÷ì♥❣ ✸✳ ▼ỉ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣ ✷✾ ✸✳✷ ❚➼♥❤ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠ët ♠❛ tr➟♥ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ự ỵ s ỵ tr A M at(nìn, K) ❑❤✐ ✤â ❞ò♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝➜♣ ❝❤♦ ❤➔♥❣ ✈➔ ❝ët ❝õ❛ ♠❛ tr➟♥ t❛ ❝â t❤➸ ✤÷❛ ♠❛ tr➟♥ xEn − A ∈ M (n × n, K[x]) ✈➲ ❞↕♥❣ s❛✉ ✤➙②              ✳✳✳ a1 (x) a2 (x) ✳✳✳ am (x)      ,      tr♦♥❣ ✤â a1 (x), , am (x) ∈ K[x] ❧➔ ❝→❝ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠❛ tr➟♥ A✳ ❈❤♦ V ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì n ❝❤✐➲✉ tr➯♥ K ✱ ❣å✐ v , v , , v ❧➔ ♠ët ❝ì sð ✭❝â t❤ù tü✮ ❝õ❛ V ✳ ●✐↔ sû T : V → V ởt tỹ ỗ tr V õ tr ❧➔ A tr♦♥❣ ❝ì sð v , v , , v ✳ ◆❤÷ tr➯♥ V ❧➔ ♠ët K[x]ổ tữỡ ự ợ ổ ữợ ❜ð✐ T ✳ ●å✐ ξ , ξ , , ξ ❧➔ ♠ët ❝ì sð ❝õ❛ K[x]✲♠ỉ✤✉♥ tü K[x] t K[x]ỗ 1 2 n n n n ϕ : K[x]n → V ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ ❝ì sð ❝õ❛ K[x] ❜ð✐ ϕ(x ) = v ✈ỵ✐ ♠å✐ ≤ i ≤ n✳ ❉♦ x · v = T (v ) = a v ♥➯♥ xv − a v = 0✱ tù❝ ❧➔ n j j n i=1 ij i j i n i=1 i ij i −a1j v1 − · · · − aj−1,j vj−1 + (x − ajj )vj − aj+1,j vj+1 − · · · − anj = ❱ỵ✐ ≤ j ≤ n✱ ❦➼ ❤✐➺✉ η ❧➔ ♣❤➛♥ tû j −a1j ξ1 − · · · − aj−1,j ξj−1 + (x − ajj )ξj − aj+1,j ξj+1 − · · · − anj ξn tr♦♥❣ K[x] ✳ ❱ỵ✐ ✤➥♥❣ t❤ù❝ ✭✸✳✹✮✱ t❛ ✈ø❛ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲ s❛✉ ✤➙②✳ ✭✸✳✹✮ ✭✸✳✺✮ n ❇ê ✤➲ ✸✳✽✳ ❱ỵ✐ ♠å✐ ≤ j ≤ n✱ ηj ❧➔ ♠ët ♣❤➛♥ tỷ ker ỗ t ♣❤➛♥ tû fj ∈ Kξ1 + Kξ2 + Kξn s❛♦ ❝❤♦ xξj = ηj + fj ❈❤÷ì♥❣ ✸✳ ▼ỉ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣ ✸✵ ✭❜✮ ✣➦t W := (K[x]η1 + K[x]η2 + · · · + K[x]ηn )+Kξ1 +Kξ2 +· · ·+Kξn ✳ ❑❤✐ ✤â W = K[x]ξ1 + K[x]ξ2 + · · · + K[x]ξn ✭❛✮ ❚ø ✭✸✳✺✮ s✉② r❛ xξ = η + a ξ + · · · + a ξ ✳ ✣➦t f = ✳ ❚❛ ❝â f ∈ Kξ + Kξ + · · · + Kξ ✈➔ xξ = η + f ✳ ✭❜✮ ❚❤❡♦ ❝➙✉ ✭❛✮ t❛ ❝â✿ xξ = η + c ξ + · · · + c ξ ∈ W, ✭✸✳✻✮ tr♦♥❣ ✤â c , , c ∈ K ✳ ●✐↔ sû x ξ ∈ W ✳ t x ữợ ự j n i=1 aij ξi i i j j j 1j n 1j m n nj n j j j nj n m j j xm ξj = δ1 (x)η1 + · · · + δn (x)ηn + d1 ξ1 + · · · + dn ξn , ✈ỵ✐ δ (x) ∈ K[x]✱ d ∈ K ✳ ❙✉② r❛ i i xm+1 ξj = xδ1 (x)η1 + · · · + xδn (x)ηn + xd1 ξ1 + · · · + xdn ξn ❚❤❡♦ ✭✸✳✻✮ t❛ ❝â d (xξ ) + · · · + d (xξ ) ∈ W ✳ ❙✉② r❛ x ξ ∈ W ✳ ❇➡♥❣ q✉② ♥↕♣ t❛ ❝â x ξ ∈ W ✈ỵ✐ ♠å✐ m ∈ N✱ ✈ỵ✐ ♠å✐ ≤ j ≤ n✳ ❚ø ✤â s✉② r❛ K[x]ξ + K[x]ξ + · · · + K[x]ξ ⊆ W ✳ ▼➦t ❦❤→❝ t❤❡♦ ✭✸✳✺✮✱ W ⊆ K[x]ξ + K[x]ξ + · · · + K[x]ξ ✳ ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ 1 n m+1 n j m j 1 n n ❇ê ✤➲ ✸✳✶✵✳ ❈→❝ ♣❤➛♥ tû η1, η2, , ηn ❧➟♣ ♥➯♥ ♠ët ❤➺ s✐♥❤ ❝õ❛ ker ϕ✳ ●✐↔ sû y ∈ ker ϕ ⊆ K[x]ξ ❚❤❡♦ ❇ê ✤➲ ✸✳✾ t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ + K[x]ξ2 + · · · + K[x]ξn ✳ y ∈ (K[x]η1 + K[x]η2 + · · · + K[x]ηn ) + Kξ1 + Kξ2 + · · · + Kξn , ♥➯♥ y ❝â ❞↕♥❣✿ y = γ1 (x)η1 + · · · + γn (x)ηn + λ1 ξ1 + · · · + λn ξn , tr♦♥❣ ✤â γ (x) ∈ K[x]✱ λ ϕ(y) = 0✱ ❞♦ ✤â i tù❝ ❧➔ λ v i ✱ ✳ ▼➦t ❦❤→❝ ✈➻ y ∈ ker ϕ ♥➯♥ ∈K 1≤i≤n γ1 (x)η1 + · · · + γn (x)ηn + λ1 (ξ1 ) + · · · + λn (ξn ) = 0, ✳ ❱➻ v , v , , v ❧➔ ♠ët ❝ì sð ❝õ❛ V ♥➯♥ ✳ ❱➻ ✈➟② y = γ (x)η + · · · + γ (x)η ✳ ❱➟② ❧➔ ♠ët ❤➺ s✐♥❤ ❝õ❛ ker ϕ✳ + · · · + λn = λ1 = λ2 = · · · = λn = η1 , η2 , , ηn 1 n 1 n n ❈❤÷ì♥❣ ✸✳ ▼ỉ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣ ✸✶ ❇ê ✤➲ ✸✳✶✶✳ ▼❛ tr➟♥ q tữỡ ự ợ ỡ s 1, 2, , ξn ❝õ❛ K[x]n ✈➔ ❤➺ s✐♥❤ η1 , η2 , , ηn ❝õ❛ ker ϕ ❧➔ ♠❛ tr➟♥ xEn − At ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✭✸✳✶✵✮✱ ✈ỵ✐ ≤ j ≤ n✱ t❛ ❝â ηj = −a1j ξ1 − · · · − aj−1,j ξj−1 + (x − ajj )ξj − aj+1,j ξj+1 − · · · − anj ξn ❉♦ õ tr q tữỡ ự ợ ỡ s ξ , ξ , , ξ ❝õ❛ K[x] ✈➔ ❤➺ s✐♥❤ η , η , , η ❝õ❛ ker ϕ ❧➔ 1 2 n n n x − a11 −a21  −a12 x − a22 B=  −a1n −a2n  −an1 −an2   = xEn − At  x − ann  ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ R = K[x] ✱ V = K ✳ ●å✐ e , e , , e ❧➔ ❝ì sð ❝❤➼♥❤ t➢❝ ❝õ❛ K ✳ ●å✐ ξ , ξ , , ξ ❧➔ ♠ët ❝ì sð ❝õ❛ K[x] ✳ ❳➨t →♥❤ ①↕ T : V V ữ s ự ỵ ✸✳✼✳ n n n n n n n T (ej ) = aji ei i=1 ❙✉② r❛ T ❧➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ t✉②➳♥ t➼♥❤ ❝â ♠❛ tr➟♥ ❧➔ A tr♦♥❣ ❝ì sð e , e , , e ✳ ❳➨t →♥❤ ①↕ t✉②➳♥ t➼♥❤ t n ϕ : K[x]n → V = K n s❛♦ ❝❤♦ ϕ(ξ ) = e ✈ỵ✐ ♠å✐ ≤ i ≤ n✳ ✣➦t i i ηi = −ai1 ξ1 − · · · − ai,i−1 ξi−1 + (x − aii )ξi − ai,i+1 ξi+1 − · · · − ain ξn , ✈ỵ✐ ≤ i ≤ n✳ ❚❤❡♦ ❜ê ✤➲ ✭✸✳✶✶✮✱ xE − A tr q tữỡ ự ợ , ξ , , ξ ❧➔ ❝ì sð ❝õ❛ K[x] ✱ ✈➔ η , η , , η ❧➔ ❤➺ s✐♥❤ ❝õ❛ ker ϕ✳ ❚❤❡♦ ỵ t õ t ũ ✤ê✐ ❝➜♣ ❝õ❛ ❤➔♥❣ ✈➔ ❝ët ❝õ❛ ♠❛ tr➟♥ ✤➸ ✤÷❛ ♠❛ tr➟♥ xE − A ✈➲ ♠❛ tr➟♥ B ❝â ❞↕♥❣ s❛✉ n n n n n  b1 (x)  b2 (x) B=  0   ,  bn (x) ❈❤÷ì♥❣ ✸✳ ▼ỉ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣ ✸✷ tr♦♥❣ ✤â b (x) ❧➔ ❝→❝ ✤❛ t❤ù❝ ❧ã✐✱ b (x) | b (x) | · · · | b (x) ✈➔ deg b (x) ú ỵ r V ởt K[x]ổ ①♦➢♥ ♥➯♥ b (x) = 0✱ ❞♦ ✤â t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ b (x) = 0✳ ●✐↔ sû b (x) = · · · = b (x) = 1✱ deg b (x) ≥ 1✱ ✈ỵ✐ ♠å✐ i ≥ l + 1✳ ❑❤✐ ✤â✱ ✤➦t a (x) = b (x), , a (x) = b (x) ✈➔ t❛ ❝â ♠❛ tr➟♥     ✳✳✳     i k i m i 1 l l+1 i m         a1 (x) a2 (x) tr♦♥❣ ✤â a (x) | · · · | a ✳✳✳ am (x) n   ,      ✱ a (x) ❧➔ ❝→❝ ✤❛ t❤ù❝ ❧ã✐✳ m (x) i ✸✳✸ Ù♥❣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ t➻♠ ✤❛ t❤ù❝ tè✐ t✐➸✉ ❱➼ ❞ö ✸✳✶✷✳ ❚➻♠ ❞↕♥❣ ❝❤✉➞♥ ❤ú✉ t✛ ❝õ❛ ♠❛ tr➟♥ s❛✉  422 465 15 −30  −420 −463 −15 30   A=  840 930 32 −60  −140 −155 −5 12  ●✐↔✐✿ ❳➨t ♠❛ tr➟♥  x − 422 −465 −15 30  420 x + 463 15 −30   xE4 − A =   −840 −930 x − 32 60  140 155 x − 12 ỏ tự t ợ ỏ tự tữ tr tr xE t ữủ tr ợ ữ s −A  140 155 x − 12  420 x + 463 15 −30     −840 −930 x − 32 60  x − 422 −465 −15 30  ❝❤♦ ♥❤❛✉ t❛ ❈❤÷ì♥❣ ✸✳ ▼ỉ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣ ✸✸ ▲➜② ❞á♥❣ t❤ù ❤❛✐ ❝ë♥❣ ✈ỵ✐ −3 ❧➛♥ ❞á♥❣ t❤ù ♥❤➜t✱ ❧➜② ❞á♥❣ t❤ù ❜❛ ❝ë♥❣ ✈ỵ✐ ✻ ❧➛♥ ❞á♥❣ t❤ù ♥❤➜t✱ ❞á♥❣ t❤ù t❛ ❝ë♥❣ ✈ỵ✐ ✸ ❧➛♥ ❞á♥❣ t❤ù ♥❤➜t t❛ t❤✉ ✤÷đ❝ ♠❛ tr➟♥  140 155 x − 12  x−2 −3(x − 2)     0 x − 6(x − 2)  x−2 0 3(x − 2)  ✣ê✐ ❝ët t❤ù ❜❛ ✈ỵ✐ ❝ët t❤ù ♥❤➜t tr♦♥❣ ♠❛ tr➟♥ tr➯♥ ❝❤♦ ♥❤❛✉ t❛ t❤✉ ✤÷đ❝ ♠❛ tr➟♥ ợ ữ s 155 140 x 12  x−2 −3(x − 2)     x−2 0 6(x − 2)  0 x − 3(x − 2)  ▲➜② ❞á♥❣ t❤ù ❜❛ ❝ë♥❣ ✈ỵ✐ − (x − 2) ❧➛♥ ❞á♥❣ t❤ù ♥❤➜t t❛ t❤✉ ✤÷đ❝ ♠❛ tr➟♥  155 140 x − 12 0  x−2 −3(x − 2)    −31(x − 2) −28(x − 2) (x − 2) − (x − 12)  0 x−2 3(x − 2)  ▲➜② ❝ët t❤ù ❤❛✐ ❝ë♥❣ ✈ỵ✐ −31 ❧➛♥ ❝ët t❤ù ♥❤➜t✱ ❧➜② ❝ët t❤ù ❜❛ ❝ë♥❣ ✈ỵ✐ −28 ❧➛♥ ởt tự t ởt tự tữ ợ (x − 12) ❧➛♥ ❝ët t❤ù ♥❤➜t t❛ t❤✉ ✤÷đ❝ ♠❛ tr➟♥   0 0  x−2 −3(x − 2)    −31(x − 2) −28(x − 2) (x − 2) − (x − 12)  0 x−2 3(x − 2) ▲➜② ❞á♥❣ t❤ù ❜❛ ❝ë♥❣ ✈ỵ✐ ✸✶ ❧➛♥ ❞á♥❣ t❤ù ❤❛✐ t❛ t❤✉ ✤÷đ❝ ♠❛ tr➟♥   0  x−2  −3(x − 2)   0 −28(x − 2) (x − 2) − 87 − (x − 12)  0 x−2 3(x − 2) ❈❤÷ì♥❣ ✸✳ ▼æ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ư♥❣ ✸✹ ❚✐➳♣ tư❝✱ t❛ ❧➜② ❝ët t❤ù t÷ ❝ë♥❣ ợ ởt tự t t ữủ tr➟♥  0   x−2 0   0 −28(x − 2) (x − 2) − 87 − 15 (x − 12)  0 x−2 3(x − 2)  ✣ê✐ ❞á♥❣ t❤ù tữ ợ ỏ tự t t ữủ ♠❛ tr➟♥  0   x−2 0    0 x−2 3(x − 2) 0 −28(x − 2) (x − 2) − 87 − 51 (x − 12)  ▲➜② ❞á♥❣ tự tữ ợ ỏ tự t t❤✉ ✤÷đ❝ ♠❛ tr➟♥  0   x−2 0    0 x−2 3(x − 2) 0 (x − 2) − − 51 (x − 12)  ▲➜② ❝ët tự tữ ợ ởt tự t t❤✉ ✤÷đ❝ ♠❛ tr➟♥   0  x−2 0  0 x−2 0 0 (x − 2) 51 x +    27 ❈❤✐❛ ❞á♥❣ t❤ù ♥❤➜t ❝❤♦ ✺ ✈➔ ♥❤➙♥ ✺ ✈➔♦ ❞á♥❣ t❤ù t÷ t❛ t❤✉ ✤÷đ❝ ♠❛ tr➟♥   0  x−2  0    0 x−2 0 0 (x − 2)(x + 27) ❚❤❡♦ ỵ tỷ t tr➟♥ A ❧➔ a1 (x) = x − 2, a2 (x) = x − 2, a3 (x) = (x − 2)(x + 27)   0  x−2  0   0  x−2 0 0 (x − 2)(x + 27) ❈❤÷ì♥❣ ✸✳ ▼ỉ✤✉♥ tr➯♥ ✈➔♥❤ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ✈➔ ù♥❣ ❞ö♥❣ ❈→❝ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠❛ tr➟♥ A ❧➔ a1 (x) = x − 2, a2 (x) = x − 2, ❉♦ ✤â✿ • ✣❛ t❤ù❝ ✤➦❝ tr÷♥❣ ❝õ❛ A ❧➔ a3 (x) = (x − 2)(x + 27) PA (x) = (x − 2)3 (x + 27); • ✣❛ t❤ù❝ tè✐ t✐➸✉ ❝õ❛ A ❧➔ mA (x) = a3 (x) = (x − 2)(x + 27) ✸✺ ❑➌❚ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ ✤➣ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ❝→❝ ❝ỉ♥❣ ✈✐➺❝ s❛✉ ✤➙②✿ • ✣å❝ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② ự ỵ trú ởt ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ♠ët ♠✐➲♥ ❝❤➼♥❤✱ t❤❡♦ ❈❤÷ì♥❣ ✶✷ tr♦♥❣ ❝✉è♥ s→❝❤ ❆❜str❛❝t ❆❧❣❡❜r❛ ❝õ❛ ❉✉♠♠✐t ✈➔ ❋♦♦t❡ ✭①❡♠ ❬✷❪✮✳ • ❚r➻♥❤ ❜➔② ♠ët t❤✉➟t t♦→♥ t➻♠ ❝→❝ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠ët ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ❊✉❝❧✐❞✳ • Ù♥❣ ❞ư♥❣ ❝õ❛ t❤✉➟t t♦→♥ t➻♠ ❝→❝ ♥❤➙♥ tû ❜➜t ❜✐➳♥ ❝õ❛ ♠ët ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ❊✉❝❧✐❞ ✤➸ t➻♠ ✤❛ t❤ù❝ tè✐ t✐➸✉ ởt tỹ ỗ tr ởt ổ tỡ ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ▼➦❝ ❞ò ✤➣ ❤➳t sù❝ ❝è ❣➢♥❣ s♦♥❣ ❞♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥✱ ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚ỉ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ ✈➔ õ õ ỵ ổ ✤➸ ❦❤â❛ ❧✉➟♥ ♥➔② ✤÷ì❝ ✤➛② ✤õ ✈➔ ❤♦➔♥ t❤✐➺♥ ỡ rữợ t tú õ ởt ♥ú❛ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ ❝→❝ ❚❤➛②✱ ❈ỉ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥✱ ✤➦❝ t ỵ ữớ t t ữợ ❞➝♥ ✈➔ ❣✐ó♣ ✤ï tỉ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣✱ ✣↕✐ sè ✤↕✐ ❝÷ì♥❣✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ö❝✱ ✶✾✾✽✳ ❬✷❪ ❉✳ ❉✉♠♠✐t✱ ❘✳ ❋♦♦t❡✱ ❆❜str❛❝t ❆❧❣❡❜r❛✱ ❏♦❤♥ ❲✐❧❡② ✫ ❙♦♥s✱ ■♥❝✳✱ ✷✵✵✹✳ ✸✼ ...TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* VŨ THỊ NGỌC ANH MƠĐUN TRÊN MIỀN CHÍNH KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Đại số Người hướng dẫn khoa học TS LÊ QUÝ