Về tính chẵn lẻ của số nhân tử bất khả quy modulo p của đa thức hệ số nguyên

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ✣⑨▼ ❚❍➚ ◆●➴❈ ❚❹▼ ❱➋ ❚➑◆❍ ❈❍➂◆ ▲➈ ❈Õ❆ ❙➮ ◆❍❹◆ ❚Û ❇❻❚ ❑❍❷ ◗❯❨ ▼❖❉❯▲❖ P ❈Õ❆ ✣❆ ❚❍Ù❈ ❍➏ ❙➮ ◆●❯❨➊◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆✱ ✺✴✷✵✶✾ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ✣⑨▼ ❚❍➚ ◆●➴❈ ❚❹▼ ❱➋ ❚➑◆❍ ❈❍➂◆ ▲➈ ❈Õ❆ ❙➮ ◆❍❹◆ ❚Û ❇❻❚ ❑❍❷ ◗❯❨ ▼❖❉❯▲❖ P ❈Õ❆ ✣❆ ❚❍Ù❈ ❍➏ ❙➮ ◆●❯❨➊◆ ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ ❚♦→♥ ❝➜♣ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ●■⑩❖ ❱■➊◆ ❍×❰◆● ❉❼◆ ❚❙✳ ◆●❯❨➍◆ ❉❯❨ ❚❹◆ ❚❍⑩■ ◆●❯❨➊◆✱ ✺✴✷✵✶✾ ✐✐✐ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ✶✳✷ ✶✳✸ ❑➳t t❤ù❝ ❝õ❛ ❤❛✐ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❇✐➺t t❤ù❝ ❝õ❛ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ ỗ rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❈❤÷ì♥❣ ỵ trr ◆❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr♦♥❣ Fp [x] ỵ trr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣❛ t❤ù❝ ♥❣✉②➯♥ ❦❤↔ q✉② ♠♦❞✉❧♦ số p tố ữỡ tỹ ỵ ❙t✐❝❦❡❧❜❡r❣❡r ❝❤♦ ✤❛ t❤ù❝ t❤ü❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỵ ❙t✐❝❦❡❧❜❡r❣❡r ✈➔ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ✷✶ ✸✳✶ ✸✳✷ ỵ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ❙t✐❝❦❡❧❜❡r❣❡r ✈➔ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ỵ trr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✷ ✸✸ ✶ ▼ð ✤➛✉ ❈❤♦ f (x) ∈ Z[x] ❧➔ ♠ët ✤❛ t❤ù❝ ❝❤✉➞♥ ✭♠♦♥✐❝✮ ❤➺ sè ♥❣✉②➯♥ ❜➟❝ n ✈➔ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♣❤ù❝ ❦➨♣✳ ●å✐ D(f ) ❧➔ ❜✐➺t t❤ù❝ ❝õ❛ f ✳ ❈❤♦ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè ❧➫ ✈➔ ❣å✐ Fp = Z/pZ ❧➔ tr÷í♥❣ ❤ú✉ ❤↕♥ ❝â p ♣❤➛♥ tû✳ ●å✐ f¯(x) ∈ Fp [x] ❧➔ ✤❛ t❤ù❝ ♥❤➟♥ ✤÷đ❝ tø f ❜➡♥❣ ❝→❝❤ t❤✉ ❣å♥ ❤➺ sè ♠♦❞✉❧♦ p✳ ●å✐ r ❧➔ sè ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ❝õ❛ f¯✳ ❑❤✐ õ ởt ỵ trr r r ✈➔ n ❝â ❝ò♥❣ t➼♥❤ ❝❤➤♥ ❧➫✱ tù❝ ❧➔ r ≡ n (mod 2)✱ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ D(f ) ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦ p✳ ▼ư❝ t✐➯✉ ❝õ❛ ❧✉➟♥ ✈➠♥ t ự ỵ t ❜❡r❣❡r ♥➔② ❝ơ♥❣ ♥❤÷ ù♥❣ ❞ư♥❣ ❝õ❛ ♥â tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐✳ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❜è ❝ö❝ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❜❛ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❦➳t t❤ù❝ ❝õ❛ ❤❛✐ ✤❛ t❤ù❝✱ t tự tự ỗ rs ữỡ ỵ trr ữỡ tr ỵ trr ởt số ởt tữỡ tỹ ỵ tự tỹ ữỡ ỵ trr t t ữỡ tr ỵ r t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ✈➔ ♠ët ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❧✉➟t sỷ ỵ trr ữủ t❤ü❝ ❤✐➺♥ ✈➔ ❤♦➔♥ t❤➔♥❤ ✈➔♦ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ◗✉❛ ✤➙②✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s s tợ ữớ t t ữợ tr sốt q tr ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚→❝ ❣✐↔ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❑❤♦❛ ❚♦→♥✲❚✐♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ✤➸ ❣✐ó♣ t→❝ ❣✐↔ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ❝ơ♥❣ ♥❤÷ ❝❤÷ì♥❣ tr➻♥❤ t❤↕❝ s➽✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ t➟♣ t❤➸ ❧ỵ♣ ❝❛♦ ❤å❝ ❑✶✶❉✱ ❦❤â❛ ✵✺✴✷✵✶✼ ✲ ✵✺✴✷✵✶✾ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t ỗ tớ t ❣✐↔ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❇❛♥ ❣✐→♠ ❤✐➺✉ ỗ t trữớ ữ ổ ❚r✐➲✉✱ ◗✉↔♥❣ ◆✐♥❤ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ❳→❝ ♥❤➟♥ ❝õ❛ ♥❣÷í✐ ữợ t ♥➠♠ ✷✵✶✾ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ✣➔♠ ❚❤à ◆❣å❝ ❚➙♠ ✸ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❦➳t t❤ù❝ ❝õ❛ ❤❛✐ ✤❛ t❤ù❝✱ ❜✐➺t t❤ù❝ ❝õ❛ ✤❛ t❤ù❝ ✈➔ ỗ rs t sỷ ❝❤÷ì♥❣ ♥➔② ❧➔ t➔✐ ❧✐➺✉ ❬✷✱ ❙❡❝t✐♦♥ ✻✳✻❪ ✈➔ ❬✸✱ ❈❤❛♣t❡r ✶✺❪✳ ✶✳✶ ❑➳t t❤ù❝ ❝õ❛ ❤❛✐ ✤❛ t❤ù❝ ●✐↔ sû f, g ❧➔ ❤❛✐ ✤❛ t❤ù❝ ❜✐➳♥ x ✈ỵ✐ ❝→❝ ❤➺ sè tr♦♥❣ ♠ët tr÷í♥❣ F ✳ ●✐↔ sû K ❧➔ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè ❝❤ù❛ F ✳ ●å✐ α1 , , αn ❧➔ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ✭❦➸ ❝↔ ❜ë✐✮ ❝õ❛ f tr♦♥❣ K ✱ tù❝ ❧➔ f (x) = a(x − α1 )(x − α2 ) (x − αn ), ✈ỵ✐ a ∈ K ♥➔♦ ✤â ❚÷ì♥❣ tü✱ ❣å✐ β1 , , βm ❧➔ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ✭❦➸ ❝↔ ❜ë✐✮ ❝õ❛ g tr♦♥❣ K ✱ tù❝ ❧➔ g(x) = b(x − β1 )(x − β2 ) (x − βm ), ✈ỵ✐ b ∈ K ♥➔♦ ✤â ❚❛ ✤à♥❤ ♥❣❤➽❛ ❦➳t t❤ù❝ ❝õ❛ f ✈➔ g ✱ R(f, g) ❧➔ n m m n (αi − βj ) (n = deg f, m = deg g) R(f, g) = a b i=1 j=1 t ữợ ởt số t➼♥❤ ❝❤➜t ❝õ❛ ❦➳t t❤ù❝✳ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶✳ R(g, f ) = (−1)mnR(f, g) ✹ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â m n n m n m m n (αi − βj ) = (−1)mn R(f, g) (βj − αi ) = a b R(g, f ) = a b j=1 i=1 i=1 j=1 ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ❚➼♥❤ ❝❤➜t ✶✳✶✳✷✳ R(f, g) = ♥➳✉ f ✈➔ g ❝â ♠ët ♥❤➙♥ tû ❝❤✉♥❣ ❜➟❝ ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ f ✈➔ g ❝â ♠ët ♥❤➙♥ tû ❝❤✉♥❣ ❧➔ h(x) ∈ F [x]✳ ❑❤✐ ✤â ❣å✐ α ∈ K ♠ët h tr K ữ tỗ t i✱ j s❛♦ ❝❤♦ αi = α ✈➔ βj = α✳ ❚❛ s✉② r❛ tr♦♥❣ t➼❝❤ ✤à♥❤ ♥❣❤➽❛ R(f, g) ❝â ♥❤➙♥ tû αi − βj = ✈➔ ❞♦ ✈➟② R(f, g) = 0✳ ❚➼♥❤ ❝❤➜t ✶✳✶✳✸✳ R(f, g) = a n m m f (βj )✳ mn n g(αi ) = (−1) i=1 b j=1 ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ g(x) = b nj=1 (x − βi )✱ ♥➯♥ t❛ ❝â g(αi ) = b ✈ỵ✐ ♠å✐ i = 1, , n✳ ❉♦ ✈➟② n a m n n m n (αi − βj ) = R(f, g) g(αi ) = a b i=1 n j=1 (αi − βj )✱ i=1 j=1 ❚÷ì♥❣ tü ✭❤♦➦❝ sû ❞ư♥❣ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶✮ t❛ s✉② r❛ m R(f, g) = (−1) mn n b f (βj ) j=1 ❚➼♥❤ ❝❤➜t ✶✳✶✳✹✳ ◆➳✉ g(x) = f q + r✱ t❤➻ R(f, g) = am−deg r R(f, r)✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❚➼♥❤ ❝❤➜t ✶✳✶✳✸✱ t❛ ❝â n R(f, g) = a deg g n g(αi ) = a i deg g [f (αi )q(αi ) + r(αi )] i=1 ✺ ❱➻ αi ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❝õ❛ f ✱ ♥➯♥ f (αi ) = ✈➔ ❞♦ ✈➟② f (αi )q(αi ) + r(αi ) = r(α)✳ ❉♦ ✤â t❛ ❝â n R(f, g) = a deg g r(αi ) i=1 ▼➦t ❦❤→❝✱ ❝ô♥❣ t❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✸ R(f, r) = adeg r n i=1 r(αi )✳ ❉♦ ✈➟② n R(f, g) = a deg g r(αi ) = adeg g−deg r R(f, r) i=1 ❚➼♥❤ ❝❤➜t ✶✳✶✳✺✳ R(f, b) = bdeg f b ổ ữợ ự t g(x) = b✳ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✸ n R(f, g) = a g(αi ) = bn i=1 ❈→❝ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶✱✶✳✶✳✹✱ ✶✳✶✳✺ ❝❤♦ ♣❤➨♣ t❛ t➼♥❤ t♦→♥ ❦➳t t❤ù❝ ❝õ❛ ❜➜t ❦➻ ❤❛✐ ✤❛ t❤ù❝ ♥➔♦ ❜➡♥❣ t❤✉➟t t♦→♥ ❝❤✐❛ ❝õ❛ ❊✉❝❧✐❞✳ ❈→❝ t➼♥❤ ❝❤➜t ♥➔② ❝ô♥❣ ❝❤♦ ♣❤➨♣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ❦➳t t❤ù❝ R(f, g) ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ tr÷í♥❣ F ♠➦❝ ❞ị ♥â ✤÷đ❝ ✤à♥❤ ỹ t tỷ tr trữớ ợ ỡ K ✳ ❚➼♥❤ ❝❤➜t ✶✳✶✳✻✳ ❚❛ ❝â R(f, g) ♥➡♠ tr♦♥❣ F ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ deg f ✳ ◆➳✉ g = b ❧➔ ❤➡♥❣ sè t❤✉ë❝ F ✳ ❚❤➻ t❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶ ✈➔ ✶✳✶✳✺✱ R(f, g) = R(b, f ) = R(f, b) = bn t❤✉ë❝ F ●✐↔ sû ❦❤➥♥❣ ✤à♥❤ ✤➣ ✤ó♥❣ ✈ỵ✐ ♠å✐ ♠å✐ ✤❛ t❤ù❝ f ✈➔ g ✈ỵ✐ f ❝â ❜➟❝ ♥❤ä ❤ì♥ ❤♦➦❝ ❜➡♥❣ n − 1✳ t f g tự tũ ỵ ✈ỵ✐ deg f = n ≥ 1✳ ❑❤✐ ✤â t❤❡♦ tt t tự tỗ t tự q ✈➔ r tr♦♥❣ F [x] s❛♦ ❝❤♦ g = f q + r, ✈ỵ✐ r = ❤♦➦❝ deg r < deg f = n✳ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✹✱ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶ ✈➔ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❛ ❝â R(f, g) = R(f, r) = ±R(r, f ) t❤✉ë❝ F ✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✻ ❚➼♥❤ ❝❤➜t ✶✳✶✳✼✳ ❚❛ ❝â ✶✳ ◆➳✉ f = f1 f2 t❤➻ R(f, g) = R(f1 , g)R(f2 , g)✳ ✷✳ ◆➳✉ g = g1 g2 t❤➻ R(f, g) = R(f, g1 )R(f, g2 )✳ ❈❤ù♥❣ ♠✐♥❤✳ ❙✉② r❛ tø ❚➼♥❤ ❝❤➜t ✶✳✶✳✸✳ ✶✳✷ ❇✐➺t t❤ù❝ ❝õ❛ ✤❛ t❤ù❝ ❈❤♦ f = f (x) ∈ F [x] ❧➔ ✤❛ t❤ù❝ ợ số tr trữớ F K ởt tr÷í♥❣ ✤â♥❣ ✤↕✐ sè ❝❤ù❛ F ✳ ❇✐➺t t❤ù❝ ❝õ❛ f ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ D(f ) = (−1)n(n−1)/2 R(f, f ), ð ✤➙② f ❧➔ ✤↕♦ ❤➔♠ ❝õ❛ f ✈➔ n = deg f ✳ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✷✱ t❛ ❝â D(f ) = ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f ✈➔ f ❦❤ỉ♥❣ ❝â t❤ø❛ sè ❝❤✉♥❣✳ ❈❤ó♥❣ t❛ ❝â t❤➸ t➼♥❤ t♦→♥ D(f ) ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ tt t tr f f ữợ ❧➔ ♠ët sè ✈➼ ❞ö✳ ❱➼ ❞ö ✶✳✷✳✶✳ ❳➨t f (x) = x − a✳ ❑❤✐ ✤â f (x) = 1, ✈➻ ✈➟② D(f ) = (−1)(1.0)/2 R(f, 1) = R(f, 1) = 1deg f = ❱➼ ❞ö ✶✳✷✳✷✳ ❳➨t f (x) = x2 + ax + b✳ ❑❤✐ ✤â f (x) = 2x + a ✈➔ D(f ) = −R(f, f )✳ ❚❛ ❝â a2 x a x + ax + b = (2x + a) + + (b − ) 4 a2 ✣➦t r = b − ✳ ❚❛ ❝â D(f ) = −R(f, f ) = −R(f , f ) (t❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶) = 2deg f −deg r (−1)R(f , r) ( t❤❡♦ ❚➼♥❤ ❝❤➜t 1.1.4) = −22−0 R(f , r) = −4r = a2 − 4b ✼ ❱➼ ❞ö ✶✳✷✳✸✳ ❈❤♦ f (x) = x3 + qx + r✳ ❚❤➻ f (x) = 3x2 + q ✈➔ t❤ü❝ ❤✐➺♥ t❤✉➟t t♦→♥ ❊✉❝❧✐❞✱ t❛ ❝â x 2q + x+r , 3 2q 9x 27r 27r2 x+r − + q+ 2q 4q 4q x3 + qx + r = (3x2 + q) 3x2 + q = ❉♦ ✤â D(f ) = (−1)3·2/2 R(f, f ) = −R(f, f ) = −R(f , f ) (t❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶) 2qx = −3deg f −1 R(f , + r) (t❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✹) 2qx = −9R( + r, f ) (t❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶) 2q 2qx 27r2 R( = −9 + r, q + ) 3 4q 27r2 ) = −4q − 27r2 = −4q (q + 4q ❱➼ ❞ö ✶✳✷✳✹✳ ❳➨t f (x) = xn − ∈ F [x]✳ ❚❛ ✤✐ t➼♥❤ ❜✐➺t t❤ù❝ ❝õ❛ f (x)✳ ●å✐ α1 , , αn ❧➔ n ♥❣❤✐➺♠ tr♦♥❣ K ✭♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè ❝❤ù❛ F ✮ ❝õ❛ ✤❛ t❤ù❝ f (x) = xn − 1✳ ❚❛ ❝â f (x) = nxn−1 ✳ ❉♦ ✈➟② n D(f ) = (−1) n(n−1)/2 R(f, f ) = (−1) n(n−1)/2 f (αk ) k=1 n−1 n = (−1) n(n−1)/2 n n αk k=1 n(n−1) = (−1)n(n−1)/2 nn (−1) = (1)n(n1)/2 nn t ỵ t à · · αn = (−1)n ✳ ✣❛ t❤ù❝ f (x) ∈ F [x] ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤❛ t❤ù❝ ❝❤✉➞♥ ✭♠♦♥✐❝✮ ♥➳✉ ❤➺ sè ù♥❣ ✈ỵ✐ sè ♠ơ ❝❛♦ ♥❤➜t ❝õ❛ ♥â ❜➡♥❣ ✶✳ ▼➺♥❤ ✤➲ ✶✳✷✳✺✳ ❈❤♦ f ❧➔ ♠ët ✤❛ t❤ù❝ ♠♦♥✐❝ ✈➔ α1, , αn ❧➔ ❝→❝ ♥❣❤✐➺♠ ✶✽ ❇ê ✤➲ ✷✳✸✳✷✳ ❈❤♦ f (x) = x4 + ax2 + b ∈ F [x]✳ ❑❤✐ ✤â ❜✐➺t t❤ù❝ ❝õ❛ f ❧➔ 2 D = 16b(a − 4b) ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ α✱ −α ✈➔ β ✱ −β ❧➔ ✹ ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ f (x) ✭tr♦♥❣ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè K ♥➔♦ ✤â ❝❤ù❛ F ✮✳ ❚❛ ❝â α2 = u, β = v ❧➔ ❤❛✐ ♥❣❤✐➺♠ ❝õ❛ x2 + ax + b✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✺✱ t❛ ❝â D = [(−α − α)(β − α)(−β − α)(β + α)(−β + α)(−β − β)]2 = 16α2 β (β − α2 )2 = 16uv (u + v)2 − 4uv) = 16b(a2 − 4b) ▼➺♥❤ ✤➲ ✷✳✸✳✸✳ ✣❛ t❤ù❝ x4 + ❧➔ ❜➜t ❦❤↔ q✉② tr➯♥ Z ♥❤÷♥❣ ❧➔ ❦❤↔ q✉② ♠♦❞✉❧♦ p ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ tè p✳ ❈❤ù♥❣ ♠✐♥❤✳ ❇✐➺t t❤ù❝ ❝õ❛ x4 + ❧➔ D = 16 · 42 ❧➔ ♠ët sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ❉♦ ✈➟② t❤❡♦ ❤➺ q✉↔ tr➯♥ t❤➻ x4 + ❧➔ ❦❤↔ q✉② ♠♦❞✉❧♦ p ✈ỵ✐ ♠å✐ p ♥❣✉②➯♥ tè ♠➔ p = 2✳ ❚❛ ❝â x4 + = (x + 1)4 (mod 2)✳ ◆❤÷ ✈➟② x4 + ❧➔ ❦❤↔ q✉② ♠♦❞✉❧♦ p ✈ỵ✐ ♠å✐ p ♥❣✉②➯♥ tè✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ x4 + ❜➜t ❦❤↔ q✉② tr➯♥ Z✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû x4 + ❧➔ ❦❤↔ q✉② tr➯♥ Z✳ ❑❤✐ ✤â (x4 + 1) = (x2 + ax + c)(x2 + bx + d), ✈ỵ✐ a, b, c, d ∈ Z ♥➔♦ ✤â✳ ❚❛ ❝â (x2 +ax+c)(x2 +bx+d) = x4 +(a+b)x3 +(ab+c+d)x2 +(ad+bc)x+cd ❉♦ ✈➟② a + b = 0✱ ab + c + d = 0✱ ad + bc = ✈➔ cd = 1✳ ❚ø cd = t❛ s✉② r❛ c = d = −1 ❤♦➦❝ c = d = 1✳ ❙✉② r❛ ab = −(c + d) = ±2✳ ❉♦ ✈➟② a2 = ±2✱ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥✳ ▼➺♥❤ ✤➲ ✷✳✸✳✹✳ ✣❛ t❤ù❝ x4 + 3x2 + ❧➔ ❜➜t ❦❤↔ q tr Z ữ q p ợ ♠å✐ sè ♥❣✉②➯♥ tè p✳ ❈❤ù♥❣ ♠✐♥❤✳ ❇✐➺t t❤ù❝ ❝õ❛ x4 + 3x2 + ❧➔ D = 16 · 52 ❧➔ ♠ët sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ❉♦ ✈➟② t❤❡♦ ❤➺ q✉↔ tr➯♥ t❤➻ x4 + x2 + ❧➔ ❦❤↔ q✉② ♠♦❞✉❧♦ p ✶✾ ✈ỵ✐ ♠å✐ p ♥❣✉②➯♥ tè ♠➔ p = ✈➔ p = 5✳ ❚❛ ❝â x4 + 3x2 + = (x + 1)4 (mod 2) ✈➔ x4 + 3x2 + = (x − 1)2 (x − 1)2 (mod 5) ◆❤÷ ✈➟② x4 + ❧➔ ❦❤↔ q✉② ♠♦❞✉❧♦ p ✈ỵ✐ ♠å✐ p ♥❣✉②➯♥ tè✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ x4 + 3x2 + ❜➜t ❦❤↔ q✉② tr➯♥ Z✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû x4 + 3x2 + ❧➔ ❦❤↔ q✉② tr➯♥ Z✳ ❑❤✐ ✤â (x4 + 1) = (x2 + ax + c)(x2 + bx + d), ✈ỵ✐ a, b, c, d ∈ Z ♥➔♦ ✤â✳ ❚❛ ❝â (x2 +ax+c)(x2 +bx+d) = x4 +(a+b)x3 +(ab+c+d)x2 +(ad+bc)x+cd ❉♦ ✈➟② a + b = 0✱ ab + c + d = 3✱ ad + bc = ✈➔ cd = 1✳ ❚ø cd = t❛ s✉② r❛ c = d = −1 ❤♦➦❝ c = d = 1✳ ❙✉② r❛ ab = − (c + d) = ❤♦➦❝ ✺✳ ❉♦ ✈➟② a2 = −1 ❤♦➦❝ −5✱ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥✳ ◆❤➟♥ ①➨t ✷✳✸✳✺✳ ❍❛✐ ♠➺♥❤ ✤➲ tr➯♥ ❧➔ ✈➼ ❞ö ✈➲ ✤❛ t❤ù❝ trị♥❣ ♣❤÷ì♥❣ ❜➟❝ ✹ ❜➜t ❦❤↔ q✉② tr➯♥ Z ữ q p ợ p tè✳ ❇↕♥ ✤å❝ ❝â t❤➸ t❤❛♠ ❦❤↔♦ ❬✶❪ ❝❤♦ ♥❣❤✐➯♥ ❝ù✉ ✤➛② ✤õ ❤ì♥ ✈➲ ❝❤õ ✤➲ ♥➔②✳ ✣➸ →♣ ỵ trr ú t õ ❦✐➸♠ tr❛ ①❡♠ D(f ) ❧➔ ♠ët ❜➻♥❤ ♣❤÷ì♥❣ mod p ❤❛② ❦❤ỉ♥❣✳ ❈❤ó♥❣ t❛ ❝â ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❤ú✉ ❤✐➺✉ ✤➸ ❧➔♠ ✤✐➲✉ ♥➔② ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐✳ ▼ët ✤✐➲✉ t❤ó ✈à r➡♥❣✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ❜➡♥❣ sỷ ỵ trr ỳ s ✤÷đ❝ tr➻♥❤ ❜➔② ð ❝❤÷ì♥❣ s❛✉✳ ✷✳✹ ❚÷ì♥❣ tü ❝õ❛ ỵ trr tự tỹ ỵ ❈❤♦ f (x) ❧➔ ✤❛ t❤ù❝ ❝❤✉➞♥ ✭♠♦♥✐❝✮ ❤➺ sè t❤ü❝ ✈ỵ✐ ❜➟❝ d ✈➔ ❜✐➺t t❤ù❝ D(f ) = 0✳ ●å✐ r ❧➔ sè ♥❤➙♥ tû ♠♦♥✐❝ ❜➜t ❦❤↔ q✉② t❤ü❝ ❝õ❛ f ✳ ❑❤✐ ✤â d ≡ r (mod 2) ⇔ D(f ) > ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f (x) = f1 (x) · · · fm (x)fm+1 (x) · · · fn (x) ❧➔ ♣❤➙♥ t➼❝❤ ❝õ❛ f t❤➔♥❤ t➼❝❤ ❝→❝ ✤❛ t❤ù❝ ♠♦♥✐❝ ❜➜t ❦❤↔ q✉② t❤ü❝✱ tr♦♥❣ ✤â f1 (x), , fm (x) ✷✵ ❧➔ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ ✷✱ ✈➔ fm+1 (x), , fm+n ❧➔ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ ✶✳ ❚❛ ❝â D(fi ) < D(fi ) = ✈ỵ✐ ♠å✐ i = 1, m, ✈➔ ✈ỵ✐ ♠å✐ i = m + 1, m + n ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✽ D(f ) = D(f1 · · · fm fm+1 · · · fm+n ) = D(f1 ) · · · D(fm )a2 , ✈ỵ✐ a ∈ R ♥➔♦ ✤â✳ ❉♦ ✈➟② D(f ) > ⇔ m ❧➔ sè ❝❤➤♥ ⇔ d = 2m + n ≡ r = m + n (mod 2) ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ ✷✳✹✳✷✳ ❈❤♦ f (x) ❧➔ ✤❛ t❤ù❝ ❝❤✉➞♥ ✭♠♦♥✐❝✮ ❤➺ sè t❤ü❝ ✈ỵ✐ ❜➟❝ d ✈➔ ❜✐➺t t❤ù❝ D(f ) = 0✳ ❑❤✐ ✤â ✭❛✮ ◆➳✉ D(f ) > t❤➻ f ❝â d − 4k ♥❣❤✐➺♠ t❤ü❝✱ ✈ỵ✐ k ≥ ♥➔♦ ✤â❀ ✭❜✮ ◆➳✉ D(f ) < t❤➻ f ❝â d − − 4k ♥❣❤✐➺♠ t❤ü❝✱ ✈ỵ✐ k ≥ ♥➔♦ ✤â✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ m ❧➔ sè ❝➦♣ ♥❣❤✐➺♠ ♣❤ù❝ ✭❦❤æ♥❣ t❤ü❝✮ ❝õ❛ f ✈➔ ❣å✐ n ❧➔ sè ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ f õ t ỵ tr D(f ) > ⇔ m ≡ (mod 2) ●✐↔ sû D(f ) > 0✳ ❑❤✐ ✤â m ❧➔ sè ❝❤➤♥✳ ❱✐➳t m = 2k ✈ỵ✐ k ≥ ♥➔♦ ✤â✳ ❑❤✐ ✤â✱ sè ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ f ❧➔ d − 2m = d − 4k ✳ ●✐↔ sû D(f ) < 0✳ ❑❤✐ ✤â m ❧➔ sè ❧➫✳ ❱✐➳t m = 2k + ✈ỵ✐ k ≥ ♥➔♦ ✤â✳ ❑❤✐ ✤â✱ sè ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ f ❧➔ d − 2m = d − − 4k ✳ ✷✶ ❈❤÷ì♥❣ ✸✳ ỵ trr t t ữỡ tr ỵ r t t ❜➟❝ ❤❛✐ ✈➔ ♠ët ❝❤ù♥❣ ♠✐♥❤ ❧✉➟t ♥➔② ❜➡♥❣ ❝→❝❤ sỷ ỵ trr t sỷ ❞ư♥❣ ❝❤♦ ❝❤÷ì♥❣ ♥➔② ❧➔ t➔✐ ❧✐➺✉ ❬✸✱ ❈❤❛♣t❡r ✶✻❪ t ỵ r ✸✳✶✳✶✳ ❈❤♦ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè ❧➫✱ ✈➔ a ❧➔ ♠ët sè ♥❣✉②➯♥ a ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ ổ t p õ ỵ r p s❛✉✳ a ♥➳✉ a ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦ p = p −1 ♥➳✉ a ❦❤ỉ♥❣ ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦ p ▼ët sè t➼♥❤ ❝❤➜t ❈❤♦ p ❧➔ sè ♥❣✉②➯♥ tè ❧➫✱ a ✈➔ b ❧➔ ❤❛✐ sè ♥❣✉②➯♥ ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ p✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉✳ ✶✳ a2 p = ✷✳ ab p = ✸✳ a p a p b ✳ p p−1 ≡ a (mod p) ✭❚✐➯✉ ❝❤✉➞♥ ❊✉❧❡r✮✳ ✷✷ a p ✹✳ ◆➳✉ a ≡ b ✭♠♦❞ p✮ t❤➻ ✺✳ −1 p ❜➡♥❣ ❤♦➦❝ −1 tò② t❤❡♦ p ≡ ✭♠♦❞ 4✮ ❤❛② p ≡ ✭♠♦❞ 4✮✳ ✻✳ ❑❤✐ ✤â p b ✳ p = p = ✈➔ ♥➳✉ p ≡ ✭♠♦❞ 8✮ ❤♦➦❝ p ≡ ✭♠♦❞ 8✮❀ ✈➔ = −1 ♥➳✉ p ≡ ✭♠♦❞ 8✮ ❤♦➦❝ p ≡ ✭♠♦❞ ỵ r ❚❛ ❝â 45 37 = 37 45 37 = 37 37 = 37 = −1 ỵ t t ss sû p ✈➔ q ❧➔ ❝→❝ sè ♥❣✉②➯♥ tè ❧➫ ♣❤➙♥ ❜✐➺t✳ ❑❤✐ ✤â t❤➻ p q =− p q = q p trø ❦❤✐ p ≡ q ≡ q p ỵ ❤✐➺✉ ▲❡❣❡♥❞r❡ ▲í✐ ❣✐↔✐✳ 1234 199 = 1234 ✳ 199 40 199 199 199 199 199 = (−1) = (−1) 5 = −1 = ✸✳✷ ỵ trr t t r ♠ư❝ ♥➔② ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ❧✉➟t t❤✉➟t ♥❣❤à❝❤ ❜➟❝ ỵ trr p ✈➔ q ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ❧➫ ♣❤➙♥ ❜✐➺t✳ ●å✐ e ❧➔ ❝➜♣ ❝õ❛ q mod p✱ tù❝ ❧➔ e ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥❤ä ♥❤➜t s❛♦ ❝❤♦ ✷✸ q e ≡ (mod p)✱ ❝ô♥❣ tù❝ ❧➔ e ợ ỗ ữ [q] tr õ (Z/pZ)ì s ỵ trr tự xp tr Fq [x] rữợ t t ❝â ❜ê ✤➲ s❛✉✳ ❇êp ✤➲ ✸✳✷✳✶✳ ●å✐ f (x) ❧➔ ♠ët ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ❜➜t ❦ý ❝õ❛ ✤❛ t❤ù❝ (x − 1)/(x − 1) tr♦♥❣ Fq [x]✳ ❑❤✐ ✤â ❜➟❝ ❝õ❛ f (x) ❜➡♥❣ e✱ ❝➜♣ ❝õ❛ q mod p✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ K ❧➔ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè ❝❤ù❛ Fp ✳ ●å✐ α ❧➔ ♠ët ♥❣❤✐➺♠ tr♦♥❣ K ❝õ❛ f (x)✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✳✷✱ t❛ ❝â f (x) = (a − α)(x − αq )(x − αq ) (x − αq n−1 ), n ð ✤➙② n ❧➔ ♠ët sè tü ♥❤✐➯♥ ♠ô ♥❤ä ♥❤➜t ♠➔ αq = α✳ ❚❛ s➩ ❝❤➾ r❛ r➡♥❣ n = e✳ ❱➻ αp = ✈➔ α = 1✱ ♥➯♥ ❝➜♣ ❝õ❛ α ✭tr♦♥❣ ♥❤â♠ K × ✮ ♣❤↔✐ ❜➡♥❣ p✳ ✣✐➲✉ ♥➔② s✉② r❛✱ ♥➳✉ αr = ✈ỵ✐ r ∈ N ♥➔♦ õ t p ữợ r n n αq = α✱ ♥➯♥ αq −1 = 1✳ ❉♦ ✈➟② p | q n − 1✱ tù❝ ❧➔ q n ≡ (mod p)✳ ❉♦ ✈➟② e ≤ n✳ e ▼➦t ❦❤→❝✱ ✈➻ pe ≡ (mod p)✱ ♥➯♥ p | pe − 1✳ ❉♦ ✈➟② αp −1 = 1✳ ❚❛ s✉② e r❛ αp = α ✈➔ n ≤ e✳ ◆❤÷ ✈➟② n = e ✈➔ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✸✳✷✳✷✳ ❙è ❝→❝ ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ♣❤➙♥ ❜✐➺t r ❝õ❛ xp − tr♦♥❣ Fq [x] ❧➔ r = + (p − 1)/e✳ ❈❤ù♥❣ ♠✐♥❤✳ ❇➟❝ ❝õ❛ (xp − 1)/(x − 1) ❧➔ p − 1✱ ✈➔ ♠é✐ ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ❝õ❛ (xp − 1)/(x − 1) ✤➲✉ ❝â ❜➟❝ e t trữợ õ (p − 1)/e ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ✭♠♦♥✐❝✮ ♣❤➙♥ ❜✐➺t ❝õ❛ (xp − 1)/(x − 1)✳ ❉♦ ✤â r = + (p − 1)/e✳ ❇ê ✤➲ ✸✳✷✳✸✳ ●å✐ (−1)(p−1)/2 pp ✳ D ❧➔ ❜✐➺t t❤ù❝ ❝õ❛ xp − ∈ Fq [x]✳ ❑❤✐ ✤â D = ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❱➼ ❞ö ✶✳✷✳✹✱ D = (−1)p(p−1)/2 pp ✳ ❱➻ p ❧➔ sè ❧➫ ♥➯♥ p(p − 1) p − (p − 1)2 − = ❧➔ sè ❝❤➤♥✳ ❉♦ ✈➟② (−1)p(p−1)/2 = (−1)(p−1)/2 2 ✈➔ D = (−1)(p−1)/2 pp ✳ ❈❤ù♥❣ ♠✐♥❤ ❝õ❛ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ú t s ỵ trr t❤ù❝ xp − tr♦♥❣ Fq [x]✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ỵ õ r rp (mod 2) D ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ mod p ✷✹ p−1 ❧➔ sè ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ♠♦♥✐❝ ❝õ❛ f tr♦♥❣ Fq [x]✱ e D = (−1)(p−1)/2 pp ❧➔ ❜✐➺t t❤ù❝ ❝õ❛ xp − ∈ Fp [x]✳ ❱➻ p ❧➫ ♥➯♥ Ð ✤➙② r = + r =1+ p−1 ≡p e (mod 2) ⇔ | p−1 p−1 ⇔e| e ❱➻ e ❧➔ ❝➜♣ ❝õ❛ q (mod p) ♥➯♥ e| p−1 ⇔ q (p−1)/2 ≡ (mod p) ❱➻ p p−1 ❝❤✐❛ ♥➳✉ e (−1)(p−1)/2 pp q = ❈❤ó♥❣ t❛ ♣❤➙♥ t➼❝❤ ♣❤➼❛ ❜➯♥ tr→✐✿ ❝❤✐❛ p−1 p−1 ♥➳✉ e ❝❤✐❛ ❝❤♦ e ❑❤✐ e ❧➔ t❤ù tü ❝õ❛ q ❝õ❛ p✱ e ❝❤✐❛ p−1 ♥➳✉ q (p−1)/2 ≡ 1(♠♦❞ p) ❚❤❡♦ t✐➯✉ ❝❤✉➞♥ ❊✉❧❡r✱ q p ≡ q (p−1)/2 (♠♦❞ p), ❞♦ ✈➟② q (p−1)/2 ≡ (mod p) ⇔ ❚â♠ ❧↕✐✱ t❛ ❝â r≡p (mod 2) ⇔ q p q p = = ✷✺ ▼➦t ❦❤→❝ D ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ mod p ⇔ ⇔ D p −1 q =1 (p−1)/2 p q ⇔ (−1)((q−1)/2)((p−1)/2) p2 q p q (p1)/2 =1 =1 ữ tứ ỵ ❙t✐❝❦❡❧❜❡r❣❡r ❝❤ó♥❣ t❛ ❝â q p = (−1)((q−1)/2)((p−1)/2) p q ▲✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✸✳✷✳✹✳ ❚➻♠ ❜➟❝ ❝õ❛ ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ❝õ❛ (x257 − 1)/(x − 1) tr➯♥ F19 ✳ ●✐↔✐✳ ❚❛ ❝â 19 257 = 257 19 19 = 19 19 = (−1) = (−1) = 10 19 = −1 ❉♦ ✈➟② 19128 ≡ −1 (mod 257) ✭t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❊✉❧❡r✮✳ ●å✐ e ❧➔ ❜➟❝ 19 õ e ữợ 256 = 28 ✭✈➻ 19256 = (mod 257)✱ t❤❡♦ ỵ rt ọ ữ e ổ ữợ 128 = 25 ✈➻ 19128 ≡ −1 (mod 257)✳ ❉♦ ✈➟② e = 256 ✈➔ ✤❛ t❤ù❝ x257 − = x256 + · · · + x + x−1 ❧➔ ❜➜t ❦❤↔ q✉② ♠♦❞✉❧♦ p✳ ❱➼ ❞ö ✸✳✷✳✺✳ ❚➻♠ ❜➟❝ ❝õ❛ ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ❝õ❛ f (x) = (x257 −1)/(x− 1) tr➯♥ F11 ✳ ✷✻ ●✐↔✐✳ ●å✐ e ❧➔ ❜➟❝ ❝õ❛ ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ❝õ❛ (x257 − 1)/(x − 1) tr➯♥ F11 ✳ ❑❤✐ ✤â e ❝❤➼♥❤ ❧➔ ❜➟❝ ❝õ❛ 11 ♠♦❞✉❧♦ 257✳ ❚❛ ❝â 114 ≡ −8 (mod 257)✳ ❉♦ ✈➟② 1132 ≡ 88 ≡ −1 (mod 257)✳ ❙✉② r❛ 1164 ≡ (mod 257) ữ e ữợ 64 ữ ổ ữợ 32 e = 64 ✈➔ f (x) ❝â ✹ ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② tr F11 ỵ trr ♠♦❞✉❧♦ ✷ ◆❤÷ ð ❱➼ ❞ư ✷✳✷✳✶ t❛ ✤➣ t❤➜②✱ t ỵ ổ ỏ ú ỳ ❝❤♦ tr÷í♥❣ ❤đ♣ p = 2✳ ❚✉② ♥❤✐➯♥ t❛ ❝â ỵ s t số ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ❝õ❛ ✤❛ t❤ù❝ ♥❣✉②➯♥ ♠♦❞✉❧♦ ỵ f (x) ởt t❤ù❝ ♠♦♥✐❝ ❜➟❝ m ✈ỵ✐ ❝→❝ ❤➺ sè ♥❣✉②➯♥✳ ●✐↔ sû D(f ) ≡ (mod 2)✳ ●å✐ r ❧➔ sè ❝→❝ ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ❝õ❛ f (x) ♠♦❞✉❧♦ ✷✳ ❑❤✐ ✤â D ≡ (mod 4) ✈➔ r ≡ m (mod 2) ⇔ D(f ) ≡ (mod 8) ✣à♥❤ ♥❣❤➽❛ ✸✳✸✳✷✳ ❈❤♦ f (x) ∈ F [x] ❜➟❝ m ✈➔ ❝â m ♥❣❤✐➺♠ α1, , αm tr♦♥❣ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè K ❝❤ù❛ F ✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ δ (f ) = (αi + αj ) 1≤i