tài liệu ôn thi toán chuyên đề số phức

T
T
T
T
C
C[x]
R[x],
R,
C.
T = R ×R = {(a, b)|a, b ∈ R}
(a, b) = (c, d) a = c, b = d
(a, b) + (c, d) = (a + c, b + d)
(a, b) . (c, d) = (ac − bd, ad + bc)
(a, b).(c, d) (a, b)(c, d).
i = (0, 1) ∈ T i
2
= i.i = (0, 1)(0, 1) = (−1, 0)
(a, b)(1, 0) = (1, 0)(a, b) = (a, b)
(a, b) = (a, 0) + (0, b) = (a, 0) + (b, 0)(0, 1), ∀ (a, b) ∈ T.
´
A φ : R → T, a → (a, 0),
φ(a + a

) = φ(a) + φ(a

), φ(aa

) = φ(a)φ(a


) a, a

∈ R.
(a, 0) ∈ T a ∈ R. (a, b) = (a, 0) +
(b, 0)(0, 1) = a + bi i
2
= (−1, 0) = −1.
C T C =
{a + bi|a, b ∈ R, i
2
= −1}
a + bi = c + di a = c, b = d
a + bi + c + di = a + c + (b + d)i
(a + bi)(c + di) = ac −bd + (ad + bc)i.
z = a + bi ∈ C a,
Re z, b, Im z; i
a − bi z = a + bi
z = a + bi. zz = (a + bi)(a − bi) = a
2
+ b
2
, z
1
z
2
= z
1
z
2

|z| =
√
zz z. z

= c + di −z

= −c − di
z − z

= (a + bi) − (c + di) = a − c + (b −d)i.
(Oxy). z = a + bi
M(a; b). C → R×R, z = a+bi →
M(a; b). C (Oxy) z M,
C R
C
z = a + bi = 0. a
2
+ b
2
> 0. z

= x + yi ∈ C
zz

= 1

ax −by = 1
bx + ay = 0.
x =
a

a
2
+ b
2
, y = −
b
a
2
+ b
2
.
z

=
a
a
2
+ b
2
−
b
a
2
+ b
2
i z. C
a ∈ R a+0i ∈ C R C.
z = 0 z
−1
=

z
|z|
2
z

z
= z

z
−1
=
z

z
|z|
2
.
z = 0. M
z.
Ox OM z
arg(z). ∠xOM z Arg z.
α z z
α + k.2π k ∈ Z. z = 0, α + k.2π z.
r =
√
zz. z = a + bi a = r cos α, b = r sin α.
z = 0 z = r

cos α + i sin α


z.
z
1
= r
1

cos α
1
+ i sin α
1

, z
2
= r
2

cos α
2
+ i sin α
2

r
1
, r
2
 0
|z
1
z
2

| = |z
1
||z
2
| |
z
1
z
2
| =
|z
1
|
|z
2
|
.
z
1
z
2
= r
1
r
2

cos

α
1

+ α
2

+ i sin

α
1
+ α
2

z
1
z
2
=
r
1
r
2

cos

α
1
− α
2

+ i sin

α

1
− α
2

r > 0.
z
1
z
2
z
1
= z
2
⇔ |z
1
| = |z
2
|, arg z
1
= arg z
2
+ 2kπ, k ∈ Z.
arg(z
1
z
2
) = arg(z
1
) + arg(z
2

) + 2kπ, k ∈ Z.
arg(
z
1
z
2
) = arg(z
1
) −arg(z
2
) + 2kπ, k ∈ Z.
Arg(z
1
z
2
) = Arg(z
1
) + Arg(z
2
).
Arg(
z
1
z
2
) = Arg(z
1
) −Arg(z
2
).

a + bi =

x + iy

n
a
2
+ b
2
=

x
2
+ y
2

n
.
a + bi =

x + iy

n
a − bi =

x − iy

n
.
a

2
+ b
2
=

x
2
+ y
2

n
.
z = r(cos α + i sin α)
n z
n
= r
n

cos

nα

+ i sin

nα

.
n.
n z = r


cos α + i sin α

= 0
n z
k
= r
1/n

cos
α + 2kπ
n
+ i sin
α + 2kπ
n

k = 1, 2, . . . , n.
z = r(cos α + i sin α) z = re
iα
.
r = 1 z = e
iα
r = 1, α = 0 e
0
= 1.
e
iα
e
iβ
= e
i(α+β)

e
iα
e
iβ
= e
i(α−β)
.
e
iα
e
iβ
= (cos α + i sin α)(cos β + i sin β) = cos(α + β) +
i sin(α + β) e
iα
e
iβ
= e
i(α+β)
.
e
iα
e
iβ
=
cos α + i sin α
cos β + i sin β
=
cos(α − β) + i sin(α − β)
e
iα

e
iβ
= e
i(α−β)
.

e
iα
= cos α + i sin α
e
−iα
= cos α − i sin α





cos α =
e
iα
+ e
−iα
2
sin α =
e
iα
− e
−iα
2
cos α =

e
iα
+ e
−iα
2
sin α =
e
iα
− e
−iα
2
.

1+i

n
= 2
n/2

cos
nπ
4
+i sin
nπ
4


1 + i tan α
1 −i tan α


n
=
1 + i tan nα
1 −i tan nα
.

1+i

n
= 2
n/2

cos
π
4
+sin
π
4

n
= 2
n/2

cos
nπ
4
+i sin
nπ
4


.

1 + i tan α
1 −i tan α

n
=

cos α + i sin α

n

cos α − i sin α

n
=
cos nα + i sin nα

cos nα − i sin nα
=
1 + i tan nα
1 −i tan nα
.
z = 0 z +
1
z
= 2 cos α z
n
+
1

z
n
= 2 cos nα
n.
z
2
− 2z cos α + 1 = 0 z = z
1
= cos α +
i sin α,
1
z
= z
2
= cos α − i sin α. z
n
= z
n
1
= cos nα + i sin nα,
1
z
n
=
z
n
2
= cos nα + i sin nα. z
n
+

1
z
n
= 2 cos nα
n.
C
C[x]
C.
K
K[x] K.
K[x]
K
R[x]
R.
f(x) = a
0
x
2s+1
+ a
1
x
2s
+ ··· + a
2s
x + a
2s+1
∈ R[x]
a
0
= 0. a

0
f(x) +∞ x → +∞ a
0
f(x)
−∞ x → −∞. α > 0
β < 0 a
0
f(α) > 0, a
0
f(β) < 0. a
2
0
f(α)f(β) < 0
f(α)f(β) < 0. f(x) R
f(α)f(β) < 0 f(x)
(α, β).
C[x] C.
z
z
1
, z
2
z
2
1
= z, z
2
2
= z. z = a+bi = 0 z
1

= x+yi
a, b, x, y ∈ R z
2
1
= z

x
2
− y
2
= a
2xy = b.
b = 0 b = 0
b = 0 x = 0.



y =
b
2x
4x
4
− 4ax
2
− b
2
= 0






x
1,2
= ±

a +
√
a
2
+ b
2
2
= 0
y =
b
2x
.
z
1
= x
1
+
bi
2x
1
z
2
= x
2

+
bi
2x
2
z
2
1
= z
2
2
= z.
z
1
z
2
z
2
1
= z
2
2
= b
2
−4ac.
−b + z
1
2
−b + z
2
2

.
C[x] C.
f(x) = x
n
+ a
1
x
n−1
+ ···+ a
n
f(x) = x
n
+ a
1
x
n−1
+ ··· + a
n
. g(x) = f(x)f(x) ∈ R[x].
g(α) = 0 f(α) = 0 f(α) = 0. f(α) = 0
0 = f (α) = f (α). g(x) f(x)
f(x) = x
n
+a
1
x
n−1
+···+a
n
∈ R[x]

K R K[x]
f(x) = (x − α
1
)(x −α
2
) . . . (x − α
n
).
n = 2
d
 
α
i
∈ C d.
d = 0 f(x) C
d > 0, R[x] m
m = 2
e
p, p e < d, C.
c
β
ij
= α
i
α
j
+ c(α
i
+ α
j

)
i, j = 1, . . . , n, i < j. (i, j)
n(n −1)
2
= 2
d−1
(2
d
 − 1) = 2
d−1
q q 2
d−1
q
g(x) =

1i<jn
(x −β
ij
)
α
h
β
ij
.
g(x) β
ij
α
i
.
g(x) α

i
c g(x)
g(x) C,
β
ij
∈ C.
n(n −1)
2
i, j i < j.
n(n −1)
2
+ 1
c. (x)
n(n −1)
2
i, j i < j,
n(n −1)
2
+ 1 g(x)
(i, j) c
1
c
2
a = α
i
α
j
+ c
1
(α

i
+ α
j
), b = α
i
α
j
+ c
2
(α
i
+ α
j
) C.





α
i
+ α
j
=
a −b
c
1
− c
2
,

α
i
α
j
=
bc
1
− ac
2
c
1
− c
2
.
f(x)
α
i
, α
j
∈ C.
C[x] :
C[x] n > 0 n
C C[x]
f(x) ∈ R[x] \ R. f(x)
f(x) = ax + b a = 0 f(x) = ax
2
+ bx + c a = 0
b
2
− 4ac < 0.

f(x) = ax + b a = 0 f(x) =
ax
2
+ bx + c a = 0 b
2
− 4ac < 0 f(x)
f(x) ∈ R[x] deg f(x)  1.
deg f(x) = 1 f(x) = ax + b a = 0.
deg f(x) = 2. f(x) = ax
2
+bx +c a = 0. ∆ = b
2
−4ac  0
f(x) α
1
, α
2
∈ R f(x) = a(x −α
1
)(x −α
2
) :
b
2
−4ac < 0. deg f(x) > 2. C
f(x) = 0 α ∈ C
α. f(x) (x −α)(x −α) ∈ R[x]
f(x) f(x)
f(x) = ax + b a = 0 f(x) = ax
2

+ bx + c
a = 0 b
2
− 4ac < 0.
R[x]
f(x) ∈ R[x] \R
f(x) = a(x − a
1
)
n
1
. . . (x −a
s
)
n
s
(x
2
+ b
1
x + c
1
)
d
1
. . . (x
2
+ b
r
x + c

r
)
d
r
b
2
i
− 4c
i
< 0 i = 1, . . . , r r  1.
R[x] f(x)
R[x].
R[x] ax + b a = 0
ax
2
+ bx+ c a = 0, b
2
−4ac < 0 f(x)
f(x) = a(x − a
1
)
n
1
. . . (x −a
s
)
n
s
(x
2

+ b
1
x + c
1
)
d
1
. . . (x
2
+ b
r
x + c
r
)
d
r
b
2
i
− 4c
i
< 0 i = 1, . . . , r r  1.
x
1
, . . . , x
n
n n
f(x) = x
n
− δ

1
x
n−1
+ δ
2
x
n−2
− ··· + (−1)
n
δ
n
.











δ
1
= x
1
+ x
2
+ ··· + x

n
δ
2
= x
1
x
2
+ x
1
x
3
+ ··· + x
n−1
x
n

δ
n
= x
1
x
2
. . . x
n
.
f(x) = x
n
−δ
1
x

n−1
+···+(−1)
n
δ
n
= (x−x
1
) . . . (x−x
n
),
x
1
, . . . , x
n
n f(x), δ
1
= x
1
+ x
2
+ ···+ x
n
, . . . ,
δ
n
= x
1
x
2
. . . x

n
.
cos
π
7
+ cos
3π
7
+ cos
4π
7
1
cos
4
π
7
+
1
cos
4
2π
7
+
1
cos
4
3π
7
.
−1 = cos π + i sin π =


cos
π
7
+ i sin
π
7

7
cos
π
7
64x
7
− 112x
5
+ 56x
3
− 7x + 1 = 0
(x − 1)(8x
3
− 4x
2
− 4x + 1)
2
= 0. cos
π
7
8x
3

− 4x
2
− 4x + 1 = 0. cos
3π
7
cos
5π
7
8x
3
− 4x
2
− 4x + 1 = 0. cos
π
7
, cos
3π
7
cos
5π
7
8x
3
− 4x
2
− 4x + 1 = 0.
cos
π
7
+ cos

3π
7
+ cos
5π
7
= −
1
2
.

cos x + i sin x

7
= cos 7x + i sin 7x
tan 7x =
cos 7x
sin 7x
tan x. tan
6
x −21 tan
4
x + 35 tan
2
x −7 = 0
x
1
= tan
2
π
7

, x
2
= tan
2
2π
7
, x
3
= tan
2
3π
7
x
3
− 21x
2
+ 35x − 7 = 0. T =
1
cos
4
π
7
+
1
cos
4
2π
7
+
1

cos
4
3π
7
.
T = (1 + x
1
)
2
+ (1 + x
2
)
2
+ (1 + x
3
)
2
= 416.
a
n
=
1
cos
n
π
7
+
1
cos
n

3π
7
+
1
cos
n
5π
7
, n = 1, 2, . . . .
a
n
n  4.
(cos
π
7
+ i sin
π
7
)
7
= −1. x =
cos
π
7
. x
64x
7
−112x
5
+ 56x

3
−7x +1 = 0 (x + 1)(8x
3
−4x
2
−4x +1)
2
= 0.
x = −1 8x
3
− 4x
2
− 4x + 1 = 0. cos
3π
7
, cos
5π
7
y
1
=
1
cos
π
7
, y
2
=
1
cos

3π
7
, y
3
=
1
cos
5π
7
y
3
− 4y
2
− 4y + 8 = 0. a
n
= y
n
1
+ y
n
2
+ y
n
3
n






y
1
+ y
2
+ y
3
= 4,
y
1
y
2
+ y
2
y
3
+ y
3
y
1
= −4,
y
1
y
2
y
3
= −8.
a
1
= 4, a

2
= (y
1
+ y
2
+ y
3
)
2
−2(y
1
y
2
+ y
2
y
3
+ y
3
y
1
) = 24,
a
3
= (y
1
+ y
2
+ y
3

)(y
2
1
+ y
2
2
+ y
2
3
−y
1
y
2
−y
2
y
3
−y
3
y
1
) + 3y
1
y
2
y
3
= 88,
a
n+3

= 4a
n+2
+ 4a
n+1
− 8a
n
, n  1.
n a
n
n  4.
n
(4 + cot
2
π
n
)(4 + cot
2
2π
n
) ···(4 + cot
2
(n −1)π
2n
) ∈ Q.
3
2011
4022
> (4 + cot
2
π

2011
)(4 + cot
2
2π
2011
) ···(4 + cot
2
(2010)π
4022
).
(x + i)
n
= (x − i)
n

x + i
x −i

n
= 1.
n C
x + i
x −i
= cos
k2π
n
+ i. sin
k2π
n
k = 1, . . . , n −1. x = cot

kπ
n
k = 1, . . . , n −1.
p(x) = (x + i)
n
− (x − i)
n
cot
kπ
n
= −cot
(n −k)π
n
p(x) = 2ni(x − cot
π
n
)(x −cot
2π
n
) . . . (x − cot
(n −1)π
n
)
= 2ni(x
2
− cot
2
π
n
)(x

2
− cot
2
2π
n
) . . . (x
2
− cot
2
(n −1)π
2n
).
x = 2i, (4+cot
2
π
n
)(4+cot
2
2π
n
) ···(4+cot
2
(n −1)π
2n
) =
3
n
− 1
2n
∈ Q.

n = 2011.
ax
2
+ bx + c = 0
ax
2
+ bx + c = 0 x
1,2
=
−b ±
√
b
2
− 4ac
2a
.
x
1
+ x
2
= −
b
a
, x
1
x
2
=
c
a

.
ax
3
+ bx
2
+ cx + d = 0
f(x) = ax
3
+ bx
2
+ cx+ d g(x) = x
3
+ ux
2
+ vx +t.
y = x+
u
3
h(y) = y
3
+py +q.  =
−1 + i
√
3
2
,
h(y) C :






















y
1
=
3

−
q
2
+

q
2

4
+
p
3
27
+
3

−
q
2
−

q
2
4
+
p
3
27
y
2
= 
3

−
q
2
+


q
2
4
+
p
3
27
+ 
2
3

−
q
2
−

q
2
4
+
p
3
27
y
3
= 
2
3

−

q
2
+

q
2
4
+
p
3
27
+ 
3

−
q
2
−

q
2
4
+
p
3
27
.
x
1
, x

2
, x
3
ax
3
+ bx
2
+ cx + d = 0
x
1
+ x
2
+ x
3
= −
b
a
, x
1
x
2
+ x
2
x
3
+ x
3
x
1
=

c
a
, x
1
x
2
x
3
= −
d
a
.
ax
4
+ bx
3
+ cx
2
+ dx + e = 0
a y = x +
b
2a
y
4
+ py
2
+ qy + r = 0.
(y
2
+ z)

2
= (2z −p)y
2
−qy + z
2
−r. y
y
4
+ py
2
+ qy + r = 0. z (2z −p)y
2
−qy + z
2
−r =
(sy + t)
2
. ∆ = q
2
−4(2z −p)(z
2
−r) = 0.
z. z
0
(y
2
+ z
0
)
2

= (sy + t)
2
.
ax
4
+ bx
3
+ cx
2
+ dx + e = 0.
x
3
−3x−1 = 0 2 cos 20
0
, −2 cos 40
0
−2 cos 80
0
.
x
4
+ x
2
+ 1 = 0,
C C
Q R.
f(x), g(x) ∈ R[x]. f(x)
g(x) h(x) ∈ R[x] f(x) = g(x)h(x).
f(x) = a
0

x
n
+ a
1
x
n−1
+ ··· + a
n
∈ Z[x], a
0
= 0.
p
q
(p, q) = 1 f(x) = 0
p a
n
q a
0
.
p −mq f (m) m.
p
q
(p, q) = 1 f(x) = 0.
a
0
p
n
+ a
1
p

n−1
q + ··· + a
n
q
n
= 0.
(p, q) = 1 p a
n
q a
0
.
f(x) x −m
f(x) = a
0
(x −m)
n
+ b
1
(x −m)
n−1
+ ··· + b
n−1
(x −m) + f(m) ∈ Z[x].
x =
p
q
a
0
(p − mq)
n

+ b
1
(p − mq)
n−1
q + ··· + b
n−1
(p −
mq)q
n−1
+ f (m)q
n
= 0. (p, q) = 1 p −mq f(m)
m.
f(x) = x
n
+ a
1
x
n−1
+ ···+ a
n
∈
Z[x]
α =

2 +

2 +
√
3 −


6 −3

2 +
√
3
α
4
−16α
2
+ 32 = 0 f(x) = x
4
−16x
2
+ 32 ∈ Z[x]
f(α) = 0. f(x). α
f(x) = a
0
x
n
+a
1
x
n−1
+···+ a
n
∈ Z[x], a
0
= 0. cont(f) =
d = (a

0
, . . . , a
n
).
1
d
f
g, h ∈ Z[x] cont(gh) = cont(g) cont(h).
cont(g) = cont(h) = 1
g h
g
cont(g)
h
cont(h)
g(x) = a
0
x
n
+ a
1
x
n−1
+ ···+ a
n
h(x) = b
0
x
m
+ b
1

x
m−1
+
···+b
∈
Z[x], a
0
b
0
= 0, cont(g) = cont(h) = 1. cont(gh) = d > 1.
p d. gh
p g h p.
a
r
b
s
g h
p. c
r+s
gh





a
r−1
≡ a
r−2
≡ ··· ≡ a

0
≡ 0(mod p)
b
s−1
≡ b
s−2
≡ ··· ≡ b
0
≡ 0(mod p)
c
r+s
= a
r
b
s
+ a
r+1
b
s−1
+ ··· + a
r−1
b
s+1
+ ··· ≡ a
r
b
s
≡ 0(mod p).
cont(gh) = 1.
f ∈ Z[x] Z

Q.
f ∈ Z[x] f = gh g, h ∈ Q[x].
cont(f) = 1. g m
mg ∈ Z[x]. n = cont(mg) r =
m
n
. rg ∈ Z[x]
cont(rg) = 1. s h sh ∈ Z[x]
cont(sh) = 1. f = (rg)(sh) f Z.
1 = cont(f) = cont(rg) cont(sh) = cont(rsgh) = cont(rsf).
rs = 1.
Z.
f(x) = a
n
x
n
+ a
n−1
x
n−1
+
··· + a
0
, a
n
= 0, p
a
n
p a
i

, i < n, p a
0
p
2
. f(x) Z.
f = gh = (
r

i=0
b
i
x
i
)(
s

j=0
c
j
x
j
) g, h ∈ Z[x]
r = deg g, s = deg h > 0, r + s = n. b
0
c
0
= a
0
p
b

0
c
0
p, b
0
p.
a
0
p
2
c
0
p. b
i
p a
n
p :
b
i
p. i b
i
p. 0 < i  r. a
i
= b
i
c
0
+ b
i−1
c

1
+ ···+ b
0
c
i
p b
i−1
c
1
, . . . , b
0
c
i
p b
i
c
0
p : f
Z.
n f(x) = 1+x+
x
2
2!
+···+
x
n
n!
Q.
n!f(x) = n! + n!x +
x

2
2!
+ ···+ x
n
Z. p p  n < 2p n p,
n! p
2
. n!f
Z.
p f(x) = 1 + x + ···+ x
p−1
Z.
f(x + 1) = x
p−1
+

p
1

x
p−2
+
··· +

p
p−1

Z. f Z.
f(x) = b
0

x
n
+ b
1
x
n−1
+ ··· + b
n
p b
0
p
b
k+1
, . . . , b
n
p, b
n
p
2
. f(x)
 n − k.
f(x)
g(x) = c
0
x
m
+ c
1
x
m−1

+ ··· + c
m
∈ Z[x]
c
m
p. f(x) = g(x)h(x) h(x) = d
0
x
h
+
d
1
x
h−1
+ ··· + d
h
∈ Z[x]. d
h
p. c
i
g(x) p b
m
, . . . , b
i+1
p. c
m
d
h
= b
n

p, p
2
.
b
h+i
= c
i
d
h
+ b
i+1
d
h−1
+ ··· p h + i  k
n −m + i  k. m  n + i − k  n −k.
f(x) ∈ R[x] \R. f(x)
f(x) = ax + b a = 0 f(x) = ax
2
+ bx + c a = 0
b
2
− 4ac < 0.
f(x) = ax + b a = 0 f(x) =
ax
2
+ bx + c a = 0 b
2
− 4ac < 0 f(x)
f(x) ∈ R[x] deg f(x)  1.
deg f(x) = 1 f(x) = ax + b a = 0.

deg f(x) = 2. f(x) = ax
2
+bx +c a = 0. ∆ = b
2
−4ac  0
f(x) α
1
, α
2
∈ R f(x) = a(x −α
1
)(x −α
2
) :
b
2
−4ac < 0. deg f(x) > 2. C
f(x) = 0 α ∈ C
α. f(x) (x −α)(x −α) ∈ R[x]
f(x) f(x)
f(x) = ax + b a = 0 f(x) = ax
2
+ bx + c
a = 0 b
2
− 4ac < 0.
R[x]
f(x) ∈ R[x] \R
f(x) = a(x − a
1

)
n
1
. . . (x −a
s
)
n
s
(x
2
+ b
1
x + c
1
)
d
1
. . . (x
2
+ b
r
x + c
r
)
d
r
b
2
i
− 4c

i
< 0 i = 1, . . . , r r  1.
R[x] f(x)
R[x].
R[x] ax + b a = 0
ax
2
+ bx + c a = 0, b
2
−4ac < 0. f(x)
f(x) = a(x − a
1
)
n
1
. . . (x −a
s
)
n
s
(x
2
+ b
1
x + c
1
)
d
1
. . . (x

2
+ b
r
x + c
r
)
d
r
b
2
i
− 4c
i
< 0 i = 1, . . . , r r  1.
f(x) ∈ Z[x]
√
2 +
3
√
3
x =
√
2+
3
√
3. 3 = (x−
√
2)
3
= x

3
+6x−(3x
2
−2)
√
2.
f(x) = (x
3
+6x−3)
2
−2(3x
2
+2)
2
= x
6
−6x
4
−6x
3
+12x
2
−36x+1
x
1
=
√
2 +
3
√

3
p = 2, f(x + 1)
x
6
−6x
4
−6x
3
+ 12x
2
−36x + 1
x
1
=
√
2 +
3
√
3
f(x) ∈ Z[x]
tan
π
16
, tan
5π
16
, tan
9π
16
, tan

13π
16
T = tan
4
π
16
+ tan
4
5π
16
+
tan
4
9π
16
+ tan
4
13π
16
+ 4(tan
3
π
16
+ tan
3
5π
16
+ tan
3
9π

16
+ tan
3
13π
16
).
tan 2x =
2 tan x
1 −tan
2
x
, tan
π
4
= 1
tan
π
8
=
√
2 − 1. tan
π
16
= −
√
2 − 1 +

4 + 2
√
2.

x = −
√
2 −1 +

4 + 2
√
2 x
4
+ 4x
3
− 6x
2
− 4x + 1 = 0.
f(x) = x
4
+ 4x
3
− 6x
2
− 4x + 1 ∈ Z[x]
x
1
= tan
π
16
x
2
= tan
5π
16

, x
3
= tan
9π
16
, x
4
= tan
13π
16
.

x
1
+ x
2
+ x
3
+ x
4
= −4
x
1
x
2
+ x
1
x
3
+ x

1
x
4
+ x
2
x
3
+ x
2
x
4
+ x
3
x
4
= −6.
x
2
1
+ x
2
2
+ x
2
3
+ x
2
4
= 28. x
1

, x
2
, x
3
, x
4
f(x) = 0
x
4
1
+ 4x
3
1
− 6x
2
1
− 4x
1
+ 1 = 0
x
4
2
+ 4x
3
2
− 6x
2
2
− 4x
2

+ 1 = 0
x
4
3
+ 4x
3
3
− 6x
2
3
− 4x
3
+ 1 = 0
x
4
4
+ 4x
3
4
− 6x
2
4
− 4x
4
+ 1 = 0
T =
4

i=1
x

4
i
+ 4
4

i=1
x
3
i
= 6
4

i=1
x
2
i
+ 4
4

i=1
x
i
− 4 = 148.
x
2n
+ x
n
+ 1
R[x]
n−1


k=0
sin
(3k + 1)π
3n
.
α
r
= cos
r2π
3n
+ i sin
r2π
3n
r = 0, 1, 2, . . . , 3n.
f(x) = (x
n
−1)(x
2n
+ x
n
+ 1) = x
3n
−1
f(x) =
n

k=0
(x −α
3k

)
n−1

k=0

x −α
3k+1
)
n−1

k=0

x −α
3k+2
).
n

k=0
(x −α
3k
) =
n

k=0
(x −cos
k2π
n
− i sin
k2π
n

) = x
n
− 1
x
2n
+ x
n
+ 1 =
n−1

k=0

x −α
3k+1
)
n−1

k=0

x −α
3k+2
).
(x − α
3k+1
)(x − α
3(n−k−1)+2
) = x
2
− 2x cos
(3k + 1)2π

3n
+ 1
R k = 0, 1, . . . , n − 1
x
2n
+x
n
+1 =
n−1

k=0

x
2
−2x cos
(3k + 1)2π
3n
+1

x = 1 3 =
n−1

k=0
4 sin
2
(3k + 1)π
3n
.
n−1


k=0
sin
(3k + 1)π
3n
=
√
3
2
n
.
x
2n
−2x
n
+ 2
R[x]
n−1

k=0
sin
(8k + 1)π
8n
.
f(x) = x
2n
−2x
n
+2 = (x
n
−1)

2
+1 = (x
n
−1−i)(x
n
−1+i).
f(x) =

x
n
−
√
2(cos
π
4
+ i sin
π
4
)

x
n
−
√
2(cos
π
4
− i sin
π
4

)

.
f(x) =
n−1

k=0

x −
2n
√
2(cos
π
4
+ 2kπ
n
+ i sin
π
4
+ 2kπ
n
)

n−1

k=0

x −
2n
√

2(cos
π
4
+ 2kπ
n
− i sin
π
4
+ 2kπ
n
)

=
n−1

k=0

x
2
− 2
2n
√
2x cos
(8k + 1)π
4n
+
n
√
2


.
= x
2n
− 2x
n
+ 2 =
n−1

k=0

x
2
− 2
2n
√
2x cos
(8k + 1)π
4n
+
n
√
2

R. x =
2n
√
2 4 − 2
√
2 =
n−1


k=0
4
n
√
2 sin
2
(8k + 1)π
8n
.
n−1

k=0
sin
(8k + 1)π
8n
=

2 −
√
2
2
n
.
K k α ∈ K
k.
p(x) ∈ k[x] α
f(x) ∈ k[x] f(α) = 0 f(x) p(x).
g(x) α
α k. f(x) ∈ k[x].

f(x) = q(x)g(x) + r(x), deg r(x) < deg g(x).
f(α) = 0 r(α) = 0. f(x) g(x)
f(α) = 0.
m n, x
m
+ x
n
+ 1
x
2
+ x + 1 mn −2 ˙: 3. x
10
+ x
5
+ 1, x
20
+ x
4
+ 1
x
2
+ x + 1.
x
2
+ x + 1 Q α =
−1 + i
√
3
2
, α

3
= 1. m = 3h + r, n = 3k + s r, s ∈ {0, 1, 2}.
p(x) = x
m
+x
n
+1 x
2
+x+1 α
m
+α
n
+1 = 0
α
r
+ α
s
+ 1 = 0. 0 =

−1 + i
√
3
2

r
+

−1 + i
√
3

2

s
+ 1 =
cos
r2π
3
+ i sin
r2π
3
+ cos
s2π
3
+ i sin
s2π
3
+ 1.





sin
r2π
3
+ sin
s2π
3
= 0
cos

r2π
3
+ cos
s2π
3
+ 1 = 0



r + s ˙: 3 ↔

r=s=0
r+s=3
2 cos
(r + s)π
3
cos
(r − s)π
3
+ 1 = 0.
r = s = 0 r + s = 3 {r, s} = {1, 2}.
mn = (3u + 1)(3v + 2) = 3(3uv + u + v) + 2. x
m
+ x
n
+ 1
x
2
+ x + 1 mn −2 ˙: 3.
m n, x

m
+ x
n
+ 1
x
2
−x + 1 m + n, mn −2 ˙: 6. x
m
+ x
n
+ 1
x
4
+ x
2
+ 1 m + n, mn − 2 ˙: 6.
x
2
− x + 1 Q α =
1 + i
√
3
2
, α
3
= −1. m = 6h+r, n = 6k+s r, s ∈ {0, 1, 2, 3, 4, 5}.
x
m
+ x
n

+ 1 x
2
−x + 1 α
m
+ α
n
+ 1 = 0
α
r
+ α
s
+ 1 = 0. 0 =

1 + i
√
3
2

r
+

1 + i
√
3
2

s
+ 1 =
cos
rπ

3
+ i sin
rπ
3
+ cos
sπ
3
+ i sin
sπ
3
+ 1.



sin
rπ
3
+ sin
sπ
3
= 0
cos
rπ
3
+ cos
sπ
3
+ 1 = 0






2 sin
(r + s)π
6
cos
(r − s)π
6
= 0
2 cos
(r + s)π
6
cos
(r − s)π
6
+ 1 = 0.



r + s ˙: 6 ↔

r=s=0
r+s=6
2 cos
(r + s)π
6
cos
(r − s)π
6

+ 1 = 0.
r = s = 0
r + s = 6.



r + s = 6
cos
(r − s)π
6
=
1
2

r + s = 6
r − s = ±2.
{r, s} = {2, 4}. m = 6u + 4, n = 6v + 2.
m + n = 6(u + v + 1), mn = 6(6uv + 4v + 2u + 1) + 2.
m + n, mn −2 ˙: 6. x
m
+ x
n
+ 1
x
4
+ x
2
+ 1 m + n, mn − 2 ˙: 6.
m n, x
m

+ x
n
− 2
x
4
+ x
3
+ x
2
+ x + 1 m, n ˙: 5.
x
4
+ x
3
+ x
2
+ x + 1 Q
α = cos
2π
5
+ i sin
2π
5
, α
5
= 1. m = 5h + r, n = 5k + s
r, s ∈ {0, 1, 2, 3, 4}. x
m
+ x
n

−2 x
4
+ x
3
+ x
2
+ x + 1
α
m
+ α
n
− 2 = 0 α
r
+ α
s
− 2 = 0. cos
r2π
5
+
i sin
r2π
5
+ cos
s2π
5
+ i sin
s2π
5
− 2 = 0.






sin
r2π
5
+ sin
s2π
5
= 0
cos
r2π
5
+ cos
s2π
5
− 2 = 0





2 sin
(r + s)π
5
cos
(r − s)π
5
= 0

2 cos
(r + s)π
5
cos
(r − s)π
5
− 2 = 0.



r + s ˙: 5 ↔

r=s=0
r+s=5
2 cos
(r + s)π
5
cos
(r − s)π
5
− 2 = 0.
r = s = 0
r + s = 5



r + s = 5
cos
(r − s)π
5

= −1

r + s = 5
r − s = ±5.
{r, s} = {0, 5}. x
m
+x
n
−2 x
4
+x
3
+x
2
+x+1
m, n ˙: 5.
m n, x
m
+ x
n
+ 1
x
4
+ x
3
+ x
2
+ x + 1.
x
4

+ x
3
+ x
2
+ x + 1 Q
α = cos
2π
5
+ i sin
2π
5
, α
5
= 1. m = 5h + r, n = 5k + s
r, s ∈ {0, 1, 2, 3, 4}. x
m
+ x
n
+ 1 x
4
+ x
3
+ x
2
+ x + 1
α
m
+ α
n
+ 1 = 0 α

r
+ α
s
+ 1 = 0. cos
r2π
5
+
i sin
r2π
5
+ cos
s2π
5
+ i sin
s2π
5
+ 1 = 0.





sin
r2π
5
+ sin
s2π
5
= 0
cos

r2π
5
+ cos
s2π
5
+ 1 = 0





2 sin
(r + s)π
5
cos
(r − s)π
5
= 0
2 cos
(r + s)π
5
cos
(r − s)π
5
+ 1 = 0.



r + s ˙: 5 ↔


r=s=0
r+s=5
2 cos
(r + s)π
5
cos
(r − s)π
5
+ 1 = 0.
r = s = 0
r+s = 5



r + s = 5
2 cos
(r − s)π
5
= 1.
x
m
+x
n
+1 x
4
+x
3
+x
2
+x+1.

m (x + 1)
m
+ x
m
+ 1
x
2
+ x + 1.
x
2
+ x + 1 Q α =
cos
2π
3
+ i sin
2π
3
, α
3
= 1. α
2
+ α + 1 = 0 1 + α = −α
2
.
(1 + α)
6
= 1. m = 6k + r r ∈ {0, 1, 2, 3, 4, 5}.
(x+1)
m
+x

m
+1 x
2
+x+1 (1+α)
m
+α
m
+1 = 0
(1+α)
r
+α
r
+1 = 0. cos
rπ
3
+i sin
rπ
3
+cos
r2π
3
+i sin
r2π
3
+1 = 0.






sin
rπ
3
+ sin
r2π
3
= 0
cos
rπ
3
+ cos
r2π
3
+ 1 = 0



2 sin
rπ
2
cos
rπ
6
= 0
2 cos
rπ
2
cos
rπ
6

+ 1 = 0.

r = 0, 2, 3, 4
2 cos
rπ
2
cos
rπ
6
+ 1 = 0.
r = 2, r = 4
(x + 1)
m
+ x
m
+ 1 x
2
+ x + 1
m = 6k + 2 m = 6k + 4.
m (x + 1)
m
+ x
m
+ 1
(x
2
+ x + 1)
2
. 21
2014

+ 20
2014
+ 1 421
2
.
x
2
+ x + 1 Q α =
cos
2π
3
+i sin
2π
3
, α
3
= 1. α
2
+α+1 = 0 1+α = −α
2
. (1+α)
6
=
1. m = 6k + r r ∈ {0, 1, 2, 3, 4, 5}. (x + 1)
m
+ x
m
+ 1
(x
2

+ x + 1)

(1 + α)
m
+ α
m
+ 1 = 0
m(1 + α)
m−1
+ mα
m−1
= 0

r = 2, r = 4
(1 + α)
r−1
+ α
r−1
= 0.
r = 4
(x + 1)
m
+ x
m
+ 1 (x
2
+ x + 1)
2
m = 6k + 4. x = 20 21
2014

+ 20
201
+ 1 421
2
.
m

(x + 1)
m
+ x
m
+ 1, (x + 1)
m
− x
m
− 1

(x
2
+ x + 1)
3
.