Ma38 - Polinômios e Equações Algébricas (1)

8/10/2019 Ma38 - Polinmios e Equaes Algbricas (1)

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n


2/187

= 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


3/187

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.


4/187

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.


5/187

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.


6/187

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


9/187

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


25/187

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


29/187

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| .


30/187

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


31/187

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


32/187

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


34/187

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


35/187

(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


47/187

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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50/187

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


56/187

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


57/187

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


58/187

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

= .


59/187


60/187

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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62/187

|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


63/187

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


64/187


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66/187


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68/187

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.


69/187


70/187

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,


71/187

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


72/187

C S2 N

C {} S2 N C

P


73/187


74/187

C

F[x]


75/187

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


76/187

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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78/187

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


79/187


80/187

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


81/187

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]


82/187


83/187

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)


84/187

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


85/187

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]


86/187

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


87/187

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


88/187

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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90/187

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)


91/187


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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]


93/187


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95/187

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


96/187


97/187

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


98/187


99/187


100/187

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


101/187


102/187


103/187

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.


104/187


105/187


106/187

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]


107/187

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


108/187

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)


109/187

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]


110/187

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


111/187

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]


112/187

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)


113/187

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]


114/187

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)


115/187

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)


116/187

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)


117/187

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)


118/187

h(x) F[x] f(x) g(x) h(x) d(x)

d(x) d (x) f(x) g(x) d(x) d (x)

Documents

Ma38 - Polinômios e Equações Algébricas (1)