(Luận văn thạc sĩ) 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♥❣ ♣❤→♣ ❚♦→♥ sì ❝➜♣
▼➣ 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❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
❚ü ✤ç♥❣ ❝➜✉ ❋r♦❜❡♥✐✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

❈❤÷ì♥❣ ✷✳ ✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r
✷✳✶
✷✳✷
✷✳✸
✷✳✹

✶
✸

◆❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr♦♥❣ Fp [x] ✳ ✳ ✳ ✳
✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✣❛ t❤ù❝ ♥❣✉②➯♥ ❦❤↔ q✉② ♠♦❞✉❧♦ ♠å✐ sè p ♥❣✉②➯♥ tè
❚÷ì♥❣ tü ❝õ❛ ✤à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r ❝❤♦ ✤❛ t❤ù❝ t❤ü❝

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✶✷

✶✷
✶✹
✶✼
✶✾

❈❤÷ì♥❣ ✸✳ ✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r ✈➔ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ✷✶
✸✳✶
✸✳✷
✸✳✸

❑þ ❤✐➺✉ ▲❡❣❡♥❞r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r ✈➔ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✷✷
✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r ♠♦❞✉❧♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✷
✸✸





f (x) Z[x] ởt tự số n
ổ õ ự ồ D(f ) t tự ừ f p
ởt số tố ồ Fp = Z/pZ trữớ ỳ õ p tỷ
ồ f(x) Fp [x] tự ữủ tứ f t ồ số
p ồ r số tỷ t q ừ f õ ởt ỵ ừ
trr r r n õ ũ t tự r n
(mod 2) D(f ) ữỡ p
ử t ừ t ự ừ ỵ t
rr ụ ữ ự ử ừ õ tr ự t t

t t ố ử ừ

ữủ ữỡ

ữỡ ởt số tự

ữỡ tr ởt số tự t tự ừ tự
t tự ừ tự ỗ rs

ữỡ ỵ trr

ữỡ tr ỵ trr ởt số ử ồ
ởt tữỡ tỹ ừ ỵ tự tỹ

ữỡ ỵ trr t t

ữỡ tr ỵ r t t
ởt ự ừ t sỷ ử ỵ trr
ữủ tỹ t t t
trữớ ồ ồ ồ t
tọ ỏ t ỡ s s tợ ữớ t t ữợ
tr sốt q tr t
ỷ ớ ỡ t rữớ ồ
ồ ồ t ồ ú t ồ t
t ụ ữ ữỡ tr t s ụ
ỷ ớ ỡ tợ t t ợ ồ õ
ở ú ù t tr q tr ồ t t


✷
♥➔②✳ ✣ç♥❣ t❤í✐ t→❝ ❣✐↔ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tî✐ ❇❛♥ ❣✐→♠ ❤✐➺✉ ✈➔ ❝→❝ ✤ç♥❣
♥❣❤✐➺♣ t↕✐ tr÷í♥❣ ❚❍❈❙ ❍÷♥❣ ✣↕♦✱ ✣æ♥❣ ❚r✐➲✉✱ ◗✉↔♥❣ ◆✐♥❤ ✤➣ t↕♦ ✤✐➲✉

❦✐➺♥ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳
❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳
❳→❝ ♥❤➟♥ ❝õ❛ ♥❣÷í✐ ❤÷î♥❣ ❞➝♥

❚❙✳ ◆❣✉②➵♥ ❉✉② ❚➙♥

❚❤→✐ ◆❣✉②➯♥✱ 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❝ ✈➔ ✤ç♥❣ ❝➜✉ ❋r♦❜❡♥✐✉s✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ sû ❞ö♥❣
❝❤♦ ❝❤÷ì♥❣ ♥➔② ❧➔ t➔✐ ❧✐➺✉ ❬✷✱ ❙❡❝t✐♦♥ ✻✳✻❪ ✈➔ ❬✸✱ ❈❤❛♣t❡r ✶✺❪✳

✶✳✶ ❑➳t t❤ù❝ ❝õ❛ ❤❛✐ ✤❛ t❤ù❝
●✐↔ sû f, g ❧➔ ❤❛✐ ✤❛ t❤ù❝ ❜✐➳♥ x ✈î✐ ❝→❝ ❤➺ 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 ), ✈î✐ 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 ), ✈î✐ 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 ❦➯ ❞÷î✐ ✤➙② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦➳t t❤ù❝✳

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



ự õ
m

n

n

m n

m

m n

(i j ) = (1)mn R(f, g).

(j i ) = a b

R(g, f ) = a b

j=1 i=1

i=1 j=1

õ ự

t R(f, g) = 0 f

g õ ởt tỷ

ữỡ

ự f g õ ởt tỷ h(x) F [x] õ ồ
K ởt ừ h tr K ữ tỗ t i j s i =
j = s r tr t R(f, g) õ tỷ i j = 0
R(f, g) = 0

t R(f, g) = a

n

m

m

f (j )

mn n

g(i ) = (1)
i=1

b

j=1

ự g(x) = b nj=1 (x i ) t õ g(i ) = b
ợ ồ i = 1, . . . , n
n

a

m

n

n

m n

(i j ) = R(f, g).

g(i ) = a b
i=1

n
j=1 (i j )

i=1 j=1

ữỡ tỹ sỷ ử t t s r
m

R(f, g) = (1)

mn n

b

f (j ).
j=1

t g(x) = f q + r t R(f, g) = amdeg r R(f, r)
ự ứ t t õ
n

R(f, g) = a

deg g

n

g(i ) = a
i

deg g

[f (i )q(i ) + r(i )].
i=1



i ừ ừ f f (i ) = 0 f (i )q(i ) + r(i ) =
r() õ t õ
n

R(f, g) = a

deg g

r(i ).
i=1

t ụ t t R(f, r) = adeg r

n
i=1 r(i )



n

R(f, g) = a

deg g

r(i ) = adeg gdeg 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

0

g(i ) = bn .
i=1

t t t t t tự ừ t
tự tt t ừ t t
ụ t ự ữủ r t tự R(f, g) ởt
tỷ ừ trữớ F ũ õ ữủ ỹ t tỷ tr
trữớ ợ ỡ K

t õ R(f, g) tr F

ự ự q t deg f g = b
số tở F t t R(f, g) = R(b, f ) =
R(f, b) = bn tở F.
sỷ ú ợ ồ ồ tự f g ợ f õ ọ

ỡ n 1 t f g tự tũ ỵ ợ deg f = n 1
õ t tt t tự tỗ t tự q r tr F [x]
s
g = f q + r,
ợ r = 0 deg r < deg f = n t t
t tt q t õ R(f, g) = R(f, r) = R(r, f ) tở F
õ ự


✻

❚➼♥❤ ❝❤➜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❝ ✈î✐ ❤➺ sè tr♦♥❣ tr÷í♥❣ F ✈➔ K ❧➔ ♠ët
tr÷í♥❣ ✤â♥❣ ✤↕✐ sè ❝❤ù❛ F ✳
❇✐➺t t❤ù❝ ❝õ❛ f ✤÷ñ❝ ✤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 ❦❤æ♥❣ ❝â
t❤ø❛ sè ❝❤✉♥❣✳
❈❤ó♥❣ t❛ ❝â t❤➸ t➼♥❤ t♦→♥ D(f ) ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ t❤✉➟t t♦→♥ ❊✉❝❧✐❞
tr➯♥ f ✈➔ f ✳ ❉÷î✐ ✤➙② ❧➔ ♠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 = 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.


✼

❱➼ ❞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❤❡♦ ✤à♥❤ ❧þ ❱✐➨t❡ α1 · · · αn = (−1)n ✳
✣❛ t❤ù❝ f (x) ∈ F [x] ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ✤❛ t❤ù❝ ❝❤✉➞♥ ✭♠♦♥✐❝✮ ♥➳✉ ❤➺ sè
ù♥❣ ✈î✐ sè ♠ô ❝❛♦ ♥❤➜t ❝õ❛ ♥â ❜➡♥❣ ✶✳

▼➺♥❤ ✤➲ ✶✳✷✳✺✳ ❈❤♦ f ❧➔ ♠ët ✤❛ t❤ù❝ ♠♦♥✐❝ ✈➔ α1, . . . , αn ❧➔ ❝→❝ ♥❣❤✐➺♠



ừ õ tr trữớ õ số K õ

2

n j1

(i j )2 .

(i j ) =

D(f ) =
j=2 i=1

1i
ự 1 , . . . , n K ừ f f (x) = (x
1 )(x 2 ) . . . (x n ) õ

f (x) = (x 2 )(x 3 ) . . . (x n ) + (x 1 )(x 3 ) . . . (x n )
+ ã ã ã + (x 1 ) . . . (x n1 )
n

n

(x j ).

=

i=1 j=1,j=i

ữ f (i ) =

n
j=1,j=i (i

j ) t t õ

n
n(n1)/2

D(f ) = (1)

n

f (i ) = (1)
i=1

n

n(n1)/2

(i j ).
i=1 j=1,j=i

ú ỵ r tr t ố ũ ổ tự tr õ n(n 1) tứ số
tr õ ởt ỷ ỗ tứ số i j ợ i < j ởt ỷ
tứ số i j ợ i > j ộ tứ số tự ợ
(1) t õ ự

ử t t tự ừ tự sỷ

ử ổ tự t t tự tr trữợ
t f (x) = x2 + ax + b F [x] ồ 1 , 2 ừ f
tr ởt trữớ õ số õ ự F õ t tự ừ f

D(f ) = (1 2 )2 = (1 + 2 )2 41 2 = a2 4b.
t tự f (x) = x3 +qx+r F [x] ồ 1 , 2 , 3
ừ f õ t tự ừ f

D(f ) = (2 1 )2 (3 1 )2 (3 2 )2 .
õ

x3 + qx + r = (x 1 )(x 2 )(x 3 ).


✾
▲➜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❛ ✤÷ñ❝

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 )




= R(f, f g)R(g, f g )
= R(f, f )R(f, g)R(g, f )R(g, g )
= (1)n(n1)/2 D(f )R(f, g)(1)mn R(f, g)(1)m(m1)/2 D(g)
= (1)n(n1)/2+mn+m(m1)/2 D(f )D(g)(R(f, g))2 .
ứ õ t õ ự

(m + n)(m + n 1) n(n 1)

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

f1, . . . , fr tự tr F [x] õ
D(f1 ã ã ã fr ) = D(f1 ) ã ã ã D(fr )R2 ,
R =

1i
R(fi , fj ) F

ự ự q t r ổ tự ú ợ r = 2
sỷ õ ú ợ r 1 tự ợ r 3 t
t q t õ

D(f1 ã ã ã fr ) = D(f1 ã ã ã fr1 )D(fr )R(f1 ã ã ã fr1 , fr )2
r1
2

= D(f1 ) ã ã ã D(fr1 )

R(fi , fr )2

R(fi , fj ) D(fr )
1i

i=1

R(fi , fj )2 .

= D(f1 ) ã ã ã D(fr )
1i
ỹ ỗ rs
t p ởt số tố t Fp = Z/pZ = {[0], [1], . . . , [p 1]}
trữớ õ p tỷ số p ồ K ởt trữớ t
ý ự Fp t p : K K p (a) = ap ợ ồ a K
ó r

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

ỡ ỳ
p1
p

p

p (a + b) = (a + b) = a +
k=1

p pk k
a b + bp = ap + bp = p (a) + p (b).
k

é t sỷ ử t r p |

p
k

ữ p tỹ ỗ

ừ trữớ K p ữ tr ữủ ồ tỹ ỗ rs ừ
K

ờ a ởt tỷ tr K õ p(a) = a
a tở Fp

ự a tở Fp t t ỵ rt ọ t õ p (a) =
ap = a
ữủ sỷ p (a) = a s r a K ừ tự
p
x x ỵ rt ọ p tỷ tr Fp ừ
tự xp x xp x õ p p tỷ ừ Fp tt
ừ xp x a ởt tr a tở
Fp
rở p tứ K[x] K[x] ữ s ợ
f (x) = an xn + ã ã ã + a1 x + a0 t

p (f (x)) = p (an )xn + ã ã ã + p (a1 )x + p (a0 )
= apn xn + ã ã ã + ap1 x + ap0 .

ờ f (x) K[x] õ p(f (x)) = f (x)
f (x) Fp [x]

ự r tứ ờ trữợ




ữỡ ỵ
trr
ữỡ tr ỵ trr ởt số ử ồ
ởt tữỡ tỹ ừ ỵ tự tỹ t sỷ
ử ữỡ tr t
f (x) Z[x] ởt tự số n
ồ D(f ) t tự ừ f p ởt số tố sỷ
p D(f ) ồ f(x) Fp [x] tự ữủ tứ f t
ồ số p ồ r số tỷ t q ừ f õ
ỵ trr r r n õ ũ t tự
r n (mod 2) D(f ) ữỡ p ỵ
tr ởt t q ừ trr õ ụ ữủ ự
P ố ợ p = 2 ừ ỵ trr
s ữủ tr ữỡ ử ụ tr ởt t q
ừ trr õ ụ ữủ ự rt
r

ừ tự t q tr F [x]
p

ờ p sỷ K ởt ừ tự f (x) Fp[x]
õ ụ ừ f (x)

ự t f (x) = an xn + an1 xn1 + ã ã ã + a1 x + a0 Fp

õ 0 = f () = ni=0 ai i ỷ ử ỗ rs p t õ
n

0 = p (0) = p

n

ai
i=0

i

n
i

=

ai pi = f (p ).

p (ai )p ( ) =
i=0

i=0

é tr t sỷ ử t t p (a) = a ợ ồ a Fp




f (x) ởt tự t q tr Fp[x]

ợ d K ởt trữớ ự Fp K ởt ừ
f (x) õ tr K[x] t õ
2

f (x) = (x )(x p )(x p ) . . . (x p

d1

).

2

ự ờ trữợ t õ , p , p , . . . ừ
f (x) tự f (x) õ ỳ tr K tỗ t số
k
l
l
k
ổ t k < l s p = p õ 0 = p p =
lk
k
lk
(p )p r p = ữ tỗ t số tỹ ọ t r
r
s p =
2
r1
k
h
õ , p , p . . . , p t ổ sỷ ap = p

hk
ợ h, k, 1 k < h < r õ t ữ trữợ p
=
t ợ t ọ t ừ r f õ t t r t
tr K õ r ọ ỡ ừ f tự r d
t
r1
g(x) = (x )(x p ) ã ã ã (x p ).
r1

(p

)p = ,
r1

p (g(x)) = p (x )p (x p ) ã ã ã p (x p
2

)

r

= (x p )(x p ) ã ã ã (x p )
= g(x).
g(x) tr Fp [x] ởt ừ g(x)
tự f (x) g(x) t õ

f (x) = g(x)q(x) + h(x),
ợ q(x), h(x) Fp [x] h = 0 deg h < r ợ ồ i = 0, . . . , r 1
t õ

i

i

i

i

h(p ) = f (p ) g(p )q(p ) = 0.

r1

ữ tự h õ t t r t , p , . . . , p õ
h tự 0 f (x) = g(x)q(x) f (x) t q
f (x) = g(x) r = d õ ự


✶✹

✷✳✷ ✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r

✣à♥❤ ❧þ ✷✳✷✳✶✳ ❈❤♦ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè ❧➫✱ f (x) ❧➔ ♠ët ✤❛ t❤ù❝ ♠♦♥✐❝

❜➟❝ m ✈î✐ ❝→❝ ❤➺ 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✐➯♥ ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ ❝❤♦ tr÷í♥❣ ❤ñ♣ r = 1✳
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔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 , . . . , αp
t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✳✷✳ ❈❤ó þ r➡♥❣ t❛
pm
❝ô♥❣ ❝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 ).



ữ

D(f ) ữỡ tr Fp (f ) Fp p ((f )) = (f )
(1)m1 (f ) = (f ) (1)m1 = 1.
m 1 (mod 2).
ữ ỵ ú ợ r = 1
ớ t sỷ f = f1 f2 . . . fr t ừ r tỷ t q
t (f ) (fi ) tữỡ tỹ ữ tr trữớ ủ r = 1

tr
t õ

D(f ) = D(f1 f2 . . . fr ) = D(f1 )D(f2 ) . . . D(fr )R2 ,
ợ R õ tở Fp õ tr K t õ

(f ) = (f1 )(f2 ) . . . (fr )S,
ợ S = R Fp trữớ ủ r = 1 ợ ồ i = 1, . . . , r t õ
p ((fi )) = (1)di 1 (fi ) ỡ di = deg fi õ

((f )) = (f1 )(f2 ) . . . (fr )S(1)d1 1 (1)d2 1 . . . (1)dr 1
= (f )(1)d1 +d2 +ããã+dr r
= (1)mr (f ).
ú ỵ r d1 + ã ã ã + dr = deg f = m ứ õ t õ

D(f ) ữỡ tr Fp (f ) Fp p ((f )) = (f )
(1)mr (f ) = (f ) (1)mr = 1
m r (mod 2).
õ ự
ự ỵ trr ữ r tr ự ợ
sỷ ờ t ủ ừ r

ử t f (x) = x2 + x + 1 F2[x] õ d = 2 f (0) =
f (1) = 1 = 0 tr F2 f (x) ổ õ tr F2
f (x) F2 [x] t q r trữớ ủ số tỷ t
q ừ f (x) r = 1


✶✻
▼➦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♥❣ ❤ñ♣ ♥➔②✳ ◆❤÷ ✈➟② ♣❤→t ❜✐➸✉ ❝õ❛ ✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r
❦❤æ♥❣ ❝á♥ ✤ó♥❣ ♥ú❛ ❝❤♦ tr÷í♥❣ ❤ñ♣ 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 ❦❤æ♥❣ ❧➔ ❜➻♥❤ ♣❤÷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✳
✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r ♥â✐ 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 ❦❤æ♥❣ ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦ 5✳ ❚❤❡♦ ❤➺ q✉↔
tr➯♥ t❤➻ Np (f ) = 1✳ ❚❤ü❝ t➳ x = 2 ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ x3 − x − 1


✶✼
♠♦❞✉❧♦ 5 ✈➔ ♣❤➙♥ t❛ ❝â ♣❤➙♥ t➼❝❤

(x3 − x − 1) = (x − 2)(x2 + 2x + 3)

(mod 5).

• ◆➳✉ p = 7 t❤➻ D = −23 = 5 (mod 7) ❦❤æ♥❣ ❧➔ ❜➻♥❤ ♣❤÷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) ❧➔ ❦❤æ♥❣ ❜➻♥❤ ♣❤÷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✉↔ trü❝ t✐➳♣ ❝õ❛ ✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r✳

❍➺ q✉↔ ✷✳✸✳✶✳ ❈❤♦ f (x) ❧➔ ♠ët ✤❛ t❤ù❝ ♠♦♥✐❝ ❜➟❝ ❝❤➤♥ ❤➺ sè ♥❣✉②➯♥✳
●✐↔ sû ❜✐➺t t❤ù❝ D ❝õ❛ f ❧➔ ♠ët sè ❝❤➼♥❤ ♣❤÷ì♥❣ ❦❤→❝ ✵✳ ❑❤✐ ✤â ✈î✐ ♠å✐
p ♥❣✉②➯♥ tè ❧➫ ✈➔ ❦❤æ♥❣ ❧➔ ÷î❝ ❝õ❛ 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✳ ❉♦ ✈➟② t❤❡♦ ✤à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r✱
r ≡ deg f mod 2✳ ❉♦ ✈➟② r ❧➔ sè ❝❤➤♥✳ ◆â✐ r✐➯♥❣ r = 1 ✈➔ f (x) ❧➔ ❦❤↔
q✉② ♠♦❞✉❧♦ p✳


✶✽

❇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 ✈î✐ ♠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 ✈î✐ ♠å✐ p ♥❣✉②➯♥
tè ♠➔ p = 2✳ ❚❛ ❝â x4 + 1 = (x + 1)4 (mod 2)✳ ◆❤÷ ✈➟② x4 + 1 ❧➔ ❦❤↔ q✉②
♠♦❞✉❧♦ p ✈î✐ ♠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),
✈î✐ 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 ♥❤÷♥❣ ❧➔ ❦❤↔
q✉② ♠♦❞✉❧♦ p ✈î✐ ♠å✐ sè ♥❣✉②➯♥ 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



ợ ồ p tố p = 2 p = 5 õ x4 + 3x2 + 1 = (x + 1)4
(mod 2)

x4 + 3x2 + 1 = (x 1)2 (x 1)2

(mod 5).

ữ x4 + 1 q p ợ ồ p tố
ự x4 + 3x2 + 1 t q tr Z t sỷ
x4 + 3x2 + 1 q tr Z õ

(x4 + 1) = (x2 + ax + c)(x2 + bx + d),
ợ a, b, c, d Z õ õ

(x2 +ax+c)(x2 +bx+d) = x4 +(a+b)x3 +(ab+c+d)x2 +(ad+bc)x+cd.
a + b = 0 ab + c + d = 3 ad + bc = 0 cd = 1 ứ cd = 1 t s
r c = d = 1 c = d = 1 r ab = 3 (c + d) = 1
a2 = 1 5 ữỡ tr ổ õ

t tr ử tự trũ ữỡ

t q tr Z ữ q p ợ ồ p tố
ồ õ t t ự ừ ỡ ừ
ử ỵ trr ú t õ
tr D(f ) ởt ữỡ mod p ổ ú t õ ởt

ữỡ ỳ sỷ ử t t
ởt tú r t õ t ự t t
sỷ ử ỵ trr ỳ
s ữủ tr ữỡ s

ữỡ tỹ ừ ỵ trr tự tỹ
ỵ f (x) tự số tỹ ợ d

t tự D(f ) = 0 ồ r số tỷ t q tỹ ừ f
õ
d r (mod 2) D(f ) > 0.
ự sỷ f (x) = f1 (x) ã ã ã fm (x)fm+1 (x) ã ã ã fn (x) t
ừ f t t tự t q tỹ tr õ f1 (x), . . . , fm (x)


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

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

✈î✐ ♠å✐ i = 1, . . . m, ✈➔
✈î✐ ♠å✐ i = m + 1, . . . m + n.

❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✽

D(f ) = D(f1 · · · fm fm+1 · · · fm+n ) = D(f1 ) · · · D(fm )a2 ,
✈î✐ 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❝ ✈î✐ ❜➟❝ d
✈➔ ❜✐➺t t❤ù❝ D(f ) = 0✳ ❑❤✐ ✤â

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

D(f ) > 0 ⇔ m ≡ 0

(mod 2).

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


✷✶

❈❤÷ì♥❣ ✸✳ ✣à♥❤ ❧þ
❙t✐❝❦❡❧❜❡r❣❡r ✈➔ ❧✉➟t
t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✈➲ ❦þ ❤✐➺✉ ▲❡❣❡♥❞r❡✱ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐
✈➔ ♠ët ❝❤ù♥❣ ♠✐♥❤ ❧✉➟t ♥➔② ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r✳ ❚➔✐

❧✐➺✉ t❤❛♠ ❦❤↔♦ sû ❞ö♥❣ ❝❤♦ ❝❤÷ì♥❣ ♥➔② ❧➔ t➔✐ ❧✐➺✉ ❬✸✱ ❈❤❛♣t❡r ✶✻❪ ✈➔ ❬✷✱
❙❡❝t✐♦♥ ✻✳✼❪✳

✸✳✶ ❑þ ❤✐➺✉ ▲❡❣❡♥❞r❡

✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✶✳ ❈❤♦ p ❧➔ ♠ët sè ♥❣✉②➯♥ tè ❧➫✱ ✈➔ a ❧➔ ♠ët sè ♥❣✉②➯♥
a

✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷
❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ p✳ ❑❤✐ ✤â ❦þ ❤✐➺✉ ▲❡❣❡♥❞r❡
p
s❛✉✳
a
1
♥➳✉ a ❧➔ ❜➻♥❤ ♣❤÷ì♥❣ ♠♦❞✉❧♦ p
=
p
−1 ♥➳✉ a ❦❤æ♥❣ ❧➔ ❜➻♥❤ ♣❤÷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✮✳


✷✷

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✮✳

❱➼ ❞ö ✸✳✶✳✷✳ ❚➼♥❤ ❦þ ❤✐➺✉ ▲❡❣❡♥❞r❡
▲í✐ ❣✐↔✐✳ ❚❛ ❝â

45
37

=

8
37

45
.
37
2
4
=
37
37

=

2
37

= −1.

✣à♥❤ ❧þ ✸✳✶✳✸✳ ✭▲✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐ ●❛✉ss✮✳ ●✐↔ sû p ✈➔ q ❧➔ ❝→❝
sè ♥❣✉②➯♥ tè ❧➫ ♣❤➙♥ ❜✐➺t✳ ❑❤✐ ✤â
t❤➻

p
q

=−

p
q

=

q
p

trø ❦❤✐ p ≡ q ≡ 3 ✭♠♦❞ 4✮

q
✳
p

❱➼ ❞ö ✸✳✶✳✹✳ ❚➼♥❤ ❧þ ❤✐➺✉ ▲❡❣❡♥❞r❡
▲í✐ ❣✐↔✐✳
1234
199

=

1234
✳
199

40
199
2
199

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

✸✳✷ ✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r ✈➔ ❧✉➟t t❤✉➟♥ ♥❣❤à❝❤ ❜➟❝ ❤❛✐
❚r♦♥❣ ♠ö❝ ♥➔② ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ❧✉➟t t❤✉➟t ♥❣❤à❝❤ ❜➟❝ ✷ ❜➡♥❣ ❝→❝❤
→♣ ❞ö♥❣ ✣à♥❤ ❧þ ❙t✐❝❦❡❧❜❡r❣❡r✳ ❈❤♦ p ✈➔ q ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ❧➫ ♣❤➙♥ ❜✐➺t✳
●å✐ e ❧➔ ❝➜♣ ❝õ❛ q mod p✱ tù❝ ❧➔ e ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥❤ä ♥❤➜t s❛♦ ❝❤♦