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n
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= 5+
15 e = 5
15,
10
40
x2 10x+40 = 0,
(
15)2 = 15,
+= 10 e = 40.
= 5 +
15
1, = 5
15
1,
10 40
(
1)2 = 1
10 26
x2 10x+26 = 0,
5 +
1 5
1
10 26
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a + b1 a b
1
(
1)2 = 1.
(
1)(
1) = 1, (
1)(
1) =1,
(
1)(
1) = 1, 1(
1) =
1,
(a+b
1) + (c+d
1) = (a+c) + (b+d)
1,
(a+b
1) (c+d1) = (acbd) + (ad+bc)1.
x3 = 15x+ 4
3
2+
121+
3
2
121.
4
32+111+ 32111= 4.
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2+
1)3 = 23 +3 221+3 2(1)2 + (1)3
= 8+12
16
1
= 2+11
1,
2
1)3 = 23 3 221+3 2(1)2 + (1)3
= 812
16+
1
= 2111.
3
2+11
1+
3
211
1= (2+
1) + (2
1) =4.
a+b
1
1
i
1 i
1= 1
1=
1
1=
(1)(1) =
1= 1.
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C
i2 = 1
(a+bi) + (c+di) = (a+c) + (b+d)i,
(a+bi) (c+di) = (acbd) + (ad+bc)i.
C
z= a +bi
a
bi
a
b
z z= a+bi
a= Re(z) e b= Im(z).
a+bi= a +b i a= a e b= b .
a + bi a b
a + 0i a
a R
C C
R
R C ={a+bi; a, b R i2 = 1}, ondeiC\R.
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0 = 0 +0i 1= 1 +0i
z+0 = z e z 1= z, para todoz C.
z= a + bi z C z+ z =0
z = (a) + (b)i= abi
z
z, z, z
C
z+z =z+z, z z =z z
z+ (z+z) = (z+z) +z, z (z z) = (z z) z
z (z+z) =z z+z z
C
R
z= a +bi z=0 z = a + b i z
z = 1
1= z z = (aa bb ) + (ab +ba )i,
a b
aa bb = 1
ab +ba = 0,
a = aa2 +b2
e b = ba2 +b2
,
a2 +b2 = 0 z= a+ bi= 0
a=0 b=0
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z
z1 1
z
z= a+bi=0 1
z =
a
a2 +b2
b
a2 +b2i.
C
R
x2 = a a R a < 0
b =
|a|
b2 = a > 0 x1 =
i|a| x2 = i|a| a
ax2 +bx + c = 0 a,b,c R a= 0 =b2 4ac < 0 C
x1 = b+i
2a
x2 = bi
2a .
x2 +x+ = 0 ,, C =0 C
K KK K K
: K K K(a, b) a b .
K
(+) () K
a
b
K
a+b= b+a a b= b a
a, b
c
K
a+ (b+c) = (a+b) +c a (b c) = (a b) c
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0
1
K
a K
a+0= a a 1= a
a, b c K
a (b+c) =a b+a c a K a K
K a+a = 0
b K b b
b = 1
Q R C
0 1 a b
a b
ab
a
a b b1 1b
(+) () K K
Z
ab= a + (b)
A
a, bA, a b= 0 = a= 0 oub = 0
A
a, bA \ {0},
a b=0 Z Q R C
Zp p
Zm m
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x2 = 18
x2 5x+9 = 0
(1+i)2 (1i)2
1
2+
32
i
3
1+i
i +
i
1i
5+2i
52i (2+i)(5+3i)(14i)
i 1+i
1i
2+i
1i+
3 +2i
1+i
1+i
1i =z+i (1+2i)(iz3) =2i
a b
(a2) b+ (b2 1)i= i (a2 4) + (a2)(b2 1)i
(b2 4) + (a2 1)(b+2)i
in =
1, n
0 mod 4
i, n1 mod 4
1, n2 mod 4i, n3 mod 4.
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5i127 +3i82 7i37 +i16
1+i+i2 + +in1 nN, n1 z w zw= 0 z= 0 w= 0
Un(C) ={C ; n =1} n1
Un(C)=
Un(C) Un(C)
Un(C) 1 Un(C)
Un(C)
Un(C)
Z
0 1 a a b b
a b a b a = a b = b K
a b= 0 a= 0 b= 0 ab = 0 a= 0 a1
b= 0
1
i
90
o
z = a+ bi (a, b)
1
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C
R2
a+bi= a +b i (a, b) = (a, b ),
C R2
C R2
(a, b) + (a , b ) = (a+a , b+b ),
(a, b) (a , b ) = (aa bb , ab +ba ).
O = (0, 0) A = (a, b) A = (a , b ) OA OA OC
A A
C= (a + a , b + b )
z= a +bi z = a +b i z+z = (a+a ) + (b+b )i
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z+z
R2
z= a + bi
z= a bi
z
z= a +bi
z= a bi
z= 0 z= 0
z= z zC
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|z+ w||z|+|w|
z
w
|z+w| |z|+|w|,
|z+w|
|z+w|2 (1)
= (z+w)(z+w)(2)= (z+w)(z+w)
(3)= z z+zw+w z+w w(4)= |z|2 +zw+w z+|w|2 .
zw +w z u= zw
u= zw= zw= zw
zw+w z= u+u= 2Re(u)2|u|= 2|zw|= 2|z| |w|= 2|z| |w|
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z z
z= 2 +i z= 3+4i z= 43i
z C z=0 z1 = z|z|2
z
z= 1 2i
z= 3+4i
z= 1+i
S1 ={ z C ; |z|= 1 } z w
zS1 z1 =zS1 z, wS1 zwS1 : C C (a + bi) =a bi (z+z) =(z) +(z) z, z C (z z) =(z) (z) z, z C (z) =z z
R
1
f(x) =anxn + an1x
n1 + + a1x + a0 aj R C
f() =f() f(x) f(x)
n = 2 = a21 4a2a0 < 0 f(x)
, C\R =
Re(zw) =|zw|
| |z||w| ||z+w|
|z+w|= |z||w| w= z 10
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a= c2 d2
b=
2cd a2 = (c2 d2)2
b
2 =4c
2
d
2
a2 +b2 =c4 +d4 +2c2d2 = (c2 +d2)2. c2 +d2 =
a2 +b2 c2 d2 = a
c2 =
a2 +b2 +a
2 d2 =
a2 +b2 a
2
|c |=a2 +b2 +a
2
|d |=a2 +b2 a
2
(1)
b= 0 b = 2cd c d (1)
b b > 0 c > 0 d > 0 c < 0 d < 0
b < 0
c > 0 d < 0
c < 0
d > 0
= a+ bi
x2 =
3i a =
3
b= 1 a2 +b2 =4
|c |=
4+
3
2 =
2+
3
2=
2
2+
6
2 ,
|d |=
4
3
2
= 2
3
2=
2
2
6
2
.
b < 0 2
2+
6
2
2
2
6
2 i
2
2+
6
2 +
2
2
6
2 i
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x2 = 1+ i a = 1 b = 1
a2
+b2
=2
|c |=
2+1
2 , |d |=
21
2
b > 0
2+1
2 +i
21
2
2+1
2 i
21
2
x2 +x+ = 0
, C
x2 +x+ =
x+
2
2
2
4 +
=
x+
2
2
2 4
4 .
= 2 4 C C 2 = =
=
x+
2
2
2 4
4 =0
x+
2 =
2
x1 = +
2 4
2
x2 =
2 4
2
2 4 x2 =2 4
x2 +2ix + (2 i) = 0
= (2i)2 4(2 i) =4 +4i
4+4i a= 4 b= 4 a2 +b2 =32
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z= a + bi=0 r=
a2 +b2
(0, 0)
1 2
z
OP
z= a+ bi r =
a2 +b2 arg(z) =
z= r(cos +i sen )
cis
cos +i sen
z1 = 2 z2 = 2 z3 = 2i z4 = 2i z5 = 2 2i
z6 = 1
3i
z1, z2, z3 z4
z1 z2
z1
z2
arg(z1) =0 arg(z2) =
z3 z4
z3 z4
arg(z3) =
2 arg(z4) =
32
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+2
A = (1, 0) 0
P OA OP
+2
+2
z= r(cos +i sen )
z =r (cos +i sen ) z z =rr cos(+ ) +i sen(+ )
z z = r(cos +i sen )r (cos +i sen )=rr
(cos cos sen sen ) +i(cos sen + sen cos )
=rr
cos(+ ) +i sen(+ )
cos(+ ) =cos cos sen sen
sen(+ ) =cos sen + sen cos
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z z z z
z
z z z z z
z
z =
r (cos +i sen )
r (cos +i sen )
= r
r (cos +i sen ) (cos i sen )
= r
r (cos +i sen ) (cos( ) +i sen( ))
= r
r
(cos( ) +i sen( )) .
z z z= 5+5
3i z =2
32i
r =
(5)2 + (5
3)2 =
25+25 3= 100= 10
r =
(2
3)2 + (2)2 =
4 3+4 = 16= 4 .
rr = 40 .
z z
cos = 5
10 = 1
2
sen = 5
3
10 =
3
2
=
arg(z
) = 23
cos = 2
34 =
3
2 sen = 12
=arg(z) = 116
+ = 23 + 11
6 = 15
6 =2+ 2
z z =40 cos(2+ 2 +i sen 2+ 2 ) =40 cos 2 +i sen 2
z, z z z
z z
0 = arg(z) < 2 0 = arg(z) < 2 0+ < 4 0 < 2
cos = cos(+ ) sen = sen(+ )
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arg(z) = = 34 z =
2 cos3
4 +i sen3
4 z6 = (
2)6
cos
6 3
4
+i sen
6 3
4
=8
cos 18
4 +i sen18
4
arg(z6) [0,2) 184
184 =
92 =
8+2 =4+
2
4 2
= 2 z
6
z6 =8
cos 2 +i sen2
=8i
zn
n arg(z) 2
[0,2) = arg(zn)
1
(cos +i sen )(cos +i sen ) =cos(+ ) +i sen(+ ),
ei =cos +i sen .
sen = 1! 3
3! + 5
5! ,
cos = 1 2
2! + 4
4! ,
ei =1 + i1! + (i)2
2! + (i)3
3! + (i)4
4! + .
ei =cos +i sen = 1.
0,1,e,,i
ei +1 = 0
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z= r(cos +i sen ) =rei.
(rei)(r ei
) = (r r )ei(+ ),
(rei)n =rnein.
z
z
z= 33i z= 5i z= 7
z= 2 +2i z=
3i z= 2
32i
z= 11+i z= 5 z= 2i
zC z= 0 r
zr
zei
zrei
(2+2i)5 (1+i)7 (
3i)10 (1+
3i)8
n 2 (
2+
2i)n
3
2 + 12 i
n n Z
z
w
z w
cos =zw+wz
2|z| |w| .
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n
n
n n
n
n N \ {0} n K
n1
zK
wK
wn =z n z
n= 1 1 z z
K
z = 0 xn = 0
n1
K
zK n2
n 2 z K K n z
Q 2
R 2
2
2
n R n
n R n a > 0 n
a n
a
n R n
K
n
n
K
C z= 0 n n
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w{1, 1,i, i} w4 =1
w{4i, 23 + 2i, 23 + 2i} 64i (4i)3 = (4)3 i3 = (64) (i) =64i
2
3+ 2i 2
3+ 2i
2
3+2i= 4
cos 6 +i sen6
2
3+2i = 4
cos 56 +i sen
56
(2
3+2i)3 = 43
cos 2 +i sen2
=64i,
(2
3+2i)3 = 43 cos 52
+i sen 5
2
= 43
cos
2+ 2
+i sen
2+ 2
= 64
cos 2 +i sen2
=64i .
n
z=
0 n n
n1 zk=
n
r
cos
+2kn
+i sen
+2k
n
k= 0, 1, . . . , n1
r= |z|> 0 = arg(z)
n 2 z z= r(cos +i sen ) r = |z| = arg(z)
n
w= (cos +i sen ) z= wn
wn = n(cos(n) +i sen(n)) wn = z
n =r
n= +2, Z
= n
r, R , > 0= +2n , Z.
z = nrcos+2n +i sen+2n , onde Z.
, Z
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n
z =z +2
n +2
n =2, Z
2n
2n =2, Z
n
n =, Z
= n, Z mod n.
n
n z
k= 0, 1, . . . , n1 n z
k= +2kn n
z
zk= n
r(cos k+i sen k) k= +2k
n k= 0, 1, . . . , n1
64i
z= 27i
r= 27
= arg(z) = 3
2 z =
3
27= 3
k=+2k
3 =
2 +
2k
3 , k= 0,1, 2.
z0, z1 z2 z
0 =
2 z0 = 3 cos 2 +i sen 2 =3i 1 =
76 z1 =3 cos 76 +i sen 76 =3 32 i 12 = 332 32 i
2 = 11
6 z2 =3 cos11
6 +i sen11
6 =3
32 i
12 =
3
32
32
i
w=
2+12 + i
212 w
z= 1 +i
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|z| =
2 arg(z) = 4 z=
2 cos4 +i sen
4
z
zk = 4
2
cos
4+2k
2
+i sen
4
+2k
2
= 4
2
cos
8 +k
+i sen
8 +k
,
k= 0,1.
z0 = 4
2
cos 8 +i sen8
z1 = z0
4
2
cos 8 +i sen
8
=
2+12 +i
212
42 cos 8 =2+1
2
42 sen 8 =21
2
cos 8 =
2+
2
2 sen8 =
2
2
2
z = r arg(z) = 0 n
n z k = 2k
n
k= 0, 1, . . . , n1
n
r n n
r
n
3 n
nr 4 16
2 2i 2 2i
k=2 k
4 =
k2
, k= 0, 1, 2, 3 ; = 4
16= 2 .
0 = 0 z0 =2(cos 0+i sen 0) =2 ,1 =
2
z1 =2(cos
2 +i sen
2 ) =2i ,
2 = z2 = 2(cos +i sen ) = 2 ,3 = 32 z3 = 2(cos 32 +i sen 32 ) = 2i .
16
2=
4
16
1 1
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n
16 1
z= cos 15 +i sen
15
z20
n z
n= 2 z= 1
3i n= 4 z= 3
n= 3 z= 16+16i n= 6 z= 1
3
7+i
13 = cos + i sen cos k =
k +k
2 sen k =
k k
2i n N
sen3
sen4
cos5
cos6
z=
3+i
cos 12 sen 12
cos 8 +i sen8 =
2+
2
2 +
2
2
2 i
cos 16
sen 16
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(1+cos +i sen )n n1
cos n= cosn
n2
cosn2 sen2 +
n4
cosn4 sen4 +a
a=
(1)
n2 senn , n ;
(1)n12 n cos senn1 ,
n
sen n=
n1
cosn1 sen
n3
cosn3 sen3 + +b
b= (1)
n22 n cos senn1 , n ;
(1)
n1
2 senn
,
n
(cos +i sen )n
n 1 n
1
1
n2
= arg(1) =0, k= 2k
n , onde k= 0, 1, . . . , n1,
n
zk= cos2k
n +i sen
2k
n , k= 0, 1, . . . , n1,
n z0 =1
n n
1
C
1 n
{1, 1}
{1,i, 1, i}
1, 12 +
3
2 i, 12
3
2 i
1
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q < n S {m ; m Z}=Un(C)
n
mdc(, n) =1
S=
s Z ; s > 0 ()s =1 ()n =(n) =1 =1 nS S= S
s0
s0 =n
s0> 0 (
)s0 =1 s0S s0
n
q r
s0 =nq + r 0r < n
1= ()s0 = ()nq+r = (n)q r =r
r 0 mod n mdc(, n) = 1 mod n r0 mod n 0r < n r= 0 s0 = nq ns0 nS s0n s0 = n
n= min
s Z ; s > 0 ()s =1
() =1 1 < n ()r = ()s 0r < s < n ()r = ()s 0r < s 0.
b= 0 c= 0 b2 + c2 > 0
a(x2 +y2) + bx + cy + d= 0, b2 + c2 4ad > 0.
a = 0 a
=0
b
2a,
c
2a
R=
b2 + c2 4ad2|a|
.
1
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C
(x, y) R2 z= x +iy z= x iy x y z z
x= Re(z) =z+ z
2 , y= Im(z) =
z z
2i =
iz iz
2 .
x y
C z z
x
y
azz+(b ic)
2 z+
(b + ic)
2 z+ d= 0, b2 + c2 4ad > 0.
A|z|2 + Bz+ B z+ D= 0, |B|2 AD > 0,
A = a D = d B = 21(b ic)
R2
C
A|z|2 + Bz+ B z+ D= 0,
A, D R BC |B|2 AD > 0 A=0 A= 0 C
A= 0
B + B
2A ,
B B
2iA
, R=
|B|2 AD
|A| .
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|B|2 AD > 0
|B|2 AD 0
|B|2 AD = 0
A=0 |B|2 AD < 0
(x, y)
z1 = i z2 = 1
Bz+B z+ D = 0 B C D R |B|= 0
B= a + bi
(a + bi)z+ (a bi)z+ D= 0.
z1 = i z2 = 1
(a + bi)i + (a bi)(i) + D= 0 2b + D= 0(a + bi)1 + (a bi)1 + D= 0 2a + D= 0.
a = b D = 2b B = b+ bi b= 0 b= 1 B= 1+i D= 2
(1 + i)z+ (1 i)z+ 2= 0.
b
b= 0
C
w1 =w1+ w
1 i w2 =w
2+ w
2 i
det
y w 1 x w
1
w 2 w1 w
2w
1
= 0.
x y
i(w1 w2) z i(w1 w2) z+ 2(w1w
2 w
1w
2) =0.
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C
f : S C S C
r > 0
=0
C T(z) =z+
T C
w z w = T(z) =z+
z= w T C T
z= x +yi = a + bi T(z) = u+vi
u+vi= z+ = (x + a) + (y + b)i u= x + a v= y + b. R
2
T (x, y)
(a, b)
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C
1
z =w
1
z =w
1
|z|2 =|w|2
A |w|2 + B w + B w + D =0,
A = D B = B D = A
w
z= 0 w = J(z) = 1z
|B|2 AD= |B |2 A D .
A D
J
A= 0 D = 0 B= 0 C O J C\{O}
A = 0 D= 0 B= 0
O
J C\{O}
C
A = 0 D = 0 B= 0 O \{O} J \{O}
c C (c, 0) c y (0, 0)
|z c|= c |z c|2 =c2 c2 = (z c)(z c) =|z|2 cz cz+ c2 |z|2 cz cz= 0.
J w= J(z) =1
z w= u+vi
1 cw cw= 0 c(w +w) =1 c(2u) =1 u= 12c
.
J C O= (0, 0) u= 12c
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C\{O}
xy
u= 12c
uv
c
y = c C z z = 2iy = 2ci
iz+ iz 2c= 0
J w= J(z) = 1
z w= u+vi
iw + iw 2c|w|2 =0 |w|2 + i2c
w i
2cw= 0.
u2 +v2 i2c
(w w) =0
u2 +v2
i
2c2iv= 0
u2 +v2 + vc
=0
u2 +
v + 1
2c
2=
1
2c
2.
J C
0, 12c
12c u
y= c xy u2 +
v+ 12c
2=
1
2c
2
uv
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T(z) = z+ C
= 0
S1 ={z C ; |z|= 1} J
S1
S1
D = {zC ; |z|1} (D) ={
zC ; |
z|< 1
} (
D) = {
zC ; |
z|> 1
}
J (D)\{0} (D)\{0} (D)
f(z) = az+ b
cz+ d, a, b, c, d C ad bc=0.
T(z) =z+ C a = 1 b = c = 0 d= 1
=0 H(z) =z C\{0} a=
b= 0 c= 0 d= 1
J(z) =
1
z
a= 0, b= 1, c= 1
d= 0
a,b,c,d
az+ b
cz+ d =
az+ b
cz+ d
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ad bc= 0 f
D(f) f
D(f) = cz+ d = 0 z C
c= d = 0 ad bc = 0
z, z D(f) f(z) = f(z)
(adbc)z = (adbc)z f
ad bc=0 f f
c= 0 a=0 d=0
f(z) =az+ b
d =z+ , nde =
a
d=0 = b
d.
D(f) = C f
c=0 D(f) = C \ { dc }
f(z) = z D(f)
(a c)z = d b = ac
f C \ { ac }
C
C C
C = C {}.
f(z) =
az+ b
cz+ d
ad bc= 0 c= 0 C
f
d
c
= .
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c= 0 f(z) =z+ = ad = bd
f= T H.
c=0
f(z) =ac z+
bc
z+ dc=
ac (z+
dc )
ac dc + bc
z+ dc=
a
c +
bcadc2
z+ dc.
= adbcc2
= dc = a
c
f= T
H
J
T.
c= 0 f f(z) =z+
=0
w C w = z+ w = z z= 1(w ) f
f1(z) = 1
z
, f1() = .
c=0 w= az+ bcz+ d
z= dc
z
w
z= dw + b
cw a , w= a
c.
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|z|= 1
M f : C C GL(2,C) 2
: AGL(2,C) M
A=
a b
c d
(A) =fA, fA(z) = az+ b
cz+ d.
(A B) =(A) (B) (A1) = ((A))1
(A) =(B) B= A
C\{0}
M
2(fA) = tr2(A)
det(A) fA= A
| det(A)|
tr(A) =a + d A=
|a|2 +|b|2 +|c|2 +|d|2.
z0 C
f
f(z0) =z0
C
C
1 1 J
f f(z) =z+ ,
C
=0
T H
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f(z) = az+bcz+d c= 0
f() = ac= f f f() = c = 0 f
f f() = f(z) =z+ , C =0
f
f= Id C
Id
C
f(z) = az+bcz+d ad bc= 0 f C
az+ b
cz+ d =z.
c = 0 f (a d)z= b C
a= d
b= 0
f= Id
a= d b=0 z= bda a=d
c=0 f() = ac f
f C f
dc
=
z= dc f
az+ b= (cz+ d)z
cz2 + (d a)z b= 0,
C
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S2 (0,0,0) 1 R3
S2 ={(x,y,z) R3 ; x2 +y2 + z2 =1}.
N= (0,0,1)S2 x +yi C (x,y,0) R3 R
3 z= 0
z
R3
z= 0
z > 0
N
z < 0
C N v P(v) z
v S2\{N}
N
v
R3
(av, bv, 0)
P : S2\{N} C P(v) =av+ bvi
P
N v S2\{N}
v N
N v= (x,y,z)
Q
Q = N + t(v N)
= (0,0,1) + t(x,y,z 1)
= (tx,ty,1 + t(z 1)), tR.
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2w
|w|2
+ 1
P1(w) =
2w
|w|2 + 1,
2w
|w|2 + 1,|w|2 1
|w|2 + 1
, w= w +w i.
v N = (0,0,1) z 1 |w|= |P(v)| P(N) = P
S2
P : S2 C P : S2 C C
C
P
S2
N C S2 N
C {} S2 S2 R3 S2
ax + by + cz+ d= 0 n= (a,b,c)= (0,0,0) O R3
d(O, ) = |d|a2 + b2 + c2
.
S
2
d(O, ) 1
d(O, ) =1
ax + by + cz+ d= 0 C
w= w +w i (x,y,z)
a 2w
|w|2 + 1+ b
2w
|w|2 + 1+ c
|w|2 1
|w|2 + 1+ d= 0.
w
w
a w +w|w|2 + 1
+ biw iw|w|2 + 1
+ c|w|2
1|w|2 + 1
+ d= 0.
A|w|2 + B w + B w + D= 0,
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A= c + d, B= a bi, D= d c. C
a,b,c d
A = c +d = 0
ax + by + cz+ d= 0
N= (0,0,1)
|B|2 AD= a2 +b2 +c2 d2 >
0 d(O, ) < 1
S2
|B|2 AD = a2 +
b2 +c2 d2 = 0 d(O, ) = 1
S2
|B|2 AD = a2 +
b2 +c2 d2 < 0 d(O, ) > 1
S2
P1 : C S2
C S2 N C
{} S2 N
C
S2
P C S2
S2
z= c 1 < c < 1
P(C) c 1 P(C) c 1
P
C
S2
S
2
x=
c
1 < c < 1
P(C) c 1 P(C) c 1 P1 : C S2
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C S2 N
C {} S2 N C
P
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C
F[x]
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C
Z Q R C
a A
a A a a =1 a a a1
1 1 1 1
1 1 Z
1 1
R
Z[
2] ={a+b
2; a, b Z}.
Z[
2]
(a+b
2) + (a +b
2) = (a+a ) + (b+b )
2
(a+b
2) (a +b
2) =
(a a +2b b ) + (a b +b a )
2
Z[
2] a+a , b+b Z a a +2b b , a b +b a Z
Z[
2] Z[
2]
R
0= 0+0
2 Z[
2] 1= 1+0
2 Z[
2]
a+b
2 R
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C
ab
2= (a) + (b)
2 Z[
2]
a, b Z Z[
2] R
B A A
A
Z Q R C B A B A C
B A A
0 B 1 B
a, b B a+b B a, b B a b B
a B a B
Z
Z[
2]R
C Z Z[
2]
Z[2] R C A C Z 0
1 Z
A C
C
K L
L L
Q R C R C Q R Q C
Z[i] ={a + bi; a, b Z } C
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j f(x) a0
A[x]
A
A[x] ={a0+a1x+ +anxn ; aj A, 0 j n, n N} f(x) = a0 f(x) = 0
f(x)
f(x) =0 + 0x+ + 0xn n N ajx
j aj = 0
f(x) j x
j
R[x] f(x) = 12 +x+
2x2 +2x3 g(x) =
2
3x+x3 25 x5
Z[x] h(x) = x+ 3x2 3x4 r(x) = 3+
2x+x2 s(x) =2x+3x3 3x5 t(x) =2x+3x2 3x4
f(x) = a0 + a1x+ +anxn A[x] f(x) = a0+ a1x+ +anxn +0xn+1 +0xn+2 + +0xm m > n
f(
x), g
(x
) A
[x]
x
f(x) =a0+ a1x1 + a2x
2 + + anxn g(x) =b0+ b1x
1 +b2x2 + +bnxn A[x]
aj = bj 0 j n f(x) =g(x)
h(x) = x + 3x2 3x4 t(x) = 2 x + 3x2 3x4
h(x) =t(x)
f(x) =2x4 + x5 +4x2 3 x g(x) = 3 +4x2
x+x5 +2x4 Z[x] aj j
xj a0 = 3 a1 = 1 a2 = 4 a3 = 0 a4 =2 a5 =1
x
f(x) g(x)
f(x) =g(x) = 3x+4x2 +2x4 +x5
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2x2 6x 1
(+) 3x
3 + 5x
2 3x
+ 2 3x3 + 7x2 9x + 1
g(x) +h(x) = 3x3 +7x2 9x+1
f(x) =0 g(x) =0 f(x) +g(x) =0
gr(f(x) +g(x)) max{ gr(f(x)), gr(g(x))} gr(f(x)) = gr(g(x))
f(x) =
nj=0
ajxj
g(x) =
mj=0
bjxj
A[x]
f(x) g(x) =n+mj=0
cjxj
c0 =a0 b0c1 =a0 b1+a1 b0c2 =a0 b2+a1 b1+a2 b0
cj =a0 bj+a1 bj1+ +aj b0=
+=j
a b
cn+m =an bm.
j, k N xj xk =xj+k f(x) =a g(x) =b0+b1x+ +bmxm
f(x) g(x) =a g(x) =a
mk=0
bkxk
=
mk=0
(a bk)xk
= (a
b0) + (a
b1)x+
+ (a
bm)xm,
a0 =a n= 0 cj = a0 bj = a bj j N A[x]
A
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A[x]
f(x)
g(x)
h(x)
A[x]
(f(x) +g(x)) +h(x) =f(x) + (g(x) + h(x))
(f(x) g(x)) h(x) =f(x) (g(x) h(x)) f(x) +g(x) =g(x) +f(x)
f(x) g(x) =g(x) f(x) f(x) (g(x) +h(x)) =f(x) g(x) +f(x) h(x) f(x) =
0+f(x) f(x) A[x] f(x) =a0+ a1x + + anxn f(x)
f(x) = (a0) + (a1)x+ + (an)xn 1
1 f(x) =f(x) f(x) A[x]
f(x) =
nj=0
ajxj
g(x) =
mj=0
bjxj
h(x) =
j=0
cjxj
A[x]
n=m=
f(x) g(x) h(x) x
(f(x) +g(x)) + h(x) (1)
=
nj=0
(aj+bj)xj +
nj=0
cjxj
(2)=
nj=0
(aj+ bj) +cj
xj
(3)=
nj=0
aj+ (bj+ cj)
xj
(4)=
nj=0
ajxj +
nj=0
(bj+cj)xj
(5)= f(x) + (g(x) +h(x)),
A[x]
A
A[x]
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f(x) +g(x) +h(x) =4x2 5x+6
f(x) = 2x3 +3x2 4x + 3 g(x) = x2 +2x + 3
Z[x] f(x) g(x)
f(x) g(x) = (2x3 +3x2 4x+3) (x2 +2x+3)
=2x3 (x2 +2x+3) + 3x2 (x2 +2x+3)+(4x) (x2 +2x+3) +3 (x2 +2x+3)
= (2x5 +4x4 +6x3) + (3x4 +6x3 +9x2)+(4x3 8x2 12x) + (3x2 +6x+9)
=2x5 + (4 + 3)x4 + (6 + 6 4)x3 + (9 8 + 3)x2 + (12 + 6)x + 9
=2x5 +7x4 +8x3 +4x2 6x+9 Z[x]
g(x) = x2 +2x + 3 h(x) = 2x3 x + 2 Z[x]
h(x) g(x) h(x) g(x) x
g(x)
h(x) x
x
2x3 + 0x2 x + 2
() 2x2 + 2x + 3 6x3 + 0x2 3x + 6
4x4 + 0x3 2x2 + 4x 4x5 + 0x4 2x3 + 4x2
4x5 4x4 8x3 + 2x2 + x + 6
3(2x3 x + 2) 2x (2x3 x+2) 2x2 (2x3 x+2)
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gr(f(x) g(x)) =5 = gr(f(x)) + gr(g(x))
gr(h(x) g(x)) =5 = gr(h(x)) + gr(g(x))
A[x]
f(x) g(x) A[x] A
f(x) g(x) an bm
f(x) g(x) an bm an bm
f(x) g(x)
f(x
) g
(x
)
A[x]
gr(f(x) g(x)) = gr(f(x)) + gr(g(x))
A
f(x) g(x)
x4 +4Z[x]
f(x) =x2 +ax+b g(x) =x2 +cx+d Z[x]
x4 +4 = f(x)g(x) = (x2 +ax+b)(x2 +cx+d)
= x4 + (a+c)x3 + (d+ac+b)x2 + (ad+bc)x+bd.
a+c= 0
d+ac+b= 0
ad+bc= 0
bd= 4
b
d
b = 1 d = 4 b = 2 d = 2 b = 4 d = 1 b = 1
d= 4 b= 2 d= 2 b= 4 d= 1
a= c d + b= c2
b = 2 d = 2 c = 2
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c = 2 a = 2 a = 2
f(x) =x2
2x+2
g(x) =x2
+2x+2
x2 +4= (x2 2x+2)(x2 +2x+2).
A[x] A
B B = {b B ; b B} A A[x] =A
a
A
ab =0 b A\ {0} f(x), g(x) A[x]\ {0} f(x) g(x) gr(f(x)g(x)) = gr(f(x)) + gr(g(x))
f(x) = 2x3 x2 +x+1
g(x) =3x3 + 12 x2 x2 Q[x]
a b c d
R[x]
(a2 5)x3 + (2b)x2 + (3c2)x+ (d+3) =0
3ax6
2bx5
3c2
x4
+d3
=x5
x4
+2
ax2 +bx+c= (axd)2
(b+d)x4 + (d+a)x3 + (ac)x2 + (c+b) =4x4 +2x3 +2
f(x) R[x] a R f(x) = (a2 1)x2 + (a2 4a+3)x+ (a+2)
f(x) = (a3 4a)x3 +a(a+2)x2 + (a2)3
a Z x4 +ax3 +7x2 ax+ 1
Z[x]
f(x) =2x+1 g(x) =2x2 +3 h(x) =3x+2 Z4[x]
f(x) g(x) f(x) h(x) Z4[x]
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A A[x1] A
x1 x2 A[x1]
A[x1, x2] =
A[x1]
[x2].
n
A[x1, x2, . . . , xn] =
A[x1, x2, . . . , xn1]
[xn].
n f(x1, . . . , xn) A[x1, . . . , xn]
f(x1, . . . , xn) =
0 j1 s1
0 jn sn
aj1,...,jnxj11 xjnn,
s1, . . . sn N aj1,...,jn A aj1,...,jnx
j11 xjnn
j1+ +jn aj1,...,jn=0 n
A
Q[x1, x2, x3]
f(x1, x2, x3) =x1x2 14 x1x3+ x
22 x
21
g(x1, x2, x3) = 13 +x1 2x3+ x1x2
25 x
21+ 3x1x2x
232x
42x3+ x
53
h(x1, x2, x3) =2+x1x3 34 x1x2x3+4x
323x1x3x
32+ x
52+
12 x
32x
33
gr(f(x1, x2, x3)) =2, gr(g(x1, x2, x3)) =5 gr(h(x1, x2, x3)) =6.
m
m
n
m
m
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f(x1, x2, x3)
2
h(x1, x2, x3)
0 2
2 x1x3
3 34 x1x2x3+ 4x32
5 3x1x3x32+x
52
6 12 x32x
33
A A[x1]
A[x1, . . . , xn]
Q[x1, x2, x3]
x21x22+ x
43+x
21x2 2x1x2+x
22x
23+7x1
23 x2
3x21x322x1x
43+x
21x2 2x1x2x32x
22x3+x
32x3+ 4x
41x2
Q[x, y] Q[x]
[y]
2x5y+x3y2 +2xy+3x2y2 5xy2 +5x+3y+2
2x3y3x2y2 +2xy+3x2y2 +4x3y2 +5xy2 +3x2y+2x4
F[x]
a,b,c Z a =0 a b= a c b= c a b= a c 0= a b a c= a(b c) a =0
b c = 0 b = c 1 1
Z
Q =a
b ; a, b Z b =0
a
b =
a
b a b =b a
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F[x]
a
b Q a b
Q
a
b+
c
d =
a d+b cb d
a
b c
d =
a cb d
a
b =
a
b
c
d =
c
d
a d+b cb
d =
a d +b c b
d
a cb
d =
a c b
d
Q Z
Z 1
Q Z L
Z Q L Q Z
F[x] F
F[x] f(x) g(x) = 0 f(x) = 0 g(x) = 0 F[x] F
F[x]
f(x), g(x), h(x) F[x] f(x)= 0 f(x)g(x) = f(x)h(x) g(x) =h(x)
f(x) g(x) =f(x) h(x) f(x) =0
0= f(x) g(x) f(x) h(x) =f(x)g(x) h(x), g(x) h(x) =0 g(x) =h(x)
F(x) = f(x)
g(x) ; f(x), g(x)
F[x] g(x)
=0
f(x)
g(x) =
f (x)
g (x) f(x) g (x) =g(x) f (x)
F(x)
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f(x) g(x) A[x] h(x) A[x] f(x) =g(x) h(x) f(x) g(x) g(x)= 0 g(x) f(x)
x2 2x + 2 x4 +4 Z[x] x2 + 2x + 2
x4 +4
f(x) =0 f(x) 0
A[x]
A f(x) g(x) A[x]\{0} g(x) f(x) gr(g(x))
gr(f(x))
g(x) f(x)
h(x) A[x]\{0} f(x) =g(x)h(x)
gr(f(x)) = gr(g(x)h(x))
= gr(g(x)) + gr(h(x)) gr(g(x)).
A[x]
Z
1
1
Q R C
A f(x), g(x) A[x] g(x) =0 A q(x) r(x) A[x]
f(x) =q(x)g(x) +r(x)
r(x) =0 gr(r(x))< gr(g(x))
g(x) = b0+ b1x+ +bmxm bm b
1m A
f(x) =0 q(x) =r(x) =0
f(x) =0 n= gr(f(x)) f(x) =a0+ a1x + +anxn an=0
n < m q(x) =0 r(x) =f(x)
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g(x)
f(x)
f(x)
g(x)
q(x) r(x)
f(x) g(x)
f(x) g(x)
q(x)
r(x)
f(x) =2x+5 g(x) =x2 +2x+4 Z[x]
gr(f(x)) =1 < 2 = gr(g(x))
q(x) =0 r(x) =f(x) =2x+5
2x + 5 x2 + 2x + 4
0 0
2x + 5
f(x) =2x2 +3x+3 g(x) =x2 +2x+2 Q[x]
f(x) 2x2
g(x) x2 2x2 x2 q1(x) =2
r1(x) =f(x) q1(x)g(x) = (2x2 +3x+3) 2x2 4x4= x1
2x2 + 3x + 3 x2 + 2x + 2
2x2 4x 4 2
x 1
1 = gr(r1(x)) < gr(g(x)) = 2
q(x) =q1(x) =2 r(x) =r1(x) = x1
f(x) =3x4 + 5x3 + 2x2 + x 3
g(x) =x2 +2x+1 Z[x]
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0= f() =q()() +r= r
x
f(x)
x f(x) q(x) A[x] f(x) =q(x)(x)
f() =q()() =q() 0= 0.
x
f(x) A[x] x A
f(x) = anxn +an1x
n1 + +a1x+ a0 A[x] an= 0 A q(x) A[x] r A f(x) x
f(x) =q(x)(x) +r gr(q(x)) =n1
q(x) =qn1xn1 +qn2x
n2 + +q1x+q0 f(x) = (qn1x
n1 +qn2xn2 + +q1x+q0)(x) +r
= qn1xn + (qn2qn1)x
n1 + + (q0 q1)x+(rq0),
f(x)
qn1=anqn2qn1 =an1qn3qn2 =an2
q1q2=a2q0q1=a1rq0=a0
qn1 = anqn2 = an1+qn1qn3 = an2+qn2
q1 =a2+ q2q0 =a1+ q1r= a0+ q0
q(x)
qn1 = an
q(x) r
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f(x) = 4x4 +5x2 7x+2 Q[x]
x 1
2
f(x)
2
f(x) x 12
12 4 0 5 7 2
4 2 6 4 |0
12 f(x) f(x) =
x 12
q(x) q(x) =4x3+2x2+6x4
12 q(x) q(x) x
12
= 12
q(x)
12 4 0 5 7 2
12 4 2 6 4 |0
4 4 8 |0
12 q(x) q(x) =
x 12
(4x2+4x+8) f(x) =
x 12
2(4x2+
4x+8)
12 q2(x) = 4x
2 +4x+ 8
12 q2(x)
12 4 0 5 7 2
12 4 2 6 4 |0
12 4 4 8 |0
4 6 |11
x 12 q2(x)
f(x) =
x 122
(4x2 +4x+8)
x 12
2 f(x) x
12
3 f(x)
A f(x) A[x] m (x )m f(x) (x )m+1 f(x) A[x]
q(x) A[x] f(x) = (x)mq(x) q() =0
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f(x) x2
f(x) =x5 +7x4 +16x3 +8x2 16x16 Q[x] 2 f(x)
f(x) x+2
A f(x) A[x] A f(x) m q(x) A[x]
f(x) = (x)mq(x) q() =0
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A f(x) A[x]\{0}
f(x) n f(x) n A
n= gr(f(x))
n= 0 f(x) =a =0 A
n 0 n f(x) gr(f(x)) =n+1
f(x) A f(x)
A x f(x) A[x]
q(x) A[x] f(x) =q(x)(x) gr(q(x)) =n
q(x) n A
A f(x) 0= f() =q()()() q() =0 = 0 q(x) =
A f(x)
n+1 A
x2 2 Q[x] Q x2 2 R[x]
2
2 x2 + 1
Q[x] R[x] x2 +1
i i C
n = cos 2n + i sen2
n
k k= 0, . . . , n 1
n C p(x) =xn 1
p(x) 1
x1 p(x)
xn 1= (x1)(xn1 +xn2 + +x+1). 1
xn1 +xn2 + +x+1.
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A
f(x)
g(x)
A[x]
f() = g()
A f(x) =g(x) h(x) = f(x) g(x) A h() =f() g() = 0 0 = h(x) = f(x) g(x)
f(x) =g(x)
p
Zp = { 0, 1, . . . , p1 } p Zp[x] Zp
n 1 fn(x) = xpn
x Zp[x] fn() =
0
Zp Zp
F
f(x) ax + b a =0 F[x]
ax+b f(x) f(x) F
n N n 2 n
= cos 2n +i sen2
n
1+x+x2 + +xn1 = (x)(x2) (xn1).
x = 1
sen
n sen2
n sen(n1)
n =
n
2n1.
0 1 + i 1 i
2 2 1 1
R[x] C[x]
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F f(x) F[x]\F f(x)
F[x]
f(x) = g(x) h(x) g(x), h(x) F[x] f(x) g(x)
f(x)
F[x]
f(x) F[x]
g(x), h(x) F[x] f(x) =g(x)h(x) 0
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f(x) =anxn + an1x
n1 + +a1x + a0 F[x] gr(f(x)) =n 1 a F \ {0} f(x) af(x)
f(x) =an(xn + (an)
1an1xn1 + + (an)1a1x+ (an)1a0
p(x)
)
f(x) p(x)
F[x]
f(x) R[x] R[x]
f(x) =anxn + + a1x + a0 C[x] f(x)
f(x) =anxn + +a1x+a0
aj aj j= 0, . . . , n
f(x), g(x), h(x) C[x]
f(x) =g(x) +h(x) f(x) =g(x) +h(x)
f(x) =g(x) h(x) f(x) =g(x) h(x) f(x) =f(x) f(x) R[x] C f() =f()
f(x) =
sj=0
ajxj
g(x) =
nj=0
bjxj
h(x) =
mj=0
cjxj
f(x) = g(x) h(x) aj =
+=j
bc j= 0, . . . , s
aj = +=j
bc= +=j
bc, j= 0, . . . , s,
g(x) h(x) =
f(x)
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f(x) =f(x) aj =aj, j= 0, . . . , s aj R, j= 0, . . . , s f(x) R[x]. C C
f() =
nj=0
ajj
=
nj=0
ajj =
nj=0
ajj =f().
C f(x)
C[x] m f(x) m
C f(x) C[x] m
f(x) = (x)mq(x) q()= 0 f(x) = (x)mq(x) q() = q()= 0 = 0 f(x) m
f(x) R[x] C f(x) m f(x) m
f(x)R[x] f(x) =f(x)
f(x) R[x]
R[x]
f(x) R[x] C R f() = 0 = f() = 0 f() = 0
R[x] x a a R x2 + bx + c b2 4c < 0 f(x) R[x] gr(f(x))> 2 R[x]
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x a a F
F
2 F
F[x] F x2 +bx + c
R[x] x2 + bx + c R b2 4c < 0
f(x) R[x] gr(f(x))> 2 C f(x) R x f(x) R[x] f(x) R[x]
C \ R = f(x) (x)(x) f(x) C[x]
(x)(x) = x2 (+)x+
= x2 2Re()x+| |2 R[x],
x2 2Re()x+ | |2 f(x) R[x] f(x) R[x]
R[x] f(x) n 1
f(x) =a(x1)r1 (xt)rtp1(x)n1 ps(x)ns
a R \ {0} f(x) 1, . . . , t f(x) pj(x) =x
2 +bjx+cj
bj2 4cj < 0 j = 1, . . . , s r1 rt n1
ns r1+ +rt+2n1+ +2ns = n
x4 2 = (x2
2)(x2 +
2) = (x 4
2)(x+ 4
2)(x2 +
2)
R[x]
R[x]
f(x) =x4
+1
C f(x) 1 4 +1 = 0 4 = 1
1 1 k=+2k
4 = (2k+1)
4 , k= 0, 1, 2, 3 = 4
|1|= 4
1= 1
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0 = 4
z0 = cos
4 +i sen
4 =
2
2 +
22 i
1 = 34 z1 = cos 34 +i sen 34 =
22 +
22 i
2 = 5
4 z2 = cos 54 +i sen 54 = 22 22 i3 =
74 z3 = cos 74 +i sen 74 = 22 22 i
x4 +1 = (xz0)(xz1)(xz2)(xz3) C[x]
z0 =z3 z1 = z2 z0 z3
(xz0)(xz0) =x2
2x+1 z1 z2
(xz1)(xz1) =x2 +
2x+1
x4 +1 = (x2 +
2x+1)(x2
2x+1)
R[x]
F f(x) F[x]\F f(x) af(x) a F\{0} f(x) F[x] 2 3 F f(x) F[x] f(x)
F
f(x) 2 3 F[x]
F
f(x) F[x]
gr(f(x)) =4 f(x) F f(x) F[x]
Q[x]
x2 2x1 x2 +x+1 x3 2
F
F[x]
f(x) 3
1 1i f(x)
f(x)
R[x] C[x]
g(x) 4
2 3+4i 2+i f(x)
g(x)
R[x] C[x]
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3+i
2
3+i 2
3 + i
2 3 + i
2
R[x]
x4 +5x2 +6 x4 +6x2 +9 x4 +x2 +1
f(x) =x5 5x4 +7x3 2x2 +4x8
f(x)
R[x] C[x]
f(x) C
f(x) =x5 +7x4 +16x3 +8x2 16x16
f(x)
R[x] C[x]
f(x) C
C[x] R[x]
x6 16
x6 +16
x8 1
x8 +1
f(x) =x6 4x5 +15x4 24x3 +39x2 20x+25 1+2i
f(x)
f(x) f(x)
R[x]
f(x) A[x]\F A f(x)
g(x)h(x) f(x) f(x) g(x)
h(x)
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f(x)
f(x) = g(x)h(x) g(x) h(x) A[x] f(x) g(x)h(x)
f(x) g(x) f(x) h(x) f(x)
g(x) g(x) =f(x)q(x) q(x) A[x] f(x) =g(x)h(x) =f(x)q(x)h(x)
f(x) A
1 = q(x)h(x) h(x) =a =0 a F f(x)
A
f(x), g(x), d(x) F[x] F
I(f(x), g(x)) ={ a(x)f(x) +b(x)g(x) ; a(x), b(x) F[x] }
I(d(x)) ={ c(x)d(x) ; c(x) F[x] }.
I(d(x)) =I(d(x), 0) h(x) I(f(x), g(x)) ah(x) I(f(x), g(x)) a F\ {0} f(x), g(x) F[x] F
d(x) I(f(x), g(x)) I(f(x), g(x)) =I(d(x))
S= { gr(h(x)) ; h(x) I(f(x), g(x)) h(x) =0 } S f(x) = 1 f(x) +0 g(x) I(f(x), g(x)) g(x) =0 f(x) + 1 g(x)I(f(x), g(x)) I(f(x), g(x)) SN S s d(x) I(f(x), g(x)) d(x) =0
s= gr(d(x))
d(x) I(f(x), g(x)) a0(x), b0(x) F[x]
d(x) =a0(x)f(x) +b0(x)g(x).
c(x) F[x]
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c(x)d(x) = (c(x)a0(x))f(x) + (c(x)b0(x))g(x) I(f(x), g(x)) I(d(x)) I(f(x), g(x))
h(x) I(f(x), g(x)) h(x)
d(x) q(x), r(x) F[x] h(x) =q(x)d(x) +r(x)
r(x) =0 gr(r(x))< gr(d(x))
r(x) =h(x) q(x)d(x).
h(x) I(f(x), g(x)) a(x), b(x) F[x] h(x) = a(x)f(x) + b(x)g(x)
r(x) = a(x)f(x) +b(x)g(x) q(x)(a0(x)f(x) +b0(x)g(x))= (a(x) q(x)a0(x))f(x) + (b(x) q(x)b0(x))g(x).
r(x) I(f(x), g(x)) d(x)
I(f(x), g(x)) r(x) =0
h(x) =q(x)d(x) I(d(x)) I(f(x), g(x)) I(d(x))
d(x) I(f(x), g(x)) = I(d(x))
I(f(x), g(x))
f(x) g(x) I(d(x)) f(x) = c1(x)d(x)
g(x) = c2(x)d(x) c1(x), c2(x) F[x] d(x)
f(x) g(x)
d(x) I(f(x), g(x)) a0(x), b0(x) F[x] d(x) =a0(x)f(x) +b0(x)g(x)
h(x) f(x) g(x) h(x)
a0(x)f(x) +b0(x)g(x) =d(x)
f(x) F[x]\F F f(x) f(x)
f(x)F[x]\F g(x) h(x) F[x] f(x) g(x)h(x) f(x) g(x) f(x) h(x)
I(f(x), g(x))
d(x)= 0 F[x] I(d(x)) = I(f(x), g(x)) d(x)
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f(x) a F \ {0}
d(x) =a
d(x) =af(x)
d(x) g(x) f(x) g(x) d(x) =a
a = d(x) =a0(x)f(x) + b0(x)g(x) a0(x) b0(x)
F[x] h(x)a1
h(x) =a0(x)f(x)h(x)a1 +b0(x)g(x)h(x)a
1
f(x) f(x)
h(x)
p(x)
p(x) p(x) 1 p(x) p(x)
F
f(x) F[x]
gr(f(x)) 1 p1(x) ps(x) a F\{0} n1 1 ns 1
f(x) =ap1(x)n1 ps(x)ns
p1(x) pm(x)
f(x) =ap1(x) pm(x)
p1(x) ps(x)
s m
n= gr(f(x))
gr(f(x)) =1 f(x) =ax + b= a(x+a1b) a, b F a =0 gr(f(x)) = n 2
F[x] n
f(x) f(x) =anxn + +a1x+a0 f(x)
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f(x) =an(xn + +a1n a1x+a1n a0)
p1(x)
f(x) g(x)
h(x) F[x]
f(x) =g(x)h(x)
1 gr(g(x)), gr(h(x))< n= gr(f(x)) g(x) =bp1(x) pr(x) p1(x), . . . , pr(x) b F\{0}
h(x) = cpr+1(x) pr+(x) pr+1(x), . . . , pr+(x) c F\{0}
f(x) = bp1(x)
pr(x)
cpr+1(x)
pr+(x)
= a p1(x) pr(x) pr+1(x) pr+(x),
a= b c F\{0} p1(x), . . . , pr+(x)
f(x) =a p1(x) pm(x) =b q1(x) qr(x),
a, b F\{0} p1(x), . . . , pm(x), q1(x), . . . , qr(x) a= f(x) =b
p1(x) pm(x) =q1(x) qr(x)
p1(x)
p1(x) q1(x) qr(x) p1(x) p1(x) qj(x) j = 1, . . . , r qj(x) = up1(x) uF\{0} u= 1 qj(x) = p1(x)
q1(x), . . . , qr(x)
p1(x) =q1(x)
m m= 1 r= 1 m > 1
p1(x)
p2(x) pm(x) =q2(x) qr(x)
m 1= r 1 m= r
pj(x) qj(x)
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F f(x), g(x) F[x] h(x), k(x) I(f(x), g(x)) h(x) +k(x) I(f(x), g(x))
(x) F[x] h(x) I(f(x), g(x)) (x)h(x) I(f(x), g(x)) h(x) I(f(x), g(x)) ah(x) I(f(x), g(x)) a F\{0} f(x), g(x)
d(x) F[x] I(f(x), g(x)) =I(d(x))
F
f1(x), . . . , fs(x), p(x) F[x] p(x) p(x) f1(x) fs(x) j= 1, . . . , s
p(x) fj(x)
f1(x), . . . , fs(x) p(x) f1(x)
fs(x)
j= 1, . . . , s aj=0 F p(x) =ajfj(x) f1(x), . . . , fs(x) p(x)
p(x) f1(x) fs(x) j= 1, . . . , s p(x) =fj(x) F F f(x) F[x] f(x) F[x] F
F f(x) = anxn +an1x
n1 + +a1x + a0 F[x] an
= 0 f(x) (f(x)) =
an+ an1x+ +a1xn1 +a0xn a0= 0 f(x) (f(x))
p xp x Zp[x]
f(x), g(x) F[x] F d(x) F[x] f(x) g(x)
d(x) f(x) g(x) d(x)
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h(x) F[x] f(x) g(x) h(x) d(x)
d(x) d (x) f(x) g(x) d(x) d (x)
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