Luận văn thạc sĩ Gía trị nguyên tố của đa thức bất khả quy

ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC






NGUYỄN THỊ LINH






GIÁ TRỊ NGUYÊN TỐ CỦA ĐA THỨC BẤT KHẢ QUY











LUẬN VĂN THẠC SĨ TOÁN HỌC

Thái Nguyên - 2014



ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC






NGUYỄN THỊ LINH






GIÁ TRỊ NGUYÊN TỐ CỦA ĐA THỨC BẤT KHẢ QUY

Chuyên ngành: PHƯƠNG PHÁP TOÁN SƠ CẤP
Mã số: 60.46.01.13



LUẬN VĂN THẠC SĨ TOÁN HỌC




: PGS.TS. Lê Thị Thanh Nhàn









Thái Nguyên - 2014

Q
R C
C R
f(x) ∈ Z[x]

Q {f(n) | n ∈ N, n > 0} f(x)
1 f(n) n ∈ N
f(x) 1
Q
Q
Q
C R
Q
F
f(x) ∈ F [x]
f(x) > 0 f(x)
f(x) > 0 f(x)
f(x)
f(x) = x
2
+ 1 ∈ R[x] R.
f(x) = x
2
−1 ∈ R[x] R x
2
−1 = (x−1)(x+1)
f(x) ∈ F [x]
f(x) 1 f(x)
f(x) 1 F f(x)
2 3
F
f(x) a ∈ F f(x)
f(x + a)
f(x) > 1 f(x) x = a ∈ F f(x) = (x − a)g(x)
deg g(x) = deg f(x) − 1 ≥ 1. f(x)

f(x) 2 3 f
1
f(x) F f(x) F f(x)
h(x) ∈ F[x] h
1
(x) = h(x − a) deg h
1
(x) =
deg h(x) f(x + a) = k(x)g(x) f(x + a)
f(x) = k
1
(x)g
1
(x)
f(x) f(x)
f(x + a)
x
3
− 2 Q 3
Q R
x
3
− 2 = (x −
3
√
2)(x
2
+
3
√

2x +
3
√
4) R[x].
K F a ∈ K. a
F 0 = f(x) ∈ F[x] a
a F a F α
α Q α Q
α
3
√
2 x
3
− 2 ∈ Q[x]
π
f(x) ∈ F [x]
f(x) 1 0 = f(x) ∈ F [x]
a
n
f
∗
(x) = a
−1
n
f(x)
K F a ∈ K
F p(x) ∈ F [x]
a g(x) ∈ F [x] a
p(x).
a 0 F

0 F a
p(x) ∈ F[x] a
p(x) p(x) p(x)
p(x) F [x]
a
p(x) g(x) ∈ F [x] a
p(x) g(x) p(x) gcd(g(x), p(x)) = 1
1 = p(x)q(x) + g(x)h(x) q(x), h(x) ∈ F[x] x = a
1 = 0 g(x) p(x)
q(x) ∈ F [x] a
q(x) p(x) q(x) = p(x)k(x) q(x)
k(x) = b ∈ F q(x) = bp(x)
q(x) p(x)
b = 1. p(x) = q(x).
p(x) ∈ F [x] a
a
x
4
− 10x
2
+ 1 ∈ Q[x]
√
2+
√
3 ∈ R x
4
−10x
2
+1
f(x) g(x) f(x) g(x) f(x)

g(x) f g g
x
4
−10x
2
+ 1 f
g
f g f(x) = x
2
+ ax + b g(x) = x
2
+ cx + d
a, b, c, d x
4
− 10x
2
+ 1 = f(x)g(x)
a = −c, ac + b + d = −10, ad + bc = 0, bd = 1.
a = −c ad + bc = 0 −cd + bc = 0
c(b − d) = 0 c = 0 b = d
c = 0 c ac+ b+ d = −10 b +d = −10
b = −10 − d b bd = 1
d
2
+ 10d + 1 = 0 d = −5 ± 2
√
6 /∈ Q
b = d bd = 1 b = d = 1
b = d = −1 b = d = 1 ac + b+ d = −10
c = ±

√
12 /∈ Q b = d = −1 ac + b + d = −10
c = ±
√
8 /∈ Q
a, b, c, d
x
4
− 10x
2
+ 1
√
2 +
√
3
x
2
− 2x + 2 ∈ R[x] 1 + i ∈ C.
x
2
−2x + 2
1 + i
1 + i
p > 1 p|ab p|a p|b
a, b
p(x) ∈ F [x] p(x)
F p(x)|a(x)b(x) p(x)|a(x)
p(x)|b(x) a(x), b(x) ∈ F [x].
p(x) p(x)|a(x)b(x)
p(x) a(x) b(x)

p(x) gcd(p(x), a(x)) = 1
s(x), r(x) ∈ F[x] 1 = s(x)p(x) + r(x)a(x)
e(x), f(x) ∈ F [x] 1 = e(x)p(x) + f(x)b(x)
1 = p(x)g(x) + r(x)f(x)a(x)b(x)
g(x) ∈ F[x] p(x)
1 p(x)
p(x)|a(x)b(x) p(x)|a(x) p(x)|b(x)
a(x), b(x) ∈ F[x]. deg p(x) > 0
p(x) p(x) = g(x)h(x) deg g(x), deg h(x) < deg p(x).
p(x)|p(x), p(x)|g(x)h(x). p(x)|g(x)
p(x)|h(x) p(x)
f(x) ∈ F[x]
d > 0 d = 1 f(x) f(x)
f(x) = f(x) d = 1 d > 1
d f(x) f(x)
f(x) = f(x) f(x)
f(x) = g(x)h(x) deg g(x), deg h(x) < deg f(x)
g
∗
(x) = a
−1
k
g(x) g(x) a
k
g(x) f(x) = g
∗
(x)(a
k
h(x))
1 = a

k
b
t
, b
t
h(x)
h
∗
(x) = a
k
h(x) f(x) = g
∗
(x)h
∗
(x) g
∗
(x), h
∗
(x)
d g
∗
(x) h
∗
(x)
f(x)
f(x)
f(x) = p
1
(x)p
2

(x) . . . p
n
(x) = q
1
(x)q
2
(x) . . . q
m
(x).
n n = m
p
i
(x) = q
i
(x) i = 1, . . . , n n = 1
p
1
(x) = q
1
(x)q
2
(x) . . . q
m
(x) p
1
(x)|q
1
(x)q
2
(x) . . . q

m
(x) p
1
(x)
p
1
(x) q
i
(x)
p
1
(x)|q
1
(x) q
1
(x) = p
1
(x)t
1
(x)
q
1
(x) t
1
(x) = a ∈ F
q
1
(x) = ap
1
(x) p

1
(x) q
1
(x)
1 = 1.a a = 1 p
1
(x) = q
1
(x) m > 1
1 = q
2
(x) . . . q
m
(x) n = 1
n > 1 p
1
(x)|q
1
(x)q
2
(x) . . . q
m
(x) p
1
(x)
p
1
(x)|q
1
(x) q

1
(x)
p
1
(x), q
1
(x)
p
1
(x) = q
1
(x) p
1
(x)
p
2
(x)p
3
(x) . . . p
n
(x) = q
2
(x)q
3
(x) . . . q
m
(x).
n − 1 = m − 1
q
i

(x) p
i
(x) = q
i
(x) i = 2, . . . , n.
Q
Q
f(x) ∈ Q[x] 1
Q
f(x) = a
n
x
n
+ . . . + a
1
x + a
0
∈ Z[x] r/s
r/s f(x) r a
0
s
a
n
. a
n
= 1 f(x)
a
0
.
r/s f(x) r −ms f(m) m ∈ Z.

r + s f(−1) r − s f(1).
r
s
∈ Q r, s s > 0
(r, s) = 1
r
s
f(x) f(
r
s
) = 0
0 = f(
r
s
) = a
n
(
r
s
)
n
+ a
n−1
(
r
s
)
n−1
+ + a
1

r
s
+ a
0
.
0 = a
n
r
n
+ a
n−1
r
n−1
s + + a
1
rs
n−1
+ a
0
s
n
a
n
r
n
= −(a
n−1
r
n−1
s + + a

1
rs
n−1
+ a
0
s
n
).
s (r, s) = 1 s a
n
a
0
s
n
= −(a
n
r
n
+ a
n−1
r
n−1
s + + a
1
rs
n−1
)
r (r, s) = 1 r a
0
.

a
n
= 1 r
n
= −(a
n−1
r
n−1
s + + a
1
rs
n−1
+ a
0
s
n
).
s
r
s
∈ Z
a
0
.
f(x) x − m
f(x) = a
n
(x − m)
n
+ b

n−1
(x − m)
n−1
+ + b
1
(x − m) + b
0
.
b
0
, b
1
, , b
n
∈ Z m ∈ Z f(m) = b
0
x =
r
s
f(
r
s
) = 0
0 = f(
r
s
) = a
n
(
r

s
− m)
n
+ b
n−1
(
r
s
− m)
n−1
+ + b
1
(
r
s
− m) + f(m).
0 = a
n
(r − ms)
n
+ b
n−1
(r − ms)
n−1
s + + b
1
(r − ms)s
n−1
+ f(m)s
n

.
f(m)s
n
= −{a
n
(r − ms)
n
+ b
n−1
(r − ms)
n−1
s + + b
1
(r − ms)s
n−1
}.
r −ms f(m)s
n
r −ms
r − ms f(m)
m = 1 r −s f(1), m = −1 r + s
f(−1).
Q f(x) = x
3
+ 2x
2
−
8x + 21 f(x) a
n
= 1

f(x) r ∈ Z f(x)
r|21. r ±1, ±3, ±7, ±21
f(x) f(x)
f(x) 3 Q f(x)
Q.
Q
≥ 4,
Q
(x
2
+ 1)(x
2
+ 1)
Q Q
Z[x].
p(x) ∈ Z[x] p(x) = g(x)f(x)
g(x), f(x) ∈ Q[x] f(x) = g
∗
(x)f
∗
(x) g
∗
(x), h
∗
(x) ∈ Z[x]
deg g(x) = deg g
∗
(x), deg f(x) = deg f
∗
(x) f(x)

Q
f(x) ∈ Z[x]
f(x)
f(x) = g(x)h(x),
g(x) = b
n
x
n
+ + b
1
x + b
0
h(x) = c
k
x
k
+ + c
1
x + c
0
f(x) = a
m
x
m
+ + a
1
x + a
0
∈ Z[x]
f(x) p

f h(x), g(x)
s b
s
p t c
t
p
a
s+t
= b
s+t
c
0
+ b
s+t−1
c
1
+ + b
s
c
t
+ b
s−1
c
t+1
+ + b
0
c
t+s
.
s t b

i
, c
j
p i > s j > t.
a
s+t
p b
s
c
t
p,
b
s
c
t
.
p(x) ∈ Z[x] p(x) = f(x)g(x) f, g ∈ Q[x]
f = af
1
g = bg
1
a, b ∈ Q f
1
, g
1
∈ Z[x]
f
1
g
1

∈ Z[x]
p = abf
1
g
1
∈ Z[x]. ab ∈ Z ab /∈ Z
ab =
r
s
r, s ∈ Z, s > 1 (r, s) = 1. f
1
g
1
= a
n
x
n
+ + a
1
x + a
0
f
1
g
1
(a
n
, a
n−1
, , a

1
, a
0
) = 1 p(x) ∈ Z[x]
ra
n
s
, ,
ra
1
s
,
ra
0
s
∈ Z.
s a
n
, , a
1
, a
0
, ab ∈ Z.
p = (abf
1
)g
1
. f
∗
= abf

1
g
∗
= g
1
p = f
∗
g
∗
p f
∗
g
∗
f f
∗
g g
∗
.
f(x) = x
4
+ 3x
3
+ x
2
+ 3
Q.
Q
f(x) = g(x)h(x) g(x) ∈ Z[x]
h(x) ∈ Z[x] f(x)
g(x) h(x) g(x)

f(x) 3
−3 f(3) = 0 f(−3) = 0 g(x)
h(x) g(x) = x
2
+ ax +b h(x) = x
2
+ cx +d
a, b, c, d ∈ Z f = gh
bd = 3, bc + ad = 0, ac + d + b = 1, c + a = 3.
bd = 3 b, d
b = 1, d = 3 b = −1, d = −3 b = 1, d = 3
c + 3a = 0, ac = −3, a + c = 3 a = −
3
2
/∈ Z
b = −1, d = −3 f(x)
4
f(x) Q.
Q
f(x) = a
n
x
n
+ . . . + a
1
x + a
0
p
p a
n

p a
n−1
, . . . , a
1
, a
0
p
2
a
0
f(x) Q
f Q
f = gh = (b
0
+ b
1
x + + b
m
x
m
)(c
0
+ c
1
x + + c
k
x
k
),
g, h ∈ Z[x] g = m < n h = k < n. p

a
0
= b
0
c
0
p b
0
c
0
p
2
a
0
b
0
c
0
p c
0
p b
0
p a
n
= b
m
c
k
a
n

p b
m
c
k
p r
c
r
p r c
k
p r < n p a
r
b
0
c
r
= a
r
− (b
1
c
r−1
+ b
2
c
r−2
+ + b
r
c
0
) c

0
, , c
r−1
p b
0
c
r
p b
0
c
r
p r = n n = r ≤ k ≤ n
k = n f Q.
p = 3 f(x) =
11x
11
−3x
4
+ 12x
3
+ 36x +24 Q
p
f(x) = x
p−1
+ x
p−2
+ . . . + x + 1
Q. h(x) = f(x + 1). f(x) =
x
p

− 1
x − 1
h(x) =
(x + 1)
p
− 1
x
= x
p−1
+ C
1
p
x
p−2
+ . . . + C
k
p
x
p−k−1
+ . . . + C
p−1
p
,
C
k
p
=
p!
k!(p − k)!
. p C

k
p
p
k = 1, . . . , p −1. C
p−1
p
= p p
2
h(x)
Q f(x)
Q
Q p
p p
Z
p
p f(x) = a
n
x
n
+. . .+a
1
x+a
0
∈ Z[x].
p a
i
∈ Z
p
a
i

p
a
i
p
f(x) := a
n
x
n
+ . . . + a
1
x + a
0
∈ Z
p
[x].
f(x) = 15x
3
+ 42x
2
+ 19x − 8 p = 7 f(x) =
x
3
+ 5x − 1 ∈ Z
7
[x].
f(x) ∈ Z[x].
p deg f(x) = deg f(x) f(x)
Z
p
f(x) Q.

f(x) Z
p
deg f(x) > 0 deg f(x) =
deg f(x) deg f(x) > 0. f(x) Q
f(x) Q. f(x)
f(x) = g(x)h(x) g(x), h(x) ∈ Z[x] deg f(x) > deg g(x)
deg f(x) > deg h(x). a, b ab = a b
a + b = a + b f(x) = g(x)h(x). deg f(x) =
deg g(x) + deg h(x). deg f(x) = deg f(x) deg g(x) ≥ deg g(x)
deg h(x) ≥ deg h(x) deg g(x) = deg g(x) deg h(x) =
deg h(x) f(x) g(x), h(x)
f(x) Z
p
deg f(x) = deg f(x)
f(x) = 10(x + 21)
5
+ 3(x + 21) ∈ Z[x].
Q x + 21
f(x) = 3(x + 21) ∈ Z
5
[x] Z
5
1
Q
8x
2
+ 16x + 7
13x
3
+ 17x

2
+ 20x − 15
31x
4
− 25x
3
+ 41x
2
− 14x + 11
f(x) = 8x
2
+ 16x + 7 ∈ Z[x] p = 3
f(x) = 2x
2
+ x + 1 ∈ Z
3
[x] f(x) Z
3
f(x) = 2 = f(x)
f(x) Q.
f(x) = 13x
3
+ 17x
2
+ 20x − 15 ∈ Z[x] p = 2
f(x) = x
3
+ x
2
− 1 ∈ Z

2
[x] f(x) Z
2
f(x) = 3 = f(x) f(x)
Q.
Z
5
f(x) = 31x
4
−25x
3
+41x
2
−14x+11 ∈ Z[x]
f(x) = x
4
+x
2
+x+1 ∈ Z
5
[x] f(x)
Z
5
f(x) Z
5
f(x) = (x
2
+ax+b)(x
2
+cx+d) a, b, c, d ∈ Z

5
a + c = 0, b + ac + d = 1, ad + bc = 1, bd = 1
bd = 1 b, d
(b, d) = (1, 1) (b, d) = (2, 3) (b, d) = (4, 4)
(b, d) = (1, 1) a + c = 0, ac = −1, a + c = 1 (b, d) = (2, 3)
a + c = 0, ac = −4, 3a + 2c = 1 a = 1, c = −1
ac = −4 −1 = −4 (b, d) = (4, 4)
a + c = 0, ac = −7, a + c =
1
4
f(x) Z
5
f(x) = 4 = f(x) f(x)
Q.
R C
F
f(x) ∈ F [x]
E F f(x) E
f(x) F
R
C R.
T F F
T T
f(x) ∈ F [x]
E F f(x) E f(x)
E f(x)
E
Q[
√
2] = {a + b

√
2 | a, b ∈ Q}
x
2
− 2 ∈ Q[x] Q[
√
2] Q x
2
− 2
x
2
−2 = (x −
√
2)(x +
√
2)
Q[
√
2] x
2
− 2
Q[
√
2].
f(x) ∈ F [x]
I = {f(x)g(x) | g(x) ∈ F [x]}
F [x] f(x). F [x]/I
F f(x)
E = F [x]/I. E
(g(x) + I) + (h(x) + I) = (g(x) + h(x)) + I;

(g(x) + I)(h(x) + I) = g(x)h(x) + I.
g(x) + I, h(x) + I ∈ E. E
g(x) + I ∈ E g(x) + I = 0 + I. g(x) /∈ I g(x)
f(x) f(x) gcd(f(x), g(x)) = 1.
p(x), q(x) ∈ F [x] 1 = f(x)p(x) + g(x)q(x)
f(x)p(x) ∈ I f(x)p(x) + I = 0 + I
1 + I = (f(x)p(x) + g(x)q(x)) + I = (g(x)q(x) + I)
= (g(x) + I)(q(x) + I).
g(x) + I E E
ϕ : F → E ϕ(a) = a + I. ϕ
ϕ(a) = ϕ(b) a, b ∈ F a + I = b + I. a −b ∈ I. a −b
f(x) a −b = 0 a −b 0 a −b
f(x) a − b = 0.
a = b. ϕ F
E α = x + I ∈ E. f(x) = x
n
+ a
n−1
x
n−1
+ . . . + a
1
x + a
0
.
ϕ a ∈ F a + I ∈ E
E
f(α) = (x + I)
n
+ (a

n−1
+ I)(x + I)
n−1
+ . . . + (a
0
+ I)
= (x
n
+ I) + (a
n−1
x
n−1
+ I) + . . . + (a
0
+ I)
= (x
n
+ a
n−1
x
n−1
+ . . . + a
0
) + I
= f(x) + I = 0 + I.
α f(x) E
f(x) ∈ F [x]
E F f(x) E
deg f(x) = n.
n = 1 f(x) E = F.

n > 1
n f(x) F I F[x]
f(x) I =

g(x)f(x) | g(x) ∈ F[x]

. E

= F [x]/I
F [x] I E

F E

α f(x) f(x)
E

[x] α ∈ E

f(x) f
1
(x) ∈ E

[x]
f(x) = (x − α)f
1
(x) deg f
1
(x) = n − 1.
E E


f
1
(x)
E F f(x). f(x) E
f(x) g(x), h(x) ∈ F [x]
f(x) = g(x)h(x) deg g(x), deg h(x) < n.
E
1
F g(x). h(x) ∈ F [x]
h(x) ∈ E
1
[x]. E
E
1
h(x) E F
f(x) f(x) E
E
F α
1
, . . . , α
n
∈ E. F(α
1
, . . . , α
n
)
E F α
1
, . . . , α
n

. F (α
1
, . . . , α
n
)
E F α
1
, . . . , α
n
.
f(x) ∈ F [x]
f(x)
E F f(x)
E f(x) E
α
1
, . . . , α
n
f(x) E F (α
1
, . . . , α
n
)
f(x)
x
3
− 2 ∈ Q[x].
Q(i
√
3,

3
√
2) C Q
i
√
3,
3
√
2 f(x) = x
3
−2 Q
Q(i
√
3,
3
√
2) x
3
− 2
x
1
=
3
√
2, x
2
=
3
√
2(−

1
2
+
i
√
3
2
), x
3
=
3
√
2(−
1
2
−
i
√
3
2
).
x
1
, x
2
, x
3
∈ Q(i
√
3,

3
√
2) Q
x
1
, x
2
, x
3
i
√
3
3
√
2.
C
R
C R.
R p(x) ∈ R[x] E R
p(x) E
f(x) = a
n
x
n
+ a
n−1
x
n−1
+ + a
0

, a
n
= 0, n
x,
f(x)
x a b
f(a) < 0, f(b) > 0.
f(x) c x, a
b f(c) = 0.
f(x) = ax
2
+ bx + c ∈ C[x].
ω
1
ω
2
b
2
− 4ac
−b + ω
1
2a
−b + ω
2
2a
.
F
S
n
{1, 2, . . . , n}

f(x
1
, . . . , x
n
) ∈ F [x
1
, . . . , x
n
]
f(x
1
, . . . , x
n
) = f(x
π(1)
, . . . , x
π(n)
)
π ∈ S
n
.