LUẬN văn THẠC sĩ HAY 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◆✱ ✺✴✷✵✶✾

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✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

✣⑨▼ ❚❍➚ ◆●➴❈ ❚❹▼

❱➋ ❚➑◆❍ ❈❍➂◆ ▲➈ ❈Õ❆ ❙➮ ◆❍❹◆ ❚Û ❇❻❚
❑❍❷ ◗❯❨ ▼❖❉❯▲❖ P ❈Õ❆ ✣❆ ❚❍Ù❈ ❍➏
❙➮ ◆●❯❨➊◆
❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ ❚♦→♥ sì ❝➜♣
▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✸

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
●■⑩❖ ❱■➊◆ ❍×❰◆● ❉❼◆

❚❙✳ ◆●❯❨➍◆ ❉❯❨ ❚❹◆

❚❍⑩■ ◆●❯❨➊◆✱ ✺✴✷✵✶✾

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✐✐✐

▼ư❝ ❧ư❝
▼ð ✤➛✉
❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶
✶✳✷
✶✳✸

❑➳t t❤ù❝ ❝õ❛ ❤❛✐ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
❇✐➺t t❤ù❝ ❝õ❛ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ỹ ỗ rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

ữỡ ỵ trr

ừ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr♦♥❣ Fp [x] ✳ ✳
ỵ trr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✣❛ t❤ù❝ ♥❣✉②➯♥ ❦❤↔ q✉② ♠♦❞✉❧♦ ♠å✐ sè p ♥❣✉②➯♥ tố
ữỡ tỹ ừ ỵ trr tự tỹ

ữỡ ỵ trr t t




ỵ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ỵ trr ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳
ỵ trr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✷
✸✸

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✶

▼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 ừ

trr r r ✈➔ n ❝â ❝ò♥❣ t➼♥❤ ❝❤➤♥ ❧➫✱ tù❝ ❧➔ r ≡ n
(mod 2)✱ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ D(f ) ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦ p✳
▼ư❝ t✐➯✉ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ự ừ ỵ t
rr ụ ♥❤÷ ù♥❣ ❞ư♥❣ ❝õ❛ ♥â tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤
❜➟❝ ❤❛✐✳
◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❜è ❝ư❝ ❝õ❛ ❧✉➟♥
✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❜❛ ❝❤÷ì♥❣✳

❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❦➳t t❤ù❝ ❝õ❛ ❤❛✐ tự
t tự ừ tự ỗ rs

ữỡ ỵ trr

ữỡ tr ỵ ❙t✐❝❦❡❧❜❡r❣❡r✱ ♠ët sè ✈➼ ❞ư ♠✐♥❤ ❤å❛✱
✈➔ ♠ët t÷ì♥❣ tü ừ ỵ tự tỹ

ữỡ ỵ trr t t

ữỡ tr ỵ r t t
ởt ự ừ t sỷ ử ỵ ❙t✐❝❦❡❧❜❡r❣❡r✳
▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ ✈➔ ❤♦➔♥ t❤➔♥❤ ✈➔♦ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾ t↕✐
tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ◗✉❛ ✤➙②✱ t→❝ ❣✐↔ ①✐♥ ❜➔②
tä ❧á♥❣ t ỡ 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❤➔♥❤ ❧✉➟♥ ✈➠♥

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ỗ tớ t ỷ ớ ỡ tợ ỗ
t trữớ ❍÷♥❣ ✣↕♦✱ ✣ỉ♥❣ ❚r✐➲✉✱ ◗✉↔♥❣ ◆✐♥❤ ✤➣ t↕♦ ✤✐➲✉
❦✐➺♥ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳
❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳
❳→❝ ♥❤➟♥ ❝õ❛ ♥❣÷í✐ ữợ



t
ữớ ✈✐➳t ❧✉➟♥ ✈➠♥

✣➔♠ ❚❤à ◆❣å❝ ❚➙♠

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✸

❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥
t❤ù❝ ❝❤✉➞♥ ❜à
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❦➳t t❤ù❝ ❝õ❛ ❤❛✐ ✤❛ tự
t tự ừ tự ỗ rs ❧✐➺✉ t❤❛♠ ❦❤↔♦ sû ❞ư♥❣
❝❤♦ ❝❤÷ì♥❣ ♥➔② ❧➔ t➔✐ ❧✐➺✉ ❬✷✱ ❙❡❝t✐♦♥ ✻✳✻❪ ✈➔ ❬✸✱ ❈❤❛♣t❡r ✶✺❪✳

✶✳✶ ❑➳t t❤ù❝ ❝õ❛ ❤❛✐ ✤❛ t❤ù❝
●✐↔ sû f, g ❧➔ ❤❛✐ ✤❛ t❤ù❝ x ợ số tr ởt trữớ 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 ừ ❦➳t t❤ù❝✳

❚➼♥❤ ❝❤➜t ✶✳✶✳✶✳ R(g, f ) = (−1)mnR(f, g).

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✹
❈❤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) = 0 ♥➳✉ f

✈➔ g ❝â ♠ët ♥❤➙♥ tû ❝❤✉♥❣ ❜➟❝

❞÷ì♥❣✳

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ f ✈➔ g ❝â ♠ët ♥❤➙♥ tû ❝❤✉♥❣ ❧➔ h(x) ∈ F [x]✳ ❑❤✐ ✤â ❣å✐
α ∈ K ♠ët ♥❣❤✐➺♠ ❝õ❛ 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 = 0
✈➔ ❞♦ ✈➟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

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✺
❱➻ αi ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❝õ❛ f ✱ ♥➯♥ f (αi ) = 0 ✈➔ ❞♦ ✈➟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 ❧➔ ✈æ ữợ
ự t g(x) = b t
n

R(f, g) = a

0

g(αi ) = bn .

i=1

❈→❝ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶✱✶✳✶✳✹✱ ✶✳✶✳✺ ❝❤♦ ♣❤➨♣ t❛ t➼♥❤ t♦→♥ ❦➳t t❤ù❝ ❝õ❛ ❜➜t
❦➻ ❤❛✐ ✤❛ t❤ù❝ ♥➔♦ ❜➡♥❣ t❤✉➟t t♦→♥ ❝❤✐❛ ❝õ❛ ❊✉❝❧✐❞✳ ❈→❝ t➼♥❤ ❝❤➜t ♥➔②
❝ơ♥❣ ❝❤♦ ♣❤➨♣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ❦➳t t❤ù❝ R(f, g) ❧➔ ♠ët ♣❤➛♥
tû ❝õ❛ tr÷í♥❣ F ♠➦❝ ❞ị ♥â ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❞ü❛ 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 ợ deg f = n ≥ 1✳
❑❤✐ ✤â t❤❡♦ t❤✉➟t t♦→♥ tự tỗ t tự q r tr♦♥❣ F [x]
s❛♦ ❝❤♦
g = f q + r,
✈ỵ✐ r = 0 ❤♦➦❝ 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♥❣ ♠✐♥❤✳

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✻

❚➼♥❤ ❝❤➜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] tự ợ số tr trữớ F ✈➔ K ❧➔ ♠ët
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 ) = 0 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f ✈➔ f ❦❤ỉ♥❣ ❝â
t❤ø❛ sè ❝❤✉♥❣✳
❈❤ó♥❣ t❛ ❝â t❤➸ t➼♥❤ t♦→♥ D(f ) ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ t❤✉➟t t♦→♥
tr f f ữợ ởt số ✈➼ ❞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 = 1.

❱➼ ❞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 − ).
2 4
4
2

a2

✣➦t r = b − ✳ ❚❛ ❝â
4
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.

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✼

❱➼ ❞ö ✶✳✷✳✸✳ ❈❤♦ f (x) = x3 + qx + r✳ ❚❤➻ f (x) = 3x2 + q ✈➔ t❤ü❝ ❤✐➺♥
t❤✉➟t t♦→♥ ❊✉❝❧✐❞✱ t❛ ❝â

x
2q
+
x+r ,
3
3
2q
9x 27r
27r2
x+r
− 2 + q+
3
2q
4q

4q 2

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 ✶✳✶✳✹)
3
2qx
= −9R(
+ r, f ) (t❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✶)
3
2q 2 2qx
27r2
R(
= −9
+ r, q +
)
3
3
4q 2
27r2
) = −4q 3 − 27r2 .

= −4q 2 (q +
2
4q

❱➼ ❞ö ✶✳✷✳✹✳ ❳➨t f (x) = xn − 1 ∈ 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(n−1)/2 nn .

t ỵ t 1 Ã Ã Ã n = (−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 ❧➔ ❝→❝ ♥❣❤✐➺♠

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✽
❝õ❛ ♥â tr♦♥❣ tr÷í♥❣ ✤â♥❣ ✤↕✐ sè K ✳ ❑❤✐ ✤â



2

n j−1

(αi − αj )2 .

(αi − αj ) =

D(f ) = 
j=2 i=1

1≤i
❈❤ù♥❣ ♠✐♥❤✳ ❱➻ α1 , . . . , αn ∈ K ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ f ♥➯♥ f (x) = (x −
α1 )(x − α2 ) . . . (x − αn )✳ ❉♦ ✤â

f (x) = (x − α2 )(x − α3 ) . . . (x − αn ) + (x − α1 )(x − α3 ) . . . (x − αn )
+ · · · + (x − α1 ) . . . (x − αn−1 )
n

n

(x − αj ).

=

i=1 j=1,j=i

◆❤÷ ✈➟② f (αi ) =

n
j=1,j=i (αi

− αj )✳ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✸✱ t❛ ❝â

n
n(n−1)/2

D(f ) = (−1)

n

f (αi ) = (−1)
i=1

n

n(n−1)/2

(αi j ).
i=1 j=1,j=i

ú ỵ r tr t ố ũ ð ❝æ♥❣ t❤ù❝ tr➯♥✱ ❝â n(n − 1) t❤ø❛ sè✱
tr♦♥❣ õ ởt ỷ ỗ tứ số i αj ✈ỵ✐ i < j ✈➔ ♠ët ♥û❛ ❧➔
❝→❝ t❤ø❛ sè ❞↕♥❣ αi − αj ✈ỵ✐ i > j ✳ ◆❤➙♥ ♠é✐ t❤ø❛ sè ❞↕♥❣ t❤ù ❤❛✐ ✈ỵ✐
(−1) t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

❱➼ ❞ö ✶✳✷✳✻✳ ❚❛ ✤✐ t➼♥❤ ❜✐➺t t❤ù❝ ❝õ❛ ✤❛ t❤ù❝ ♠♦♥✐❝ ❜➟❝ ✷ ✈➔ ❜➟❝ ✸ sỷ

ử ổ tự t t tự tr trữợ
❳➨t f (x) = x2 + ax + b ∈ F [x]✳ ●å✐ α1 , α2 ❧➔ ❤❛✐ ♥❣❤✐➺♠ ❝õ❛ f
✭tr♦♥❣ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè ♥➔♦ ✤â ❝❤ù❛ F ✮✳ ❑❤✐ ✤â ❜✐➺t t❤ù❝ ❝õ❛ f ❧➔

D(f ) = (α1 − α2 )2 = (α1 + α2 )2 − 4α1 α2 = a2 − 4b.
✭❜✮ ❳➨t ✤❛ t❤ù❝ f (x) = x3 +qx+r ∈ F [x]✳ ●å✐ α1 , α2 , α3 ❧➔ ❝→❝ ♥❣❤✐➺♠
❝õ❛ f ✳ ❑❤✐ ✤â ❜✐➺t t❤ù❝ ❝õ❛ f ❧➔

D(f ) = (α2 − α1 )2 (α3 − α1 )2 (α3 − α2 )2 .

❚❛ ❝â

x3 + qx + r = (x − α1 )(x − α2 )(x − α3 ).

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✾
▲➜② ✤↕♦ ❤➔♠ ❤❛✐ ✈➳ t❤❡♦ x✱ t❛ s✉② r❛

3x2 + q = (x − α1 )(x − α2 ) + (x − α1 )(x − α3 ) + (x − α2 )(x − α3 ).
❉♦ ✈➟②✱ t❤❛② x = α1 ✱ α2 ✈➔ α3 t❛ ✤÷đ❝

3α12 + q = (α1 − α2 )(α1 − α3 ),
3α22 + q = (α2 − α1 )(α2 − α3 ),
3α32 + q = (α3 − α1 )(α3 − α2 ).
❚❛ s✉② r❛

D(f ) = −(3α12 + q)(3α22 + q)(3α32 + q)
= −[27(α1 α2 α3 )2 + 9q(α12 α22 + α12 α32 + α22 α32 ) + 3q 2 (α12 + α22 + α32 ) + q 3 ].
❚❛ ❝â α1 α2 α3 = −r✱ ✈➔

α12 α22 + α12 α32 + α22 α32 = (α1 α2 + α1 α3 + α2 α3 )2 − 2α1 α2 α3 (α1 + α2 + α3 )
= q2,
✈➔

❉♦ ✈➟②

α12 + α22 + α32 = (α1 + α2 + α3 )2 − 2(α1 α2 + α1 α3 + α2 α3 )
= −2q.

D(f ) = −[27r2 + 4q 3 ] = −4q 3 − 27r2 .

▼➺♥❤ ✤➲ ✶✳✷✳✼✳ ❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ✤❛ t❤ù❝ ♠♦♥✐❝ tr♦♥❣ F [x]✳ ❑❤✐ ✤â
D(f g) = D(f )D(g)R(f, g)2 .
❈❤ù♥❣ ♠✐♥❤✳ ●å✐ n = deg f ✈➔ m = deg g ✳ ❑❤✐ ✤â m + n = deg(f g)✳ ❚❛
❝â

(−1)(m+n)(m+n−1)/2 D(f g)
= R(f g, (f g) )
= R(f g, f g + f g )
= R(f, f g + f g )R(g, f g + f g )

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✶✵

= R(f, f g)R(g, f g )
= R(f, f )R(f, g)R(g, f )R(g, g )
= (−1)n(n−1)/2 D(f )R(f, g)(−1)mn R(f, g)(−1)m(m−1)/2 D(g)
= (−1)n(n−1)/2+mn+m(m−1)/2 D(f )D(g)(R(f, g))2 .
❚ø ✤â t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ✈➻

(m + n)(m + n − 1) n(n − 1)
m(m − 1)
=
+ mn +
.
2
2

2

▼➺♥❤ ✤➲ ✶✳✷✳✽✳ ❈❤♦ f1, . . . , fr ❧➔ ❝→❝ ✤❛ t❤ù❝ ♠♦♥✐❝ tr♦♥❣ F [x]✳ ❑❤✐ ✤â
D(f1 · · · fr ) = D(f1 ) · · · D(fr )R2 ,
ð ✤➙② R =

1≤i
R(fi , fj ) ∈ F ✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❤❡♦ r✳ ❈æ♥❣ t❤ù❝ ✤ó♥❣ ✈ỵ✐ r = 2✳
●✐↔✐ sû ♥â ✤➣ ✤ó♥❣ ✈ỵ✐ r − 1 ✤❛ t❤ù❝ ✈ỵ✐ r ≥ 3✳ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✶✳✶✳✼ ✈➔
t❤❡♦ q✉② ♥↕♣ t❛ ❝â

D(f1 · · · fr ) = D(f1 · · · fr−1 )D(fr )R(f1 · · · fr−1 , fr )2
r−1
2

= D(f1 ) · · · D(fr−1 )

R(fi , fr )2

R(fi , fj ) D(fr )
1≤i
i=1

R(fi , fj )2 .

= D(f1 ) · Ã Ã D(fr )

1i
ỹ ỗ rs
t p ♠ët sè ♥❣✉②➯♥ tè ✈➔ ①➨t Fp = Z/pZ = {[0], [1], . . . , [p − 1]}
tr÷í♥❣ ❝â p ♣❤➛♥ tû ❧➔ ❝→❝ sè ♥❣✉②➯♥ ♠♦❞✉❧♦ p✳ ●å✐ K ❧➔ ♠ët tr÷í♥❣ ❜➜t
❦ý ❝❤ù❛ Fp ✳ ❳➨t φp : K → K ❧➔ →♥❤ ①↕ ❝❤♦ ❜ð✐ φp (a) = ap ✱ ✈ỵ✐ ♠å✐ a ∈ K ✳
❘ã r➔♥❣

φp (ab) = (ab)p = ap bp = φp (a)φp (b).

❍ì♥ ♥ú❛
p−1
p

p

φp (a + b) = (a + b) = a +
k=1

p p−k k
a b + bp = ap + bp = φp (a) + φp (b).
k

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✶✶
✭Ð ✤➙② t❛ ✤➣ sû ❞ö♥❣ ♥❤➟♥ ①➨t r➡♥❣ p |

p

k

ữ p tỹ ỗ

ừ trữớ K ✳ ⑩♥❤ ①↕ φp ♥❤÷ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ tü ỗ rs ừ
K

ờ a ởt ♣❤➛♥ tû tr♦♥❣ K ✳ ❑❤✐ ✤â φp(a) = a ❦❤✐ ✈➔ ❝❤➾
❦❤✐ a t❤✉ë❝ Fp ✳

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ a tở Fp t t ỵ rt ọ t õ φp (a) =
ap = a✳
◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû φp (a) = a✳ ❚❛ s✉② r❛ a ∈ K ❧➔ ♥❣❤✐➺♠ ừ tự
p
x x ỵ rt ọ p ♣❤➛♥ tû tr♦♥❣ Fp ✤➲✉ ❧➔ ♥❣❤✐➺♠ ❝õ❛
✤❛ t❤ù❝ xp − x✳ ❱➻ xp − x ❝â ❜➟❝ ❜➡♥❣ p ♥➯♥ p ♣❤➛♥ tû ❝õ❛ Fp ❝❤➼♥❤ ❧➔ t➜t
❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ xp − x✳ ❉♦ a ❧➔ ♠ët tr♦♥❣ ❝→❝ ♥❣❤✐➺♠ ♥➔② ♥➯♥ a t❤✉ë❝
Fp ✳
❚❛ ♠ð rë♥❣ →♥❤ ①↕ φp ❧➯♥ →♥❤ ①↕ tø K[x] ✈➔♦ K[x] ữ s ợ
f (x) = an xn + Ã Ã · + a1 x + a0 ✱ t❛ ✤à♥❤ ♥❣❤➽❛

φp (f (x)) = φp (an )xn + · · · + φp (a1 )x + φp (a0 )
= apn xn + · · · + ap1 x + ap0 .

❇ê ✤➲ ✶✳✸✳✷✳ ❈❤♦ f (x) ∈ K[x]✳ ❑❤✐ ✤â φp(f (x)) = f (x) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
f (x) ∈ Fp [x]

ự r tứ ờ trữợ

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ữỡ ỵ
trr
ữỡ tr ỵ ❙t✐❝❦❡❧❜❡r❣❡r✱ ♠ët sè ✈➼ ❞ư ♠✐♥❤ ❤å❛✱
✈➔ ♠ët t÷ì♥❣ tü ừ ỵ tự tỹ t❤❛♠ ❦❤↔♦ sû
❞ư♥❣ ❝❤♦ ❝❤÷ì♥❣ ♥➔② ❧➔ ❬✸✱ ❈❤❛♣t❡r ✶✺❪ ✈➔ ❬✷✱ ❙❡❝t✐♦♥ ✻✳✻❪✳
❈❤♦ f (x) ∈ Z[x] ❧➔ ♠ët ✤❛ t❤ù❝ ❝❤✉➞♥ ✭♠♦♥✐❝✮ ❤➺ sè ♥❣✉②➯♥ ❜➟❝ n
✈➔ ❣å✐ D(f ) ❧➔ ❜✐➺t t❤ù❝ ❝õ❛ f ✳ ❈❤♦ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè ❧➫ ✈➔ ❣✐↔ sû
p D(f )✳ ●å✐ f¯(x) ∈ Fp [x] ❧➔ ✤❛ t❤ù❝ ♥❤➟♥ ✤÷đ❝ tø f ❜➡♥❣ ❝→❝❤ t❤✉
❣å♥ ❤➺ sè ♠♦❞✉❧♦ p✳ ●å✐ r ❧➔ sè ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ❝õ❛ f õ
ỵ trr r r n ❝â ❝ò♥❣ t➼♥❤ ❝❤➤♥ ❧➫✱ tù❝ ❧➔
r ≡ n (mod 2)✱ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ D(f ) ❧➔ ❜➻♥❤ ữỡ p ỵ
tr ởt t q ❝õ❛ ❙t✐❝❦❡❧❜❡r❣❡r ❬✹❪✳ ◆â ❝ơ♥❣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤
❜ð✐ ❙❦♦❧❡♠ ✭✶✾✺✷✮✱ P ố ợ p = 2 ừ ỵ ❙t✐❝❦❡❧❜❡r❣❡r
✭s➩ ✤÷đ❝ tr➻♥❤ ❜➔② ð ❈❤÷ì♥❣ ✸✱ ♠ư❝ ✸✮ ❝ơ♥❣ ♥➡♠ tr♦♥❣ ♠ët ❦➳t q✉↔
❝õ❛ ❙t✐❝❦❡❧❜❡r❣❡r✱ ✈➔ ♥â ❝ơ♥❣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❈❛r❧✐t③ ✭✶✾✺✸✮✱ ❉❛❧❡♥
✭✶✾✺✺✮✱ ❙✇❛♥ ✭✶✾✻✷✮✱ ❇❡r❧❡❦❛♠♣ ✭✶✾✻✽✮✳

✷✳✶ ◆❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr♦♥❣ F [x]
p

❇ê ✤➲p ✷✳✶✳✶✳ ●✐↔ sû α ∈ K ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ f (x) ∈ Fp[x]✳ ❑❤✐
✤â α ❝ô♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ f (x)✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✈✐➳t f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ Fp ✳ ❑❤✐
✤â 0 = f (α) = ni=0 ai αi ✳ ❙û ❞ö♥❣ ỗ rs p t õ
n

0 = p (0) = φp

n

ai α
i=0

i

n
i

=

ai αpi = f (αp ).

φp (ai )φp (α ) =
i=0

i=0

Ð tr➯♥ t❛ ✤➣ sû ❞ö♥❣ t➼♥❤ ❝❤➜t φp (a) = a ✈ỵ✐ ♠å✐ a ∈ Fp ✳

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✶✸

▼➺♥❤ ✤➲ ✷✳✶✳✷✳ ❈❤♦ f (x) ❧➔ ♠ët ✤❛ t❤ù❝ ♠♦♥✐❝ ❜➜t ❦❤↔ q✉② tr♦♥❣ Fp[x]

✈ỵ✐ ❜➟❝ d✳ ❈❤♦ K ❧➔ ♠ët tr÷í♥❣ ❝❤ù❛ Fp ✱ ✈➔ α ∈ K ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛
f (x)✳ ❑❤✐ ✤â tr♦♥❣ K[x]✱ t❛ ❝â
2

f (x) = (x − α)(x − αp )(x − αp ) . . . (x − αp

d−1

).

2

❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ờ trữợ t õ , p , p , . . . ✤➲✉ ❧➔ ♥❣❤✐➺♠ ❝õ❛
f (x)✳ ❱➻ ✤❛ t❤ù❝ f (x) ❝❤➾ ❝â ❤ú✉ ❤↕♥ ♥❣❤✐➺♠ tr♦♥❣ K tỗ t số
k
l
l
k
ổ t k < l s❛♦ ❝❤♦ αp = αp ✳ ❚❛ ❝â 0 = αp −αp =
l−k
k
l−k
(αp − α)p ✳ ❙✉② r p = ữ tỗ t số tỹ ♥❤✐➯♥ ♥❤ä ♥❤➜t r
r
s❛♦ ❝❤♦ αp = α✳
2
r−1
k
h
❚❛ ❝â α, αp , αp . . . , αp ✤➲✉ ♣❤➙♥ ❜✐➺t ✭♥➳✉ ❦❤ỉ♥❣ ❣✐↔ sû ap = αp

h−k
✈ỵ✐ h, k, 1 ≤ k < h < r ♥➔♦ ✤â❀ t❤➻ ữ trữợ p
=
t ợ t ọ ♥❤➜t ❝õ❛ r✮✳ ❱➻ ✈➟② f ❝â ➼t ♥❤➜t r ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t
tr♦♥❣ K ✳ ❉♦ ✤â r ♣❤↔✐ ♥❤ä ❤ì♥ ❤♦➦❝ ❜➡♥❣ ❜➟❝ ❝õ❛ f ✱ tù❝ ❧➔ r ≤ d✳
✣➦t
r−1
g(x) = (x − α)(x − αp ) · · · (x − αp ).
r−1

❱➻ (αp

)p = α, ♥➯♥
r−1

φp (g(x)) = φp (x − α)φp (x − αp ) · · · φp (x − αp
2

)

r

= (x − αp )(x − αp ) · · · (x − αp )
= g(x).
❉♦ ✈➟② g(x) ♥➡♠ tr♦♥❣ Fp [x]✳ ❍✐➸♥ ♥❤✐➯♥ α ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ g(x)✳ ❈❤✐❛
✤❛ t❤ù❝ f (x) ❝❤♦ g(x) t❛ ❝â

f (x) = g(x)q(x) + h(x),
✈ỵ✐ q(x), h(x) ∈ Fp [x] ✈➔ h = 0 ❤♦➦❝ deg h < r✳ ❱ỵ✐ ♠å✐ i = 0, . . . , r − 1
t❛ ❝â

i

i

i

i

h(αp ) = f (αp ) − g(αp )q(αp ) = 0.

r−1

◆❤÷ ✈➟② ✤❛ t❤ù❝ h ❝â ➼t ♥❤➜t r ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t α, αp , . . . , αp ✳ ❉♦ ✤â
h ♣❤↔✐ ❧➔ ✤❛ t❤ù❝ 0 ✈➔ ❞♦ ✈➟② f (x) = g(x)q(x)✳ ❱➻ f (x) ❧➔ ❜➜t ❦❤↔ q✉② ✈➔
♠♦♥✐❝ ♥➯♥ f (x) = g(x) ✈➔ r = d✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

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ỵ trr

ỵ p ởt sè ♥❣✉②➯♥ tè ❧➫✱ f (x) ❧➔ ♠ët ✤❛ t❤ù❝ ♠♦♥✐❝

❜➟❝ m ✈ỵ✐ ❝→❝ ❤➺ sè tr♦♥❣ Fp ✳ ●✐↔ sû D(f ) = 0✳ ●å✐ r ❧➔ sè ❝→❝ ♥❤➙♥ tû
❜➜t ❦❤↔ q✉② ❝õ❛ f (x) tr♦♥❣ Fp [x]✳ ❑❤✐ ✤â r ≡ m (mod 2) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
D(f ) ❧➔ ♠ët ❜➻♥❤ ♣❤÷ì♥❣ tr♦♥❣ Fp ✳

❈❤ù♥❣ ♠✐♥❤✳ t ú t ự ỵ trữớ ❤đ♣ r = 1✳

❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② f (x) ❧➔ ❜➜t ❦❤↔ q✉② ❜➟❝ m✳ ●å✐ K ❧➔ ♠ët tr÷í♥❣
✤â♥❣ ✤↕✐ sè ❝❤ù❛ Fp ✳ ●å✐ α ∈ K ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ f ✳ ❑❤✐ ✤â t➜t ❝↔ ❝→❝
2
m−1
♥❣❤✐➺♠ ❝õ❛ f ❧➔ α, αp , αp , . . . , p
t ú ỵ r t
pm
ụ ❝â α = α✳
✣➦t
m−1 j−1

i

j

i

(αp − αp ) =

δ(f ) =
j=1 i=0

j

(αp − αp ).
0≤i
❑❤✐ ✤â δ(f ) ∈ K ✈➔ D(f ) = (δ(f ))2 ✳ ❉♦ ✈➟② D(f ) ❧➔ ♠ët ❜➻♥❤ ♣❤÷ì♥❣
tr♦♥❣ Fp ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ δ(f ) ♥➡♠ tr♦♥❣ Fp ✳ ❚❛ ❦✐➸♠ tr❛ δ(f ) ♥➡♠ tr♦♥❣
Fp ❜➡♥❣ ❝→❝❤ ❦✐➸♠ tr❛ ✤➥♥❣ t❤ù❝ φp (δ(f )) = δ(f )✳ ❚❛ ❝â

i

j

φp (αp ) − φp (αp )

φp (δ(f )) =
0≤i
i+1

(αp

=

j+1

− αp

).

0≤i
❚❤❛② ✤ê✐ ❝❤➾ sè t❛ ❝â
i

j

(αp − αp )

φ(δ(f )) =

1≤im−1
pi

pj

i

m

(αp − αp )

(α − α ) ·

=

1≤i
i=1
m−1

m−1

pi

pj

i

j

j

(α − αp )

(α − α ) ·

= (−1)

1≤i
= (−1)m−1

j=1

(αp − αp )
0≤i
= (−1)m−1 δ(f ).

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✶✺
◆❤÷ ✈➟②

D(f ) ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ tr♦♥❣ Fp ⇔ δ(f ) ∈ Fp ⇔ φp (δ(f )) = δ(f )
⇔ (−1)m−1 δ(f ) = δ(f ) ⇔ (−1)m−1 = 1.

⇔ m 1 (mod 2).
ữ ỵ ú ợ r = 1✳
❇➙② ❣✐í t❛ ❣✐↔ sû f = f1 f2 . . . fr ❧➔ t➼❝❤ ❝õ❛ r ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ♣❤➙♥
❜✐➺t✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ δ(f ) ✈➔ δ(fi ) t÷ì♥❣ tü ♥❤÷ tr♦♥❣ tr÷í♥❣ ❤đ♣ r = 1 ð
tr➯♥✳
❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✽✱ t❛ ❝â

D(f ) = D(f1 f2 . . . fr ) = D(f1 )D(f2 ) . . . D(fr )R2 ,
✈ỵ✐ R ♥➔♦ ✤â t❤✉ë❝ Fp ✳ ❉♦ ✤â✱ tr♦♥❣ K t❛ ❝â

δ(f ) = δ(f1 )δ(f2 ) . . . δ(fr )S,
✈ỵ✐ S = ±R ∈ Fp ✳ ❚❤❡♦ tr÷í♥❣ ❤đ♣ r = 1✱ ✈ỵ✐ ♠å✐ i = 1, . . . , r✱ t❛ ❝â
φp (δ(fi )) = (−1)di −1 δ(fi )✱ ð ♥ì✐ di = deg fi ✳ ❉♦ ✤â

φ(δ(f )) = δ(f1 )δ(f2 ) . . . δ(fr )S(−1)d1 −1 (−1)d2 −1 . . . (−1)dr −1
= δ(f )(−1)d1 +d2 +ÃÃÃ+dr r
= (1)mr (f ).
ú ỵ r d1 + Ã · · + dr = deg f = m✳✮ ❚ø ✤â t❛ ❝â

D(f ) ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ tr♦♥❣ Fp ⇔ δ(f ) ∈ Fp ⇔ φp (δ(f )) = δ(f )
⇔ (−1)m−r δ(f ) = δ(f ) ⇔ (−1)m−r = 1
⇔ m ≡ r (mod 2).
❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣
ự ỵ trr ữ r tr ❝❤ù♥❣ ♠✐♥❤ ✭✈ỵ✐
sû❛ ✤ê✐ t❤➼❝❤ ❤đ♣✮ ❝õ❛ ❙✇❛♥✭✶✾✻✷✮ ✈➔ ❇❡r❧❡❦❛♠♣ ✭✶✾✻✽✮✳

❱➼ ❞ö ✷✳✷✳✷✳ ❳➨t f (x) = x2 + x + 1 ∈ F2[x] ❝â ❜➟❝ d = 2✳ ❱➻ f (0) =
f (1) = 1 = 0 tr♦♥❣ F2 ♥➯♥ f (x) ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ tr♦♥❣ F2 ✈➔ ❞♦ ✈➟②
f (x) ∈ F2 [x] ❧➔ ❜➜t ❦❤↔ q✉②✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② sè ♥❤➙♥ tû ❜➜t ❦❤↔
q✉② ♠♦♥✐❝ ❝õ❛ f (x) ❧➔ r = 1✳

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✶✻
▼➦t ❦❤→❝ ❜✐➺t t❤ù❝ ❝õ❛ f (x) ❧➔ D = 1 − 4 · 1 = 1 ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ tr♦♥❣
F2 ✳ ❍✐➸♥ ♥❤✐➯♥ ❦❤➥♥❣ ✤à♥❤

r ≡ d ⇔ D ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ mod 2
❧➔ s❛✐ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✳ ữ t ừ ỵ trr
ổ ỏ ú ♥ú❛ ❝❤♦ tr÷í♥❣ ❤đ♣ p = 2✳

❍➺ q✉↔ ✷✳✷✳✸✳3 ❈❤♦ p 2❧➔ ♠ët sè ♥❣✉②➯♥ tè ❧➫ ✈➔ f (x) = x3 +qx+r ∈ Fp[x]✳
●å✐ D = −4q − 27r ❧➔ ❜✐➺t t❤ù❝ ❝õ❛ f ✳ ●✐↔ sû p D✳ ●å✐ Np (f ) ❧➔ sè
♥❣❤✐➺♠ ❝õ❛ f tr➯♥ Fp ✳ ❑❤✐ ✤â

Np (f ) =

✵ ❤♦➦❝ ✸ ♥➳✉ D ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ tr♦♥❣ Fp
1 ♥➳✉ D ❦❤ỉ♥❣ ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ tr♦♥❣ Fp .

❈❤ù♥❣ ♠✐♥❤✳ ●å✐ r ❧➔ sè ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ♠♦♥✐❝ ❝õ❛ f (x) tr➯♥ Fp ✳ ❘ã
r➔♥❣ r ❝❤➾ ❝â t❤➸ ♥❤➟♥ ❝→❝ ❣✐→ trà
✶✳ r = 1✱ tù❝ ❧➔ f ❜➜t ❦❤↔ q✉② tr➯♥ Fp ✈➔ Np (f ) = 0✱
✷✳ r = 2✱ tù❝ ❧➔ f ❝â ❞✉② ♥❤➜t ♠ët ♥❣❤✐➺♠ tr➯♥ Fp ✈➔ Np (f ) = 1✱
✸✳ ❤♦➦❝ r = 3✱ tù❝ ❧➔ f ❝â ✸ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t tr➯♥ Fp Np (f ) = 3
ỵ trr õ r

D ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ tr♦♥❣ Fp
⇔ r ≡ deg f (mod 2) ⇔ r = 1 ❤♦➦❝ ✸ ⇔ Np (f ) = 0 ❤♦➦❝ ✸.

❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤

❱➼ ❞ö ✷✳✷✳✹✳ ❳➨t ✤❛ t❤ù❝ f (x) = x3 − x − 1✳ ❇✐➺t t❤ù❝ ❝õ❛ f ❧➔
D = −4q 3 − 27r2 = −23.
• ◆➳✉ p = 3 t❤➻ D = −23 ≡ 12 (mod 3) ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦ ✸✳
❚❤❡♦ ❤➺ q✉↔ tr➯♥ t❤➻ Np (f ) = 0 ❤♦➦❝ ✸✳ ❚❤ü❝ t➳ x3 − x − 1 ❧➔ ❜➜t ❦❤↔
q✉② ♠♦❞✉❧♦ ✸✳
• ◆➳✉ p = 5 t❤➻ D = −23 ❦❤ỉ♥❣ ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦ 5✳ ❚❤❡♦ ❤➺ q✉↔
tr➯♥ t❤➻ Np (f ) = 1✳ ❚❤ü❝ t➳ x = 2 ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ x3 − x − 1

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✶✼
♠♦❞✉❧♦ 5 ✈➔ ♣❤➙♥ t❛ ❝â ♣❤➙♥ t➼❝❤

(x3 − x − 1) = (x − 2)(x2 + 2x + 3)

(mod 5).

• ◆➳✉ p = 7 t❤➻ D = −23 = 5 (mod 7) ❦❤ỉ♥❣ ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦
7✳ ❚❤❡♦ ❤➺ q✉↔ tr➯♥ t❤➻ Np (f ) = 1✳ ❚❤ü❝ t➳ x = 5 ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t
❝õ❛ x3 − x − 1 ♠♦❞✉❧♦ 7 ✈➔ ♣❤➙♥ t❛ ❝â ♣❤➙♥ t➼❝❤
(x3 − x − 1) = (x − 5)(x2 − 2x + 3)

(mod 7).

• ◆➳✉ p = 11 t❤➻ D = −23 = 10 (mod 11) ❧➔ ❦❤ỉ♥❣ ❜➻♥❤ ♣❤÷ì♥❣
♠♦❞✉❧♦ 11✳ ❚❤❡♦ ❤➺ q✉↔ tr➯♥ t❤➻ Np (f ) = 1✳ ❚❤ü❝ t➳ x = 6 ❧➔ ♥❣❤✐➺♠
❞✉② ♥❤➜t ❝õ❛ x3 − x − 1 ♠♦❞✉❧♦ 11 ✈➔ ♣❤➙♥ t❛ ❝â ♣❤➙♥ t➼❝❤

(x3 − x − 1) = (x − 6)(x2 + 6x + 2)

(mod 11).

• ◆➳✉ p = 13 t❤➻ D = −23 ≡ 3 = 42 (mod 13) ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦
13✳ ❚❤❡♦ ❤➺ q✉↔ tr➯♥ t❤➻ Np (f ) = 0 ❤♦➦❝ ✸✳ ❚❤ü❝ t➳ x3 − x − 1 ❜➜t
❦❤↔ q✉② ♠♦❞✉❧♦ ✶✸✳
• ◆➳✉ p = 59 t❤➻ D = −23 ≡ 36 = 62 (mod 59) ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦
59✳ ❚❤❡♦ ❤➺ q✉↔ tr➯♥ t❤➻ Np (f ) = 0 ❤♦➦❝ ✸✳ ❚❤ü❝ t➳ x3 − x − 1 ❝â ✸
♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ♠♦❞✉❧♦ ✺✾ ✈➔ t❛ ❝â ♣❤➙♥ t➼❝❤
x3 − x − 1 = (x − 4)(x − 13)(x − 42)

(mod 59).

✷✳✸ ✣❛ t❤ù❝ ♥❣✉②➯♥ ❦❤↔ q✉② ♠♦❞✉❧♦ ♠å✐ sè p ♥❣✉②➯♥
tè
❑➳t q✉↔ s❛✉ ❧➔ q trỹ t ừ ỵ trr

q ❈❤♦ f (x) ❧➔ ♠ët ✤❛ t❤ù❝ ♠♦♥✐❝ ❜➟❝ ❝❤➤♥ ❤➺ sè ♥❣✉②➯♥✳
●✐↔ sû ❜✐➺t t❤ù❝ D ❝õ❛ f ❧➔ ởt số ữỡ õ ợ ồ
p tố ổ ữợ ừ D t f (x) ❧➔ ❦❤↔ q✉② ♠♦❞✉❧♦ p✳

❈❤ù♥❣ ♠✐♥❤✳ ●å✐ r ❧➔ sè ♥❤➙♥ tû ❜➜t ❦❤↔ q✉② ♠♦♥✐❝ ♠♦❞✉❧♦ p ❝õ❛ f (x)✳
❘ã r➔♥❣ D ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦ p t ỵ trr
r deg f mod 2✳ ❉♦ ✈➟② r ❧➔ sè ❝❤➤♥✳ ◆â✐ r✐➯♥❣ r = 1 ✈➔ f (x) ❧➔ ❦❤↔
q✉② ♠♦❞✉❧♦ p✳

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✶✽

❇ê ✤➲ ✷✳✸✳✷✳
❈❤♦ f (x) = x4 + ax2 + b ∈ F [x]✳ ❑❤✐ ✤â ❜✐➺t t❤ù❝ ❝õ❛ f ❧➔
2
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 2 = v ❧➔ ❤❛✐
♥❣❤✐➺♠ ❝õ❛ x2 + ax + b✳
❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✺✱ t❛ ❝â

D = [(−α − α)(β − α)(−β − α)(β + α)(−β + α)(−β − β)]2
= 16α2 β 2 (β 2 − α2 )2
= 16uv (u + v)2 − 4uv)
= 16b(a2 − 4b).

▼➺♥❤ ✤➲ ✷✳✸✳✸✳ ✣❛ t❤ù❝ x4 + 1 ❧➔ ❜➜t ❦❤↔ q✉② tr➯♥ Z ♥❤÷♥❣ ❧➔ ❦❤↔ q✉②
♠♦❞✉❧♦ p ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ tè p✳

❈❤ù♥❣ ♠✐♥❤✳ ❇✐➺t t❤ù❝ ❝õ❛ x4 + 1 ❧➔ D = 16 · 42 ❧➔ ♠ët sè ❝❤➼♥❤ ♣❤÷ì♥❣✳
❉♦ ✈➟② t❤❡♦ ❤➺ q✉↔ tr➯♥ t❤➻ x4 + 1 ❧➔ ❦❤↔ q✉② ♠♦❞✉❧♦ p ✈ỵ✐ ♠å✐ p ♥❣✉②➯♥
tè ♠➔ p = 2✳ ❚❛ ❝â x4 + 1 = (x + 1)4 (mod 2)✳ ◆❤÷ ✈➟② x4 + 1 ❧➔ ❦❤↔ q✉②
♠♦❞✉❧♦ p ✈ỵ✐ ♠å✐ p ♥❣✉②➯♥ tè✳
❚❛ ❝❤ù♥❣ ♠✐♥❤ x4 + 1 ❜➜t ❦❤↔ q✉② tr➯♥ Z✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû x4 + 1 ❧➔
❦❤↔ 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 = 0 ✈➔ cd = 1✳ ❚ø cd = 1 t❛
s✉② r❛ c = d = −1 ❤♦➦❝ c = d = 1✳ ❙✉② r❛ ab = −(c + d) = ±2✳ ❉♦ ✈➟②
a2 = ±2✱ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥✳

▼➺♥❤ ✤➲ ✷✳✸✳✹✳ ✣❛ t❤ù❝ x4 + 3x2 + 1 ❧➔ ❜➜t ❦❤↔ q✉② tr➯♥ Z ữ
q p ợ ồ số tè p✳

❈❤ù♥❣ ♠✐♥❤✳ ❇✐➺t t❤ù❝ ❝õ❛ x4 + 3x2 + 1 ❧➔ D = 16 · 52 ❧➔ ♠ët sè ❝❤➼♥❤
♣❤÷ì♥❣✳ ❉♦ ✈➟② t❤❡♦ ❤➺ q✉↔ tr➯♥ t❤➻ x4 + x2 + 1 ❧➔ ❦❤↔ q✉② ♠♦❞✉❧♦ p

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✶✾
✈ỵ✐ ♠å✐ p ♥❣✉②➯♥ tè ♠➔ p = 2 ✈➔ p = 5✳ ❚❛ ❝â x4 + 3x2 + 1 = (x + 1)4
(mod 2) ✈➔

x4 + 3x2 + 1 = (x − 1)2 (x − 1)2

(mod 5).

◆❤÷ ✈➟② x4 + 1 ❧➔ ❦❤↔ q✉② ♠♦❞✉❧♦ p ✈ỵ✐ ♠å✐ p ♥❣✉②➯♥ tè✳
❚❛ ❝❤ù♥❣ ♠✐♥❤ x4 + 3x2 + 1 ❜➜t ❦❤↔ q✉② tr➯♥ Z✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû
x4 + 3x2 + 1 ❧➔ ❦❤↔ 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 = 0 ✈➔ cd = 1✳ ❚ø cd = 1 t❛ s✉②
r❛ c = d = −1 ❤♦➦❝ c = d = 1✳ ❙✉② r❛ ab = 3 − (c + d) = 1 ❤♦➦❝ ✺✳ ❉♦
✈➟② a2 = −1 ❤♦➦❝ −5✱ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥✳

◆❤➟♥ ①➨t ✷✳✸✳✺✳ ❍❛✐ ♠➺♥❤ ✤➲ tr➯♥ ❧➔ ✈➼ ❞ư ✈➲ ✤❛ t❤ù❝ trị♥❣ ♣❤÷ì♥❣ ❜➟❝

✹ ❜➜t ❦❤↔ q tr Z ữ q p ợ ồ p ♥❣✉②➯♥ tè✳ ❇↕♥
✤å❝ ❝â t❤➸ t❤❛♠ ❦❤↔♦ ❬✶❪ ❝❤♦ ♥❣❤✐➯♥ ❝ù✉ ✤➛② ✤õ ❤ì♥ ✈➲ ❝❤õ ✤➲ ♥➔②✳
✣➸ →♣ ử 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❝❤ ❜➟❝ ❤❛✐ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ✣à♥❤ ỵ trr ỳ
s ữủ tr ữỡ s

ữỡ tỹ ừ ỵ trr tự tỹ
ỵ f (x) tự ✭♠♦♥✐❝✮ ❤➺ 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 ) > 0.
❈❤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)

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✷✵

❧➔ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ ✷✱ ✈➔ fm+1 (x), . . . , fm+n ❧➔ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ ✶✳ ❚❛ ❝â

D(fi ) < 0
D(fi ) = 1

✈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 ) > 0 ⇔ 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 ) > 0 t❤➻ f ❝â d − 4k ♥❣❤✐➺♠ t❤ü❝✱ ✈ỵ✐ k ≥ 0 ♥➔♦ ✤â❀
✭❜✮ ◆➳✉ D(f ) < 0 t❤➻ f ❝â d − 2 − 4k ♥❣❤✐➺♠ t❤ü❝✱ ✈ỵ✐ k ≥ 0 ♥➔♦ ✤â✳
❈❤ù♥❣ ♠✐♥❤✳ ●å✐ m ❧➔ sè ❝➦♣ ♥❣❤✐➺♠ ♣❤ù❝ ✭❦❤æ♥❣ t❤ü❝✮ ❝õ❛ f ✈➔ ❣å✐ n
❧➔ sè ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ f õ t ỵ tr

D(f ) > 0 ⇔ m ≡ 0

(mod 2).

●✐↔ sû D(f ) > 0✳ ❑❤✐ ✤â m ❧➔ sè ❝❤➤♥✳ ❱✐➳t m = 2k ✈ỵ✐ k ≥ 0 ♥➔♦ ✤â✳
❑❤✐ ✤â✱ sè ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ f ❧➔ d − 2m = d − 4k ✳
●✐↔ sû D(f ) < 0✳ ❑❤✐ ✤â m ❧➔ sè ❧➫✳ ❱✐➳t m = 2k + 1 ✈ỵ✐ k ≥ 0 ♥➔♦
✤â✳ ❑❤✐ ✤â✱ sè ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ f ❧➔ d − 2m = d − 2 − 4k ✳

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ữỡ ỵ
trr t
t
ữỡ tr ỵ r t t ❤❛✐
✈➔ ♠ët ❝❤ù♥❣ ♠✐♥❤ ❧✉➟t ♥➔② ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ỵ trr
t sỷ ử ữỡ ♥➔② ❧➔ t➔✐ ❧✐➺✉ ❬✸✱ ❈❤❛♣t❡r ✶✻❪ ✈➔ ❬✷✱
❙❡❝t✐♦♥ ✻✳✼❪✳

✸✳✶ ỵ r

p ởt số ♥❣✉②➯♥ tè ❧➫✱ ✈➔ a ❧➔ ♠ët sè ♥❣✉②➯♥
a

✤÷đ❝ ✤à♥❤ ữ
ổ t p õ ỵ ▲❡❣❡♥❞r❡
p
s❛✉✳
a
1

♥➳✉ 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

= 1.

✷✳

ab
p

=

✸✳

a
p

a
p

b
✳
p

p−1
≡ a 2 (mod p) ✭❚✐➯✉ ❝❤✉➞♥ ❊✉❧❡r✮✳

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✷✷

a
p

✹✳ ◆➳✉ a ≡ b ✭♠♦❞ p✮ t❤➻
✺✳

−1
p

❜➡♥❣ 1 ❤♦➦❝ −1 tò② t❤❡♦ p ≡ 1 ✭♠♦❞ 4✮ ❤❛② p ≡ 3 ✭♠♦❞ 4✮✳

✻✳ ❑❤✐ ✤â

2
p

b

✳
p

=

2
p

= 1 ✈➔ ♥➳✉ p ≡ 1 ✭♠♦❞ 8✮ ❤♦➦❝ p ≡ 7 ✭♠♦❞ 8✮❀ ✈➔

= −1 ♥➳✉ p ≡ 3 ✭♠♦❞ 8✮ ❤♦➦❝ p 5 8

ử ỵ r
ớ õ

45
37

=

8
37

45
.
37
2
4
=
37

37

=

2
37

= 1.

ỵ t t ❤❛✐ ●❛✉ss✮✳ ●✐↔ sû p ✈➔ q ❧➔ ❝→❝
sè ♥❣✉②➯♥ tè ❧➫ ♣❤➙♥ ❜✐➺t✳ ❑❤✐ ✤â
t❤➻

p
q

=−

p
q

=

q
p

trø ❦❤✐ p ≡ q 3 4

q

p

ử ỵ r
ớ
1234
199

=

1234

199

40
199
2
199

5
4
199
199
199
4
= (1)
= (1)
5
5
= 1.
=

ỵ trr t t ♥❣❤à❝❤ ❜➟❝ ❤❛✐
❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ❧✉➟t tt
ử ỵ trr ❈❤♦ p ✈➔ q ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ❧➫ ♣❤➙♥ ❜✐➺t✳
●å✐ e ❧➔ ❝➜♣ ❝õ❛ q mod p✱ tù❝ ❧➔ e ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥❤ä ♥❤➜t s❛♦ ❝❤♦

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