Phân tích đa thức thành các đa thức bất khả quy để xây dựng các mã cyclic trên trường hữu hạn

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖ ♦✵♦ ✖✖✖✖✖ ◆●❯❨➍◆ ❚❍➚ ❍⑨ P❍❹◆ ❚➑❈❍ ✣❆ ❚❍Ù❈ ❚❍⑨◆❍ ❈⑩❈ ✣❆ ❚❍Ù❈ ❇❻❚ ❑❍❷ ◗❯❨ ✣➎ ❳❹❨ ❉Ü◆● ❈⑩❈ ▼❶ ❈❨❈▲■❈ ❚❘➊◆ ❚❘×❮◆● ❍Ú❯ ❍❸◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆✱ ✽✴✷✵✷✵ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖ ♦✵♦ ✖✖✖✖✖ ◆●❯❨➍◆ ❚❍➚ ❍⑨ P❍❹◆ ❚➑❈❍ ✣❆ ❚❍Ù❈ ❚❍⑨◆❍ ❈⑩❈ ✣❆ ❚❍Ù❈ ❇❻❚ ❑❍❷ ◗❯❨ ✣➎ ❳❹❨ ❉Ü◆● ❈⑩❈ ▼❶ ❈❨❈▲■❈ ❚❘➊◆ ❚❘×❮◆● ❍Ú❯ ❍❸◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ ❚♦→♥ ❝➜♣ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✸ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿ ❚❙✳ ◆●❯❨➍◆ ❚❘➴◆● ❇➁❈ ❚❤→✐ ◆❣✉②➯♥✱ ✽✴✷✵✷✵ ▼ö❝ ❧ö❝ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✼ ✶✳✶✳ ❚r÷í♥❣ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳ ❱➔♥❤ ✤❛ t❤ù❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸✳ ✣❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ✤➸ ①➙② ❞ü♥❣ ❝→❝ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ✶✽ ✷✳✶✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ xn − t❤➔♥❤ ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✶✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ xn − tr➯♥ Fq ❦❤✐ (n, q) = ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✷✳ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ xn − tr➯♥ Fq ❦❤✐ (n, q) = ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✳ ▼➣ ❝②❝❧✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✸✳✶✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ❦❤✐ (n, q) = ✳ ✸✷ ✷✳✸✳✷✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ❦❤✐ (n, q) = ✳ ỵ tt t ❧➛♥ ✤➛✉ t✐➯♥ ✈➔♦ ♥➠♠ ✶✾✹✽ ❜ð✐ ♠ët ❝æ♥❣ tr➻♥❤ ỵ tt t ỹ tr tổ ứ õ ỵ tt ♥➔② ✤➣ ✈➔ ✤❛♥❣ ✤â♥❣ ❣â♣ ✤➸ ❣✐↔✐ q✉②➳t ♥❤✐➲✉ ✈➜♥ ✤➲ q✉❛♥ trå♥❣ tr♦♥❣ t❤æ♥❣ t✐♥ ❧✐➯♥ ❧↕❝✳ ◆â ✤÷đ❝ ù♥❣ ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ♥❤÷✿ t❤ỉ♥❣ t tỷ t t t t ỵ tt ♠➣ ❤â❛ ❧➔ ♠ët ♥❣➔♥❤ ❝õ❛ t♦→♥ ❤å❝ ✈➔ ❦❤♦❛ ❤å❝ ✤✐➺♥ t♦→♥ ♥❤➡♠ ❣✐↔✐ q✉②➳t t➻♥❤ tr↕♥❣ ❧é✐ ❞➵ ①↔② r❛ tr♦♥❣ q✉→ tr➻♥❤ tr✉②➲♥ t❤æ♥❣ sè ❧✐➺✉ tr➯♥ ❝→❝ ❦➯♥❤ tr✉②➲♥ ❝â ✤ë ♥❤✐➵✉ ❝❛♦✱ ❞ị♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ t✐♥❤ ①↔♦ ❦❤✐➳♥ ♣❤➛♥ ❧ỵ♥ ❝→❝ ❧é✐ ①↔② r❛ õ t ữủ sỷ ỵ tt ỏ ỷ ỵ ỳ t ũ ủ ợ ỳ ự ử t ỵ tt ❤â❛ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❧➽♥❤ ✈ü❝ q✉❛♥ trå♥❣ ❝õ❛ t♦→♥ ❤å❝✱ ❝â ↔♥❤ ❤÷ð♥❣ ✤➳♥ r➜t ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤♦❛ ❤å❝✲❝æ♥❣ ♥❣❤➺ ✈➔ ❦✐♥❤ t➳✲①➣ ❤ë✐✳ ❚❤ü❝ t➳ ❝❤♦ t ỵ tt õ ổ ũ q trồ tứ ữ ợ sỹ t tr rt ♥❤❛♥❤ ❝õ❛ ❝æ♥❣ ♥❣❤➺ t❤æ♥❣ t✐♥✱ ✈➔ ♠↕♥❣ ✐♥t❡r♥❡t t❤➻ ♠➣ ❤â❛ t❤æ♥❣ t✐♥ ❝➔♥❣ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣✳ ▼➣ ❤â❛ ❧➔ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❜↔♦ ✈➺ t❤ỉ♥❣ t✐♥✱ ❜➡♥❣ ❝→❝❤ ❝❤✉②➸♥ ✤ê✐ t❤æ♥❣ t✐♥ tø ❞↕♥❣ rã ✭t❤æ♥❣ t✐♥ ❝â t❤➸ ❞➵ ❞➔♥❣ ✤å❝ ❤✐➸✉ ✤÷đ❝✮ s❛♥❣ ❞↕♥❣ ♠í ✭t❤ỉ♥❣ t✐♥ ✤➣ ❜à ❝❤❡ ✤✐✱ ♥➯♥ ❦❤ỉ♥❣ t❤➸ ✤å❝ ❤✐➸✉ ✤÷đ❝✱ ✤➸ ✤å❝ ✤÷đ❝ t❛ ❝➛♥ ♣❤↔✐ ❣✐↔✐ ♠➣ ♥â✮✳ ◆â ❣✐ó♣ t❛ ❝â t❤➸ ❜↔♦ ✈➺ t❤ỉ♥❣ t✐♥✱ ✤➸ ♥❤ú♥❣ ❦➫ ✤→♥❤ ❝➢♣ t❤ỉ♥❣ t✐♥✱ ❞ị ❝â ✤÷đ❝ t❤ỉ♥❣ t✐♥ ❝õ❛ ❝❤ó♥❣ t❛✱ ❝ơ♥❣ ❦❤ỉ♥❣ t❤➸ ❤✐➸✉ ✤÷đ❝ ♥ë✐ ❞✉♥❣ ❝õ❛ ♥â✳ ▼➣ ❤â❛ s➩ ♠❛♥❣ ❧↕✐ t➼♥❤ ❛♥ t♦➔♥ ❝❛♦ ❤ì♥ ❝❤♦ t❤ỉ♥❣ t✐♥✱ ✤➦❝ ❜✐➺t ❧➔ tr♦♥❣ t❤í✐ ✤↕✐ ✐♥t❡r♥❡t ♥❣➔② ♥❛②✱ ❦❤✐ ♠➔ t❤ỉ♥❣ t q tr tr trữợ ✤➳♥ ✤÷đ❝ ✤➼❝❤✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ tỉ✐ ❝❤➾ r❛ ♠ët ✈➔✐ ù♥❣ ❞ö♥❣ ❝õ❛ ♠ët sè ♠➣ ❝ö t❤➸✳ ▼➣ ■❙❇◆ ✭■♥t❡r♥❛t✐♦♥❛❧ ❙t❛♥❞❛r❞ ❇♦♦❦ ◆✉♠❜❡r✮ ❧➔ ♠➣ sè t✐➯✉ ❝❤✉➞♥ q✉è❝ ✷ t➳ ❝â t➼♥❤ ❝❤➜t t❤÷ì♥❣ ♠↕✐ ❞✉② ♥❤➜t ✤➸ ①→❝ ✤à♥❤ ✤÷đ❝ ❝→❝ t❤ỉ♥❣ t✐♥ ✈➲ ♠ët q✉②➸♥ s→❝❤ ❜➜t ❦ý ✭♥❣æ♥ ♥❣ú ❝õ❛ ❝✉è♥ s→❝❤✱ q✉è❝ ❣✐❛ ①✉➜t ❜↔♥✱ ❧➽♥❤ ✈ü❝ ❝õ❛ ❝✉è♥ s→❝❤✱✳✳✳✮✳ ▼➣ ❇❈❍ ✭❇♦s❡✕❈❤❛✉❞❤✉r✐✕❍♦❝q✉❡♥❣❤❡♠ ❝♦❞❡s✮ ❧➔ ♠ët ❧♦↕✐ ♠➣ ❝②❝❧✐❝ ✈➔ ❧➔ ❧♦↕✐ ♠➣ sû❛ ❧é✐ q✉❛♥ trå♥❣✱ ❝â ❦❤↔ ♥➠♥❣ sû❛ ✤÷đ❝ ♥❤✐➲✉ ộ ữủ ự rở r ợ ❝â ✷ ❧ỵ♣ ❝♦♥ ❧➔ ♠➣ ❇❈❍ ♥❤à ♣❤➙♥ ✈➔ ♠➣ ❇❈❍ ❦❤æ♥❣ ♥❤à ♣❤➙♥✳ ❚r♦♥❣ sè ♥❤ú♥❣ ♠➣ ❇❈❍ ❦❤ỉ♥❣ ♥❤à ♣❤➙♥ ♥➔②✱ ❧ỵ♣ q✉❛♥ trå♥❣ ♥❤➜t ❧➔ ♠➣ ❘❡❡❞ ✲ ❙♦❧♦♠♦♥✳ ▼➣ ❘❡❡❞ ✲ ❙♦❧♦♠♦♥ ✤÷đ❝ ❘❡❡❞ ✈➔ ❙♦❧♦♠♦♥ ❣✐ỵ✐ t❤✐➺✉ ❧➛♥ ✤➛✉ t✐➯♥ ✈➔♦ ♥➠♠ ✶✾✻✵✱ ❧➔ ♠ët ♠➣ sû❛ s❛✐ t❤✉ë❝ ❧♦↕✐ ♠➣ t✉②➳♥ t➼♥❤✳ ▼➣ ❘❡❡❞ ✲ ❙♦❧♦♠♦♥ ✤÷đ❝ sû ❞ư♥❣ ✤➸ sû❛ ❝→❝ ❧é✐ tr♦♥❣ ♥❤✐➲✉ ❤➺ t❤è♥❣ t❤ỉ♥❣ t✐♥ sè ✈➔ tr♦♥❣ ❧÷✉ trỳ ỗ tt ữ trỳ tứ ✤➽❛ ❈❉✱ ❱❈❉✱✳✳✳✮✱ t❤æ♥❣ t✐♥ ❞✐ ✤ë♥❣ ❤❛② ❦❤æ♥❣ ❞➙② ✭✤✐➺♥ t❤♦↕✐ ❞✐ ✤ë♥❣✱ ❝→❝ ✤÷í♥❣ tr✉②➲♥ ❱✐❜❛✮✱ t❤ỉ♥❣ t✐♥ ✈➺ t✐♥❤✱ tr✉②➲♥ ❤➻♥❤ sè ❉❱❇✱ ❝→❝ ♠♦❞❡♠ tè❝ ✤ë ❝❛♦ ♥❤÷✿ ❆❉❙▲✱ ❱❉❙▲ ✳✳✳ ▼➣ ❘❡❡❞ ✲ ❙♦❧♦♠♦♥ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ sû❛ ❝→❝ ❜✐t ❧é✐ ①↔② r❛ ❣➛♥ ♥❤❛✉✳ ▼➣ ❇❈❍ ✤÷đ❝ ❞ị♥❣ ❝❤♦ ❝→❝ ❝➙② ❆❚▼✱ tr♦♥❣ ❤➺ t❤è♥❣ ❣✐❛♦ ❞à❝❤ ❝õ❛ ❝→❝ ♥❣➙♥ ❤➔♥❣✱✳✳✳ ▼➣ ❍❛❞❛♠❛r❞ ✤÷đ❝ ❞ị♥❣ tr♦♥❣ ✈✐➺❝ tr✉②➲♥ t❤ỉ♥❣ t✐♥ ✈➔ ❤➻♥❤ ↔♥❤ tø ❝→❝ t➔✉ ✈ơ trư✱ ❝→❝ ✈➺ t✐♥❤ ✈➲ ❚r→✐ ✣➜t✳ ❚r♦♥❣ ♠ỉ✐ tr÷í♥❣ ♥❤✐➵✉ ❧♦↕♥ ❦❤ỉ♥❣ ❦❤➼ ❧ỵ♥ t❤➻ t❤ỉ♥❣ t✐♥ ✈➔ ❤➻♥❤ ↔♥❤ s➩ ❜à ❜â♣ ♠➨♦✱ t❤❛② ✤ê✐ ❦❤✐ ✤÷đ❝ tr✉②➲♥ tr♦♥❣ ♠ỉ✐ tr÷í♥❣ ♥❤✐➵✉ ❧♦↕♥ ❦❤æ♥❣ ❦❤➼✱ ✈➻ t❤➳ ✈❛✐ trá ❝õ❛ ♠➣ ❍❛❞❛♠❛r❞ ❧➔ r➜t q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ❦❤→♠ ♣❤→ ✈ơ trư✳ ợ ữủ ũ tr q ❝→❝ q✉è❝ ❣✐❛ ✤➣ ✤â♥❣ ❣â♣ ❧ỵ♥ tỵ✐ ✈✐➺❝ ❜↔♦ ♠➟t t❤æ♥❣ t✐♥ ✈➔ tr✉②➲♥ ✤↕t t❤æ♥❣ t✐♥ tø q✉è❝ tợ q ữủ tỷ ữủ ợ t ❧➛♥ ✤➛✉ t✐➯♥ ✈➔♦ ♥➠♠ ✶✾✾✻ ❜ð✐ ❙❤♦r ❬✻❪✳ ❚r♦♥❣ t tổ tữớ ỳ ữủ ữ ữợ ❞↕♥❣ ✵ ✈➔ ✶✱ ❝á♥ ♠→② t➼♥❤ ❧÷đ♥❣ tû sû ❞ö♥❣ q✉❜✐ts ✭q✉❛♥t✉♠ ❜✐ts✮ ❝❤♦ ♣❤➨♣ ♠→② t➼♥❤ ❣❤✐ ❞ú ❧✐➺✉ ð ♥❤✐➲✉ tr↕♥❣ t❤→✐ ❝ị♥❣ ❧ó❝ ✭✈➼ ❞ư ❝â t❤➸ ❧➔ ✵✱ ❝â t❤➸ ❧➔ ✶ ❤♦➦❝ ❝â t❤➸ ❝ị♥❣ ❧ó❝ ❧➔ ✵ ✈➔ ✶✮✱ ✤✐➲✉ ♥➔② ❝❤♦ ♣❤➨♣ t ữủ tỷ ỷ ỵ ữủ ỳ t ♣❤ù❝ t↕♣ ❤ì♥✳ ◆❣÷í✐ t❛ t➼♥❤ t♦→♥ r➡♥❣ ❝→❝ ♠→② t➼♥❤ ❧÷đ♥❣ tû s➩ ❣✐↔✐ q✉②➳t ❝→❝ ✈➜♥ ✤➲ ♣❤ù❝ t↕♣ ♥❤❛♥❤ ❤ì♥ ❜➜t ❦ý ♠→② t➼♥❤ ❝ê ✤✐➸♥ ♥➔♦✳ ▼→② t➼♥❤ ❧÷đ♥❣ tû ❝ì ✸ ❜↔♥ ❦❤❛✐ t❤→❝ ❝→❝ q✉② t➢❝ ❝õ❛ ❝ì ❤å❝ ❧÷đ♥❣ tû ✤➸ t➠♥❣ tè❝ ✤ë t➼♥❤ t♦→♥✳ ❱✐➺❝ ①➙② ❞ü♥❣ ♠ët ♠→② t➼♥❤ ❧÷đ♥❣ tỷ ởt õ ữ ữợ ✤➛✉ ✤➣ ❝â ♥❤ú♥❣ t❤➔♥❤ ❝æ♥❣ tø ❝→❝ t➟♣ ✤♦➔♥ ợ tr t ợ ữ t rst ❈❤♦ ✤➳♥ ♥❛②✱ ♠→② t➼♥❤ ❧÷đ♥❣ tû ❦❤ỉ♥❣ ❝❤➾ ❞ø♥❣ ❧↕✐ ❧➔ ❝✉ë❝ ❝↕♥❤ tr❛♥❤ ✈➲ ❝æ♥❣ ♥❣❤➺ ❣✐ú❛ ❝→❝ t➟♣ ✤♦➔♥ ❝ỉ♥❣ ♥❣❤➺ ❧ỵ♥ ♠➔ ♥â ❝á♥ ❧➔ ❝✉ë❝ ❝↕♥❤ tr❛♥❤ ❣✐ú❛ ❝→❝ ❝÷í♥❣ q✉è❝ ✤➸ ♣❤ư❝ ✈ư ❝❤♦ ❤♦↕t ✤ë♥❣ t➻♥❤ ❜→♦ ♥â✐ r✐➯♥❣ ✈➔ q✉è❝ ♣❤á♥❣ ♥â✐ ❝❤✉♥❣✳ ❙ü r❛ ✤í✐ ❝õ❛ ♠→② t➼♥❤ ❧÷đ♥❣ tû s➩ ❧➔♠ ❝❤♦ ❝→❝ ❤➺ ♠➟t ♥ê✐ t✐➳♥❣ ♥❤÷ ❉❊❙ ✭t❤❡ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞✮✱ ❘❙❆✱✳✳✳ s➩ ❜à ♣❤→ tr♦♥❣ t÷ì♥❣ ❧❛✐ ❣➛♥✳ ▼➟t ♠➣ ❉❊❙ ❝â t❤➸ ①❡♠ ❧➔ t✉②➺t ✤è✐ ❛♥ t♦➔♥ ✈➻ ✤➸ ❣✐↔✐ ✤÷đ❝ ♥â ❝➛♥ ♣❤↔✐ ❦✐➸♠ tr❛ ♠ët ❞❛♥❤ s→❝❤ r➜t ❧ỵ♥ ❝→❝ ❝❤➻❛ ❦❤♦→ ♠➣ t✐➲♠ ♥➠♥❣✳ ❱➼ ❞ư ♥➳✉ ❝❤ó♥❣ t❛ sû ❞ư♥❣ ♠ët ♠→② t➼♥❤ ❝ê ✤✐➸♥ ✈ỵ✐ ✻✹ ❜✐ts✱ ❦❤✐ ✤â s➩ ❝â 264 tr↕♥❣ t❤→✐✳ ❱ỵ✐ ♠ët ♠→② t➼♥❤ ❝ê ✤✐➸♥✱ ❝ù ❝❤♦ ❧➔ ♠é✐ ❣✐➙② ❦✐➸♠ tr❛ ✤÷đ❝ ✷ t✛ tr↕♥❣ t❤→✐ t❤➻ ❝ô♥❣ ❝➛♥ ❦❤♦↔♥❣ ✸✵✵ ♥➠♠ ❝❤↕② ♠→② tử ợ ữủ t 264 tr tõ ♠ët ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ♣❤✐ t❤ü❝ t✐➵♥✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ♠ët ♠→② t➼♥❤ ❧÷đ♥❣ tû ❞ị♥❣ t❤✉➟t t♦→♥ ❧÷đ♥❣ tû ●r♦✈❡r ❝â t❤➸ ❞➵ ❞➔♥❣ ❤♦➔♥ t➜t ✈✐➺❝ ♥➔② tr♦♥❣ t❤í✐ ❣✐❛♥ ✹ ♣❤ót✳ ❚❤✉➟t t♦→♥ ♠➣ ❤â❛ ❝ỉ♥❣ ❦❤❛✐ ❘❙❆ ✤❛♥❣ ✤÷đ❝ ù♥❣ ❞ư♥❣ rë♥❣ r➣✐ tr♦♥❣ ♥❣➙♥ ❤➔♥❣✱ ❣✐❛♦ ❞à❝❤ trü❝ t✉②➳♥ ✈➔ r➜t ♥❤✐➲✉ ù♥❣ ❞ö♥❣ ❛♥ ♥✐♥❤ ♠↕♥❣ ❦❤→❝✳ ❙ü ❛♥ t♦➔♥ ❝õ❛ ♠➣ ❘❙❆ ♥➡♠ ð ❝❤é ♠→② t➼♥❤ tr✉②➲♥ t❤è♥❣ ❦❤æ♥❣ t❤➸ ♣❤➙♥ t➼❝❤ ♥❤❛♥❤ ♠ët sè ♥û❛ ♥❣✉②➯♥ tè ✭s❡♠✐♣r✐♠❡✮ ❧ỵ♥ n t❤➔♥❤ t➼❝❤ ❝õ❛ ✷ sè ♥❣✉②➯♥ tè ❧ỵ♥ p ✈➔ q (n = pq)✳ ❱➲ ♠➦t t♦→♥ ❤å❝ ✤➙② ❧➔ ♠ët ❜➔✐ t♦→♥ ♣❤ù❝ t↕♣✱ ❝❤➥♥❣ ❤↕♥ ✤➸ ♣❤➙♥ t➼❝❤ ♠ët số ỗ ỳ số t t ❝ê ✤✐➸♥ ✤➣ ♣❤↔✐ ❤đ♣ ❧ü❝ ❧➔♠ ✈✐➺❝ ❧✐➯♥ tư❝ tr♦♥❣ ✈➔✐ t❤→♥❣✳ ❚✉② ♥❤✐➯♥✱ ♠ët ♠→② t➼♥❤ ❧÷đ♥❣ tû ❞ị♥❣ t❤✉➟t t♦→♥ ❧÷đ♥❣ tû ❙❤♦r ❝â t❤➸ ♣❤➙♥ t➼❝❤ ♠ët sè ❧ỵ♥ ❤ì♥ ❝↔ tr✐➺✉ ❧➛♥ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ♥❣➢♥ ❤ì♥ ❝ơ♥❣ ❝↔ tr✐➺✉ ❧➛♥✳ ❚r♦♥❣ ❧➽♥❤ ✈ü❝ s✐♥❤ ❤å❝✱ ❦❤→✐ ♥✐➺♠ ♠➣ ❉◆❆ ✤÷đ❝ ✤÷❛ r❛ ❧➛♥ ✤➛✉ t✐➯♥ ✈➔♦ ♥➠♠ ✷✵✵✸✱ ♥❤➡♠ ❣✐ó♣ ♥❤➟♥ ❞✐➺♥ ❝→❝ ♠➝✉ ✈➟t✳ ▼➣ ❉◆❆ sû ❞ö♥❣ ♠ët tr➻♥❤ tü ❉◆❆ ♥❣➢♥ ♥➡♠ tr♦♥❣ ❜ë ❣❡♥❡ ❝õ❛ s✐♥❤ ✈➟t ♥❤÷ ❧➔ ởt ộ ỵ tỹ t ú t s t ợ ữ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ✤à♥❤ ❞❛♥❤ ♠➔ ♥â sû ❞ư♥❣ ♠ët ✤♦↕♥ ❉◆❆ ❝❤✉➞♥ ♥❣➢♥ ♥➡♠ tr♦♥❣ ❜ë ❣❡♥❡ ✹ ❝õ❛ s✐♥❤ ✈➟t ✤❛♥❣ ♥❣❤✐➯♥ ❝ù✉ ♥❤➡♠ ①→❝ ✤à♥❤ s✐♥❤ ✈➟t ✤â t❤✉ë❝ ✈➲ ❧♦➔✐ ♥➔♦✳ ▼➣ ✈↕❝❤ ❉◆❆ r➜t ❤ú✉ ➼❝❤ tr♦♥❣ ✈✐➺❝ t➻♠ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❝→❝ ♠➝✉ ♠➦❝ ❞ị ❝❤ó♥❣ ❤➛✉ ♥❤÷ ❦❤ỉ♥❣ ❣✐è♥❣ ♥❤❛✉ ✈➲ ❤➻♥❤ t❤→✐✳ ▼➣ ✈↕❝❤ ❉◆❆ ❝ơ♥❣ ✤÷đ❝ ù♥❣ ❞ư♥❣ t↕✐ ❤↔✐ q ộ trủ ỗ ố s✐♥❤ ✈➟t sè♥❣ ❤♦➦❝ ❤➔♥❣ ♥❤➟♣ ❦❤➞✉✱ ✤➸ ♥❣➠♥ ❝↔♥ sü ✈➟♥ ❝❤✉②➸♥ tr→✐ ♣❤➨♣ ❝→❝ ❧♦➔✐ t❤ü❝ ✈➟t ✈➔ t qỵ q ợ ú ❦✐➸♠ s♦→t t→❝ ♥❤➙♥ ❣➙② ❤↕✐ tr♦♥❣ ♥ỉ♥❣ ♥❣❤✐➺♣✱ ❣✐ó♣ ✤à♥❤ ❞❛♥❤ ♥❤❛♥❤ ❝❤è♥❣ ❝→❝ ❧♦➔✐ ❣➙② ❜➺♥❤ ð ❣✐❛✐ ✤♦↕♥ t✐➲♠ ➞♥ ✭❣✐❛✐ ✤♦↕♥ ➜✉ trị♥❣✮✱ ❤é trđ ❝❤÷ì♥❣ tr st s trỗ r❛✱ ♠➣ ❉◆❆ ❣✐ó♣ ①→❝ ✤à♥❤ ✈➟t ❝❤õ tr✉♥❣ ❣✐❛♥ ❣➙② ❜➺♥❤✱ ❜↔♦ ✈➺ ❧♦➔✐ ♥❣✉② ❝➜♣ ✈➔ ❦✐➸♠ tr❛ t ữủ ữợ ởt số ❧ỵ♣ ♠➣ ❝②❝❧✐❝ ✤➣ ♥➯✉ ð tr➯♥✱ ❣✐ó♣ ❝❤ó♥❣ t❛ t❤➜② ✤÷đ❝ ♣❤➛♥ ♥➔♦ ✈❛✐ trá q✉❛♥ trå♥❣ ❝õ❛ ♠➣ ❝②❝❧✐❝ tr♦♥❣ ❝✉ë❝ sè♥❣✱ tr♦♥❣ ❦❤♦❛ ❤å❝ ❦➽ t❤✉➟t✳ ✣➛✉ t ỵ tt ữủ ự tr trữớ ỳ ❤↕♥ ✈➔ ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ ✤➣ ✤÷đ❝ ✤ó❝ ❦➳t tr♦♥❣ ❤❛✐ q✉②➸♥ s→❝❤ ❝õ❛ ❍✉❢❢♠❛♥ ✈➔ ❇❡r❧❡❦❛♠♣ ❬✺❪✳ ❙❛✉ ✤â✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ ♠ð rë♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♠➣ tr➯♥ ❝→❝ ✈➔♥❤ ❤ú✉ ❤↕♥✳ ❍➛✉ ❤➳t ❝→❝ ♥❣❤✐➯♥ ❝ù✉ t➟♣ tr✉♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤ë ❞➔✐ ❝õ❛ ♠➣ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➦❝ sè ❝õ❛ tr÷í♥❣✳ ◆➳✉ ✤ë ❞➔✐ ❝õ❛ ♠➣ ❝❤✐❛ ❤➳t ❝❤♦ ✤➦❝ sè ❝õ❛ tr÷í♥❣ t❤➻ ♠➣ ✤÷đ❝ ❣å✐ ❧➔ ♠➣ ♥❣❤✐➺♠ ❧➦♣✳ ◆➳✉ ✤ë ❞➔✐ ❝õ❛ ♠➣ ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✤➦❝ sè ❝õ❛ tr÷í♥❣ t❤➻ ♠➣ ✤â ✤÷đ❝ ❣å✐ ❧➔ ♠➣ ♥❣❤✐➺♠ ✤ì♥✳ ◆❣❤✐➯♥ ❝ù✉ ✈➲ ♠➣ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❤ú✉ ❤↕♥✱ ✤➦❝ ❜✐➺t ❧➔ ♠➣ ♥❣❤✐➺♠ ❧➦♣ tr➯♥ ❧ỵ♣ ❝→❝ ✈➔♥❤ ❝❤✉é✐ ❤ú✉ ❤↕♥ ❝ơ♥❣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ✈➔ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ❝ô♥❣ ✤➣ ✤÷❛ r❛ ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔ tèt✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ sû ❞ư♥❣ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❚♦→♥ ❤å❝ ✤➸ ①➙② ❞ü♥❣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔✿ tr➻♥❤ ❜➔② sü ♣❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ❙❛✉ ✤â sû ❞ö♥❣ ❦➳t q✉↔ ❝õ❛ sü ♣❤➙♥ t➼❝❤ ♥➔② ✤➸ ①➙② ❞ü♥❣ ❝→❝ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ▲✉➟♥ ỗ ữỡ r ữỡ ú tổ tr ❜➔② ✤à♥❤ ♥❣❤➽❛ tr÷í♥❣ ❤ú✉ ❤↕♥✱ ❝➜✉ tró❝ ❝õ❛ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ❙❛✉ ✤â ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✈➔♥❤ ✤❛ t❤ù❝ tr➯♥ tr÷í♥❣ ❤ú✉ ✺ ❤↕♥✳ ❈✉è✐ ❝❤÷ì♥❣ ✶ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉②✳ ❚r♦♥❣ ❝❤÷ì♥❣ ✷✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔✿ ♣❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✱ ♠➣ ❝②❝❧✐❝✱ ①➙② ❞ü♥❣ ❝→❝ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ✣➸ t➻♠ t➜t ❝↔ ❝→❝ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ Fq ✱ tr♦♥❣ ✤â q = pm ✭p ❧➔ sè ♥❣✉②➯♥ tè ❜➜t ❦➻✮ ❝❤ó♥❣ tỉ✐ ✤✐ t➻♠ ♥❤ú♥❣ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ Rn = Fq [X]/ xn − ◆ë✐ ❞✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❧✉➟♥ ✈➠♥ ❣➢♥ ❧✐➲♥ ✈ỵ✐ t♦→♥ ❝➜♣✱ ✤➦❝ ❜✐➺t ❧➔ ❜➔✐ t♦→♥ ♣❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû r➜t ✤÷đ❝ q✉❛♥ t➙♠ ð ❜➟❝ ❤å❝ ♣❤ê t❤ỉ♥❣✳ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ t ữợ sỹ ữợ s rồ ổ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s tợ ữớ ữợ ổ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ✕ ❚✐♥ ❝ò♥❣ ❝→❝ ❣✐↔♥❣ ✈✐➯♥ ✤➣ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕②✱ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ tỉ✐ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❙ð ●✐→♦ ❞ư❝ ✈➔ ✣➔♦ t t ỗ ♥❣❤✐➺♣ tr÷í♥❣ ❚❍P❚ ❍♦➔♥❣ ◗✉è❝ ❱✐➺t✱ ❤✉②➺♥ ❱ã ◆❤❛✐✱ t➾♥❤ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ❤♦➔♥ t❤➔♥❤ tèt ♥❤✐➺♠ ✈ư ❤å❝ t➟♣ ✈➔ ❝ỉ♥❣ t→❝ ❝õ❛ ♠➻♥❤✳ ❈✉è✐ ❝ị♥❣ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❣✐❛ ✤➻♥❤ t❤➙♥ ②➯✉✱ ❝↔♠ ì♥ ♥❤ú♥❣ ♥❣÷í✐ ❜↕♥ t❤➙♥ t❤✐➳t ✤➣ ❣✐ó♣ ✤ï ✤ë♥❣ ✈✐➯♥ ❦❤➼❝❤ ❧➺ tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✽ ♥➠♠ ✷✵✷✵ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ❍➔ ✻ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ❚r÷í♥❣ ❤ú✉ ❤↕♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❚r÷í♥❣ ❧➔ ♠ët t➟♣ ủ F ũ ợ t +, ữủ ❧➔ ❝ë♥❣✱ ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ♥❤➙♥ t❤ä❛ ♠➣♥ ♠ët sè t✐➯♥ ✤➲✳ ❚➟♣ F ❧➔ ♥❤â♠ ❦❤æ♥❣ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ 0; ❚➟♣ F∗ = F\{0} ❝ơ♥❣ ❧➔ ♥❤â♠ ❣✐❛♦ ❤♦→♥ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ❝â ♣❤➛♥ tû ✤ì♥ ✈à ❧➔ ♠ët ✈➔ ❦➼ ❤✐➺✉ ❧➔ 1; ✈➔ ố ợ ởt trữớ ❤ú✉ ❤↕♥ ♥➳✉ sè ♣❤➛♥ tû ❝õ❛ F ❧➔ ❤ú✉ ❤↕♥❀ ❙è ♣❤➛♥ tû ❝õ❛ F ✤÷đ❝ ❣å✐ ❧➔ ❝➜♣ ❝õ❛ F ❣✐❛♦ ❤♦→♥ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ❝â ♣❤➛♥ tû ✤ì♥ ✈à ❧➔ ❱➼ ❞ư ✶✳✶✳ ✭✐✮ ❚➟♣ ❤đ♣ ❝→❝ sè ♥❣✉②➯♥ Z ❦❤ỉ♥❣ ❧➔ ♠ët tr÷í♥❣ ✈➻ ∈ Z ❦❤ỉ♥❣ ❦❤↔ ♥❣❤à❝❤✳ ✭✐✐✮ ❈→❝ t➟♣ ❤đ♣ sè ❤ú✉ t➾ Q✱ sè t❤ü❝ R✱ sè ♣❤ù❝ C ❝ị♥❣ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ✈➔ ♥❤➙♥✱ t↕♦ t❤➔♥❤ ♠ët tr÷í♥❣✳ √ √ ✭✐✐✐✮ ❚➟♣ ❤ñ♣ Q[ 2] = {a + b : a, b ∈ Q} ✤â♥❣ ❦➼♥ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ tổ tữớ ũ ợ t♦→♥ ♥➔②✱ Q[ 2] ❧➔ ♠ët tr÷í♥❣✱ ♣❤➛♥ √ √ tû ❦❤ỉ♥❣ ❧➔ + 2, ♣❤➛♥ tû ✤ì♥ ✈à ❧➔ + 2, ♣❤➛♥ tû ✤è✐ ❝õ❛ ♣❤➛♥ √ √ √ √ tû a + b ❧➔ −a − b ✈➔ ♥➳✉ x = a + b = + t❤➻ ♥❣❤à❝❤ ✤↔♦ √ a b ❝õ❛ x ❧➔ − a − 2b2 a2 − 2b2 ❱➼ ❞ư ✶✳✷✳ ❚r÷í♥❣ ❤ú✉ ❤↕♥ F2 ✈ỵ✐ ❤❛✐ ♣❤➛♥ tû {0, 1}✱ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥ ✼ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ♥❤÷ s❛✉✿ ✰ ✵ ✶ ✵ ✵ ✶ ✶ ✶ ✵ ✳ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✶ ✣➙② ❝ô♥❣ ❧➔ ✈➔♥❤ ❝õ❛ ❝→❝ sè ♥❣✉②➯♥ ♠♦❞✉❧♦ ✷✳ ❱➼ ❞ö rữớ ỳ F3 ợ tỷ {0, 1, 2}✱ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥ ✤÷đ❝ ❝❤♦ ❜ð✐ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥ ♠♦❞✉❧♦ ✸✿ ✰ ✵ ✶ ✷ ✵ ✵ ✶ ✷ ✶ ✶ ✷ ✵ ✷ ✷ ✵ ✶ ✳ ✵ ✶ ✷ ✵ ✵ ✵ ✵ ✶ ✵ ✶ ✷ ✷ ✵ ✷ ✶ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✭✐✮ ◆➳✉ K ❧➔ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ E t❤➻ t❛ ❣å✐ E ❧➔ ♠ët tr÷í♥❣ ♠ð rë♥❣ ❝õ❛ K, ❦➼ ❤✐➺✉ ❧➔ E/K ✭✐✐✮ ●✐↔ sû E/K ❧➔ ♠ët ♠ð rë♥❣ tr÷í♥❣✳ ❳❡♠ E ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì tr➯♥ K ◆➳✉ E ❧➔ K ✲ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì ❤ú✉ ❤↕♥ ❝❤✐➲✉ t❤➻ t❛ ♥â✐ E ❧➔ ♠ð rë♥❣ ❜➟❝ ❤ú✉ ❤↕♥ ❝õ❛ tr÷í♥❣ K ◆➳✉ ❞✐♠ K E = n t❤➻ n ✤÷đ❝ ❣å✐ ❧➔ ❜➟❝ ❝õ❛ ♠ð rë♥❣ E/K ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ [E/K] ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ●✐↔ sû E/K ❧➔ ♠ët ♠ð rë♥❣ tr÷í♥❣ ✈➔ f (x) ∈ K[x] ❧➔ ✤❛ t❤ù❝ ❜➟❝ n ≥ ❚❛ ♥â✐ f (x) ♣❤➙♥ r➣ tr➯♥ E ♥➳✉ f (x) = a(x − α1 ) (x − αn ) ✽ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ g(x) ❧➔ ✤❛ t❤ù❝ ♠♦♥✐❝ ❝â ❜➟❝ ♥❤ä ♥❤➜t tr♦♥❣ C ❚ø C ❧➔ ❦❤→❝ ❦❤æ♥❣ ♥➯♥ tự g(x) ổ tỗ t c(x) C, t ợ ữ tr Fq [x], c(x) = g(x)h(x) + r(x), tr♦♥❣ ✤â ❤♦➦❝ r(x) = ❤♦➦❝ deg f (x) < deg g(x) ❱➻ C ❧➔ ✐✤➯❛♥ tr♦♥❣ Rn , r(x) ∈ C ✈➔ g(x) ❧➔ ✤❛ t❤ù❝ ❝â ❜➟❝ ♥❤ä ♥❤➜t tr♦♥❣ C s✉② r❛ r(x) = ❚ø r(x) = t❛ ❝â ✭✐✮ ợ ữ t õ xn − = g(x)h(x) + r(x), tr♦♥❣ ✤â r(x) = ❤♦➦❝ deg f (x) < deg g(x) tr♦♥❣ Fq [x]✳ ❱➻ xn − ù♥❣ ✈ỵ✐ tø ♠➣ ✵ tr♦♥❣ C ✈➔ C ❧➔ ✐✤➯❛♥ tr♦♥❣ Rn , t❛ ❝â r(x) ∈ C, ✤➙② ❧➔ ♠ët ♠➙✉ t❤✉➝♥ trø ❦❤✐ r(x) = ❙✉② r❛ ✤❛ t❤ù❝ s✐♥❤ g(x) ữợ xn sỷ r deg g(x) = n − k ❚ø ✭✐✐✮ ✈➔ ✭✐✐✐✮✱ ♥➳✉ c(x) ∈ C ✈ỵ✐ c(x) = ❤♦➦❝ deg c(x) < n t❤➻ c(x) = g(x)f (x) tr♦♥❣ Fq [x] ◆➳✉ c(x) = 0, t❤➻ f (x) = ◆➳✉ c(x) = ✈➔ deg c(x) < n, ✤✐➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ deg g(x) < k ✈➻ deg c(x) = deg(x) + deg f (x) ❉♦ ✤â C = {f (x)g(x)|f (x) = ❤♦➦❝ deg f (x) < k} ✈➔ ❤➺ {g(x), xg(x), , xk−1 g(x)} ❧➔ ❤➺ s✐♥❤✳ ❚ø ❤➺ s✐♥❤ ❝õ❛ C t❛ t❤➜② ❝â k ✤❛ t❤ù❝ ❝â ❜➟❝ ❦❤→❝ ♥❤❛✉ ✈➔ ❝❤ó♥❣ ✤ë❝ ❧➟♣ tr♦♥❣ Fq [x], ✤✐➲✉ ♥➔② ❝❤➾ r ỵ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ✈➼ ❞ư ✤➸ ♠✐♥❤ ❤å❛ ❝ư t❤➸✳ ✷✳✸✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ✷✳✸✳✶✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ❦❤✐ (n, q) = ❙❛✉ ✤➙② ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ ✼ tr➯♥ ♠ët sè tr÷í♥❣ ❤ú✉ ❤↕♥✳ ❱➼ ❞ư ✷✳✼✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ ✼ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ F2 ✣❛ t❤ù❝ x7 − = (1 + x)(1 + x + x3 )(1 + x2 + x3 ) ❙✉② r❛ ❝â ✽ ♠➣ ❝②❝❧✐❝ ✤ë ❞➔✐ tữỡ ự ợ tự s s ✭✐✮ = ✭✐✐✮ + x = + x ✸✷ ✭✐✐✐✮ + x + x3 = + x + x3 ✭✐✈✮ + x2 + x3 = + x2 + x3 ✭✈✮ (1 + x) + x + x3 = + x2 + x3 + x4 ✭✈✐✮ (1 + x) + x2 + x3 = + x + x2 + x4 ✭✈✐✐✮ + x + x3 + x2 + x3 = + x + x2 + x3 + x4 + x5 + x6 ✭✈✐✐✐✮ (1 + x) + x + x3 + x2 + x3 = + x7 ✣❛ t❤ù❝ s✐♥❤ g(x) = s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ❝❤➼♥❤ F72 ✣❛ t❤ù❝ s✐♥❤ g(x) = 1+x7 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ỗ tỷ {0000000} tự s g(x) = + x s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 6) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔  G6×7 1 0 0  0   0  = 0   0  1 0 0 1 0 0 1 0 0 1   0   0   0   0  0 0 1 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x3 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 4) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔  G4×7 1 0    0 1 0    =  0 1    0 1 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x2 + x3 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 4) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔  G4×7  0  = 0  0 1 0   1 0   1 0  0 1 ✸✸ ✣❛ t❤ù❝ s✐♥❤ g(x) = + x2 + x3 + x4 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 3) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G3×7   1 1 0   = 0 1 1 0   0 1 1 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x2 + x4 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 3) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G3×7   1 1 0   = 0 1 1 0   0 1 1 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x2 + x3 + x4 + x5 + x6 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 1) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G1×7 = 1 1 1 ❱➼ ❞ö ✷✳✽✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ ✼ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ F3 ✣❛ t❤ù❝ x7 − = (2 + x)(1 + x + x2 + x3 + x4 + x5 + x6 ) ❙✉② r❛ ❝â tữỡ ự ợ ✤❛ t❤ù❝ s✐♥❤ s❛✉ ✤➙②✿ ✭✐✮ = ✭✐✐✮ + x = + x ✭✐✐✐✮ + x + x2 + x3 + x4 + x5 + x6 = + x + x2 + x3 + x4 + x5 + x6 ✭✐✈✮ (2 + x) + x + x2 + x3 + x4 + x5 + x6 = + x7 ✣❛ t❤ù❝ s✐♥❤ g(x) = s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ❝❤➼♥❤ F73 ✣❛ t❤ù❝ s✐♥❤ g(x) = 2+x7 s➩ s✐♥❤ r❛ ỗ tỷ {0000000} t❤ù❝ s✐♥❤ g(x) = + x s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 6) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ✸✹ ❝②❝❧✐❝ ♥➔② ❧➔ G6×7   0   0  = 0   0  0 0   0 0   0 0   0 0   0 0  0 0 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x2 + x3 + x4 + x5 + x6 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 1) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G1×7 = 1 1 1 ❱➼ ❞ö ✷✳✾✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ ✼ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ F5 ✣❛ t❤ù❝ x7 − = (4 + x)(1 + x + x2 + x3 + x4 + x5 + x6 ) ❙✉② r❛ ❝â ✹ ♠➣ tữỡ ự ợ tự s✐♥❤ s❛✉ ✤➙②✿ ✭✐✮ = ✭✐✐✮ + x = + x ✭✐✐✐✮ + x + x2 + x3 + x4 + x5 + x6 = + x + x2 + x3 + x4 + x5 + x6 ✭✐✈✮ (4 + x) + x + x2 + x3 + x4 + x5 + x6 = + x7 ✣❛ t❤ù❝ s✐♥❤ g(x) = s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ❝❤➼♥❤ F75 ✣❛ t❤ù❝ s✐♥❤ g(x) = 4+x7 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ỗ tỷ {0000000} tự s g(x) = + x s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 6) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G6×7   0   0  = 0   0  0 0   0 0   0 0   0 0   0 0  0 0 ✸✺ ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x2 + x3 + x4 + x5 + x6 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 1) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G1×7 = 1 1 1 ✷✳✸✳✷✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ❦❤✐ (n, q) = P❤➛♥ t✐➳♣ t❤❡♦ ❝õ❛ ❧✉➟♥ ✈➠♥✱ ❝❤ó♥❣ tỉ✐ s➩ ①➙② ❞ü♥❣ ❝→❝ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ n = ps tr➯♥ tr÷í♥❣ Fpm ✳ ●✐↔ sû λ ❧➔ ♠ët ♣❤➛♥ tû ❦❤→❝ ❝õ❛ Fpm ❈❤ó♥❣ t❛ ✤➣ Fpm [x] ❜✐➳t r➡♥❣ ♠➣ λ−❝♦♥st❛❝②❝❧✐❝ ❧➔ ♥❤ú♥❣ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ ps ỵ x tt t tỗ t ỳ số ổ uq ur s❛♦ ❝❤♦ (uq +1)m−s s = uq m + ur ✈➔ ≤ ur ≤ m − ✣➦t λ0 = λp s (uq +1)m λp0 = λp m−ur = λp ❑❤✐ ✤â t❛ ❝â = λ ❚ø ✤✐➲✉ ♥➔② ❝❤ó♥❣ t❛ ❝â ❦➳t q✉↔ s❛✉✿ ▼➺♥❤ ✤➲ ✷✳✻✳ x − λ0 ❧➔ ❧ơ② ❧✐♥❤ tr♦♥❣ xFpp −[x]λ ✈ỵ✐ ❝❤➾ sè ❧ô② ❧✐♥❤ ❧➔ ps t ♠✐♥❤✳  ❈❤ù♥❣   ✣➛✉ t✐➯♥ t❛ t❤➜② r➡♥❣ ✈ỵ✐ ≤ i p t p ữợ m s pt pt m   , ❞♦ ✤â   = tr♦♥❣ Fpm [x] ❱➻ t❤➳ tr♦♥❣ ✈➔♥❤ Fps [x] t❛ ❝â✿ xp − λ i i   pt −1 t t pt t pt pt pt   (−λ0 )p −i xi = xp − λp0 (x − λ0 ) = x − λ0 + i i=1 ps s s s ❑❤✐ t = s, t❛ ❝â (x − λ0 ) = xp − λp0 = xp − λ = ❉♦ ✤â x − λ ❧➔ ❧ơ② Fpm [x] ✈ỵ✐ ❝❤➾ sè ❧ơ② ❧✐♥❤ ❧➔ ps ❧✐♥❤ tr♦♥❣ ✈➔♥❤ ps x −λ ✐✤➯❛♥ ❝❤➼♥❤ ♥➳✉ ♥â ✤÷đ❝ s✐♥❤ ❜ð✐ ♠ët ♣❤➛♥ tû✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ✐✤➯❛♥ ❝❤➼♥❤ ♥➳✉ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ❝õ❛ ♥â ✤➲✉ ❧➔ ✐✤➯❛♥ ❝❤➼♥❤✳ R ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ♥➳✉ R ❝â ❞✉② ♥❤➜t ♠ët ✐✤➯❛♥ tè✐ ✤↕✐✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ❝❤✉é✐ ♥➳✉ t➟♣ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❤ú✉ ❤↕♥✳ ▼ët ✐✤➯❛♥ I ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ❤ñ♣ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ❝õ❛ R ❧➔ ♠ët ❝❤✉é✐ s➢♣ t❤ù tü t❤❡♦ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠✳ ❚ø ✤â t❛ s✉② r❛ r➡♥❣ ♠ët ✈➔♥❤ ❝❤✉é✐ ❧➔ ♠ët ✈➔♥❤ s tữỡ ữỡ ợ ❧ỵ♣ ❝→❝ ✈➔♥❤ ❝❤✉é✐ ❣✐❛♦ ❤♦→♥ ❤ú✉ ❤↕♥✳ ▼➺♥❤ ✤➲ ✷✳✼✳ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❤ú✉ ❤↕♥✳ ❑❤✐ ✤â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✸✻ (i) R ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✈➔ ✐✤➯❛♥ tè✐ ✤↕✐ M ❝õ❛ R ❧➔ ✈➔♥❤ ❝❤➼♥❤❀ (ii) R (iii) R ❧➔ ✈➔♥❤ ✐✤➯❛♥ ❝❤➼♥❤ ✤à❛ ♣❤÷ì♥❣❀ ❧➔ ✈➔♥❤ ❝❤✉é✐✳ ❈❤ù♥❣ ♠✐♥❤✳ (i) ⇒ (ii)✿ ●å✐ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ ✲ ◆➳✉ I = R t❤➻ I ✤÷đ❝ s✐♥❤ ❜ð✐ ♣❤➛♥ tû ✶✳ ✲ ◆➳✉ I R t❤➻ I ⊂ M ✳ ❚❤❡♦ ✭✐✮ M ✤÷đ❝ s✐♥❤ ❜ð✐ ♠ët ♣❤➛♥ tû✱ t❛ ✤➦t M = a ✳ ❑❤✐ ✤â I = ak ợ k số ữỡ ✤â✳ ❙✉② r❛ R ❧➔ ✈➔♥❤ ✐✤➯❛♥ ❝❤➼♥❤ ✤à❛ ♣❤÷ì♥❣✳ (ii) ⇒ (iii) ●✐↔ sû R ❧➔ ✈➔♥❤ ✐✤➯❛♥ ❝❤➼♥❤ ữỡ ợ tố M = a A, B ❧➔ ❤❛✐ ✐✤➯❛♥ t❤ü❝ sü ❝õ❛ R✳ ❚❛ õ A, B M õ tỗ t ❝→❝ sè ♥❣✉②➯♥ k, l s❛♦ ❝❤♦ A = ak , B = al ✭l, k ♥❤ä ❤ì♥ ❝❤➾ sè ❧ô② ❧✐♥❤ ❝õ❛ a✮✳ ❙✉② r❛ A ⊆ B ❤♦➦❝ B ⊆ A✳ ❱➟② R ❧➔ ✈➔♥❤ ❝❤✉é✐✳ (iii) ⇒ (i) ●✐↔ sû R ❧➔ ✈➔♥❤ ❝❤✉é✐ ❣✐❛♦ ❤♦→♥ ❤ú✉ ró r R ữỡ tỗ t↕✐ ❞✉② ♥❤➜t ♠ët ✐✤➯❛♥ tè✐ ✤↕✐ t❤❡♦ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠✮✳ ✣➸ ❝❤➾ r❛ r➡♥❣ ✐✤➯❛♥ tè✐ ✤↕✐ ❝õ❛ R ❧➔ ❝❤➼♥❤✱ ❝❤ó♥❣ t❛ ❣✐↔ sû ♥❣÷đ❝ ❧↕✐✱ M ✤÷đ❝ s✐♥❤ ❜ð✐ ♥❤✐➲✉ ❤ì♥ ♠ët ♣❤➛♥ tû ✈➔ b, c ❧➔ ❤❛✐ ♣❤➛♥ tû s✐♥❤ ❝õ❛ M ✱ ✈ỵ✐ b ổ tở cR ỗ tớ c ụ ổ tở bR✳ ❑❤✐ ✤â b c ✈➔ c b ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t R ❧➔ ✈➔♥❤ ❝❤✉é✐✳ ❱➟② ▼ ❧➔ ✐✤➯❛♥ ❝❤➼♥❤✳ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ s➩ ❝❤➾ r❛ r➡♥❣ ✈➔♥❤ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ (x − λ0 ) Fpm [x] ❧➔ ♠ët ✈➔♥❤ ❝❤✉é✐ ✈➔ ❝→❝ xps − λ Fpm [x] ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ♥❤÷ s❛✉✿ xps − λ ⊃ (x − λ0 ) ps −1 ⊃ · · · ⊃ (x − λ0 ) ⊃ (x − λ0 ) ❳➨t ♠ët ♣❤➛♥ tû ❜➜t ❦ý f (x) = a0 + a1 (x) + · · · + aps −1 xp s ps −1 = , tr♦♥❣ ✤â a0 , a1 , , aps −1 ∈ Fpm ❑❤✐ ✤â ❝❤ó♥❣ t❛ ❝â ❝→❝ ♣❤➛♥ tû b0 , b1 , , bps −1 ∈ Fpm s❛♦ ❝❤♦ f (x) ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ ♥❤÷ s❛✉✿ f (x) = b0 + b1 (x − λ0 ) + · · · + bps −1 (x − λ0 )p s −1 ◆➳✉ b = 0, ❦❤✐ ✤â f (x) = (x − λ0 )g(x) ✈➔ ❞♦ ✈➟② f (x) = (x − λ0 ◆➳✉ b = 0, t❤➻ f (x) ✤÷đ❝ ữợ f (x) = b0 + (x − λ0 )g(x) ❚❛ ❝â f (x) ❧➔ ✸✼ Fpm [x] t❤➻ t❛ ❝â ✤✐➲✉ ❦❤➥♥❣ xps − λ Fpm [x] ✤à♥❤ s❛✉✿ ❤♦➦❝ f (x) ❧➔ ❦❤↔ ♥❣❤à❝❤ ❤♦➦❝ f (x) ∈ x − λ0 ❉♦ ✤â ps x Fpm [x] ởt ữỡ ợ ✐✤➯❛♥ tè✐ ✤↕✐ x − λ0 ❱➻ ✈➟② ps ❧➔ ♠ët ✈➔♥❤ x −λ ❝❤✉é✐✳ ❚ø ❝❤➾ sè ❧ô② ❧✐♥❤ ❝õ❛ x − λ0 ❧➔ ps t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❦❤↔ ♥❣❤à❝❤✱ tù❝ ❧➔ ✈ỵ✐ ❜➜t ❦➻ ♣❤➛♥ tû f (x) ∈ ❍➺ q✉↔ ✷✳✶✳ ❱ỵ✐ λ = 1, t❛ ❝â ✈➔♥❤ Fpm [x] xps − (x − 1) Fpm [x] xps − ❧➔ ♠ët ✈➔♥❤ ❝❤✉é✐ ✈➔ ❝→❝ ✐✤➯❛♥ ❝õ❛ ❝â ❞↕♥❣ s❛✉✿ ⊃ (x − 1) ⊃ · · · ⊃ (x − 1) ps −1 ps ⊃ (x − 1) = ❚ø ❤➺ q✉↔ tr➯♥✱ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ ❝→❝ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ ps tr➯♥ Fpm ❧➔ Ci = (x − 1)i tr♦♥❣ ✤â i = 0, , ps ✳ ❱➼ ❞ö ✷✳✶✵✳ ❳➨t p = 19; s = 1; m = 2, λ = ❚❛ ❝â ❝→❝ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ F192 [x] ❧➔ (x − 1)j , tr♦♥❣ ✤â ≤ j ≤ 19✱ ✈➔ j ∈ Z ❑❤✐ ✤â✱ t❛ t❤➜② r➡♥❣ x19 − ❝→❝ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ 19 tr➯♥ F192 ❧➔ Ci = (x − 1)j tr♦♥❣ ✤â ≤ j ≤ 19, ✈➔ j ∈ Z ❱➼ ❞ö ✷✳✶✶✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ ✸ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ F3✳ ✣❛ t❤ù❝ x3 − = (x − 1)3 = (2 + x)3 ❙✉② r❛ ❝â ✹ ♠➣ ❝②❝❧✐❝ ✤ë ❞➔✐ ✸ tữỡ ự ợ tự s s = ✭✐✐✮ + x = + x ✭✐✐✐✮ (2 + x) = + x + x2 ✭✐✈✮ (2 + x) = + x3 ✣❛ t❤ù❝ s✐♥❤ g(x) = s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ❝❤➼♥❤ F33 ✣❛ t❤ù❝ s✐♥❤ g(x) = 2+x3 s s r ỗ ♣❤➛♥ tû ❧➔ {000} ✸✽ ✣❛ t❤ù❝ s✐♥❤ g(x) = + x s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(3, 2) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔  G2×3 =  0   ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x2 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(3, 1) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G1×3 = 1 ❱➼ ❞ö ✷✳✶✷✳ ❚❛ ❝â x9 − = (x − 1)9 = (2 + x)9 ❙✉② r❛ ❝â tữỡ ự ợ ✤❛ t❤ù❝ s✐♥❤ s❛✉ ✤➙②✿ ✭✐✮ = ✭✐✐✮ + x = + x ✭✐✐✐✮ (2 + x) = + x + x2 ✭✐✈✮ (2 + x) = + x3 ✭✈✮ (2 + x) = + 2x + 2x3 + x4 ✭✈✐✮ (2 + x) = + 2x + 2x2 + x3 + x4 + x5 ✭✈✐✐✮ (2 + x) = + x3 + x6 ✭✈✐✐✐✮ (2 + x) = + x + 2x3 + x4 + 2x6 + x7 ✭✐①✮ (2 + x) = + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 ✭①✮ (2 + x) = + x9 ✣❛ t❤ù❝ s✐♥❤ g(x) = s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ❝❤➼♥❤ F93 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x9 s s r ỗ tỷ ❧➔ {000000000} ✣❛ t❤ù❝ s✐♥❤ g(x) = + x s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(9, 8) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ✸✾ ❝②❝❧✐❝ ♥➔② ❧➔  G8×9   0  0   0  = 0   0   0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0    0  0   0   0   0   0  ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x2 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(9, 7) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔  G7×9  0   0   = 0  0   0  1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1   0   0   0   0   0  0 0 1 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x3 s➩ s✐♥❤   0   0  G6×9 =  0   0  0 r❛ ♠➣ ❝②❝❧✐❝ C(9, 6)  0 0  0 0 0   0 0 0   0 0   0 0 0  0 0 ✣❛ t❤ù❝ s✐♥❤ g(x) = + 2x + 2x3 + x4 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(9, 5) ▼❛ tr➟♥ ✹✵ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔  G5×9  0   = 0  0  2 0 2 0 2 0 2 0 2   0   0   0  ✣❛ t❤ù❝ s✐♥❤ g(x) = + 2x + 2x2 + x3 + x4 + x5 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(9, 4) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔②   0  G4×9 =  0  ❧➔  2 1 0  2 1 0   2 1 0  0 2 1 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x3 + x6 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(9, 3) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔  G3×7  1 0 0 0   = 0 0 0 0   0 0 0 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + 2x3 + x4 + 2x6 + x7 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(9, 2) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔   2  G2×9 =  2 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(9, 1) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G1×9 = 1 1 1 1 ❱➼ ❞ö ✷✳✶✸✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ ✺ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ F5✳ ❚❛ ❝â x5 − = (x − 1)5 = (x + 4)5 ✳ ❙✉② r❛ ❝â tữỡ ự ợ ✤❛ t❤ù❝ s✐♥❤ s❛✉ ✤➙②✿ ✹✶ ✭✐✮ = ✭✐✐✮ + x = + x ✭✐✐✐✮ (4 + x) = + 3x + x2 ✭✐✈✮ (4 + x) = + 3x + 2x2 + x3 ✭✈✮ (4 + x) = + x + x2 + x3 + x4 ✭✈✐✮ (4 + x) = + x5 ✣❛ t❤ù❝ s✐♥❤ g(x) = s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ❝❤➼♥❤ F55 ✣❛ t❤ù❝ s✐♥❤ g(x) = 4+x5 s➩ s✐♥❤ r❛ ỗ tỷ {00000} t❤ù❝ s✐♥❤ g(x) = + x s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(5, 4) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G4×5   0  = 0   0  0   0  0 ✣❛ t❤ù❝ s✐♥❤ g(x) = + 3x + x2 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(5, 3) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G3×5   1 0   = 0 0   0 ✣❛ t❤ù❝ s✐♥❤ g(x) = + 3x + 2x2 + x3 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(5, 2) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G2×5    = ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x2 + x3 + x4 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(5, 1) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G1×5 = 1 1 ✹✷ ❱➼ ❞ö ✷✳✶✹✳ ❳➙② ❞ü♥❣ ♠➣ ❝②❝❧✐❝ ❝â ✤ë ❞➔✐ ✼ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ F7✳ ✣❛ t❤ù❝ x7 − = (x − 1)7 ❙✉② r❛ õ tữỡ ự ợ ✽ ✤❛ t❤ù❝ s✐♥❤ s❛✉ ✤➙②✿ ✭✐✮ = ✭✐✐✮ −1 + x = −1 + x ✭✐✐✐✮ (−1 + x) = + 5x + x2 ✭✐✈✮ (−1 + x) = −1 + 3x + 4x2 + x3 ✭✈✮ (−1 + x) = + 3x − x2 + 3x3 + x4 ✭✈✐✮ (−1 + x) = −1 + 5x + 4x2 + 3x3 + 2x4 + x5 ✭✈✐✐✮ (−1 + x) = + x + x2 + x3 + x4 + x5 + x6 ✭✈✐✐✐✮ (−1 + x) = −1 + x7 ✣❛ t❤ù❝ s✐♥❤ g(x) = s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ❝❤➼♥❤ F77 ✣❛ t❤ù❝ s✐♥❤ g(x) = −1 + x7 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ ỗ tỷ {0000000} tự s g(x) = −1 + x s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 6) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔  G6×7  −1 0 0 0 −1 0 0 −1 0 0 −1 0 0 −1  0   0   0   0  0 0       =      −1 ✣❛ t❤ù❝ s✐♥❤ g(x) = + 5x + x2 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 5) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔  G5×7 0 0  0   = 0  0  0 0 0 ✹✸   0   0   0  ✣❛ t❤ù❝ s✐♥❤ g(x) = −1 + 3x + 4x2 + x3 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 4) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G4×7  −1   −1  = 0 −1  0 −1  0  0   0  ✣❛ t❤ù❝ s✐♥❤ g(x) = + 3x − x2 + 3x3 + x4 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 3) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔   0 1 −1   G3×7 = 0 −1 0   0 −1 ✣❛ t❤ù❝ s✐♥❤ g(x) = −1 + 5x + 4x2 + 3x3 + 2x4 + x5 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 2) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔   −1  G2×7 =  −1 ✣❛ t❤ù❝ s✐♥❤ g(x) = + x + x2 + x3 + x4 + x5 + x6 s➩ s✐♥❤ r❛ ♠➣ ❝②❝❧✐❝ C(7, 1) ▼❛ tr➟♥ s✐♥❤ ❝õ❛ ♠➣ ❝②❝❧✐❝ ♥➔② ❧➔ G1×7 = 1 1 1 ✹✹ ❑➳t ❧✉➟♥ ✧P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ✤➸ ①➙② ❞ü♥❣ ❝→❝ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✧ ✤➣ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ❦➳t ▲✉➟♥ ✈➠♥ q✉↔ s❛✉✿ ❼ ❚r➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ tr÷í♥❣ ❤ú✉ ❤↕♥✱ ✈➔♥❤ ✤❛ t❤ù❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ❼ ❚r➻♥❤ ❜➔② ♠ët sè ✤à♥❤ ❧➼ ✈➲ t➼♥❤ ❜➜t ❦❤↔ q✉② ❝õ❛ ♠ët sè ✤❛ t❤ù❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ❼ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ ❞↕♥❣ xn − t❤➔♥❤ ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ Fq ❦❤✐ (n, q) = ✈➔ (n, q) = ❼ ❚r➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ❼ ❳➙② ❞ü♥❣ ❝→❝ ♠➣ ❝②❝❧✐❝ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ✹✺ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ▲➯ ỵ tt tự ❤å❝ q✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✷❪ ◆❣✉②➵♥ ❈❤→♥❤ ❚ó ✭✷✵✵✻✮✱ ỵ tt rở trữớ s tr tr÷í♥❣ ✣↕✐ ❤å❝ s÷ ♣❤↕♠ ❍✉➳✳ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❏✳ ▼❛❝❲✐❧❧✐❛♠s ❛♥❞ ◆✳ ❏✳ ❆✳ ❙❧♦❛♥❡✱ ❚❤❡ ❚❤❡♦r② ♦❢ ❊rr♦r ✲ ❈♦rr❡❝t✐♥❣ ❈♦❞❡s✱ 10th ✐♠♣r❡ss✐♦♥✱ ◆♦rt❤ ❍♦❧❛♥❞✱ ❆♠st❡r❞❛♠✱ ✶✾✾✽✳ ❬✹❪ ❲✳ P❡t❡rs♦♥ ❛♥❞ ❊✳ ❏✳ ❲❡❧❞♦♥✱ ❊rr♦r✲❈♦rr❡❝t✐♥❣ ❈♦❞❡s✱ ❘❡✈✐s❡❞✱ ✷♥❞ ❊❞✐✲ t✐♦♥✱ ❈❛♠❜r✐❞❣❡✱ ▼❛ss✱ ✶✾✼✷✳ ❬✺❪ ❱✳ P❧❡ss ❛♥❞ ❲✳ ❈✳ ❍✉❢❢♠❛♥✱ ❍❛♥❞❜♦♦❦ ♦❢ ❈♦❞✐♥❣ ❚❤❡♦r②✱ ❊❧s❡✈✐❡r✱ ❆♠✲ st❡r❞❛♠✱ ✶✾✾✽✳ ❬✻❪ P✳ ❲✳ ❙❤♦r✱ ❋❛✉❧t✲❚♦❧❡r❛♥t q✉❛♥t✉♠ ❝♦♠♣✉t❛t✐♦♥✱ Pr♦❝✳ ✸✼t❤ ■❊❊❊ ❙②♠♣✳ ♦♥ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✳✱ ♣♣✳ ✺✻✲✻✺✱ ✶✾✾✻✳ ✹✻ ... ❝❤➾ ❦❤✐ C ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ F[x] xn −λ ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✿ ●✐↔ sû C ❧➔ ♠ët ♠➣ λ − constacyclic n tr F ú ỵ r ộ tø ♠➣ c = (c0 , c1 , , cn−1 ) ∈ C ❝â t❤➸ ①❡♠ ♥❤÷ ởt tự ữợ c(x) =