, = = 7 6 = 42, =
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- ともあき はなだて
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1
2 , = = 7 6 = 42, = = 25, = 7 6, /9
3 = = = = = = = =
4 i ii iii iv a, b n b = na b a a b a b a b = a = 13, b = 91, n = 7 2 i ii iii. 0 iv. 1
5 4 1 1 a b q 0 r < b r a = qb + r. r a b q = = = = = = = = a, b q, r a = qb + r i. a = 12, b = 5. ii. a = 23, b = 12.
6 = 492 4, 1968 = = 425 4, 1700 = , 900 = = 500 4, 2000 = , 2000 = i1834 ii1960 iii2100. iv3600.
7 , /27 9/28 9/ / /27 7 {}}{ 9/28 9/29 9/30 10/ =
8 = 10 = / i ii iii = =
9 = , 105 = = 8 = = , 366 = = 8 = i. 7 ii. 7 7 iii. 7 6 i ii iii iv v.
10 , = = = = = = = = m, a, b q 1, q 2 0 r 1, r 2 < m { a = q1 m + r 1, b = q 2 m + r 2. r 1 = r 2 a b mod m.
11 10 1 a b m m a b mod m a b m 7 a b mod m. a b m 10 { 31 = , 34 = mod = 65 = i. 123, ii. 23, a,a, b, b m { a a mod m, b b mod m. i. a + b a + b mod m.
12 ii. ab a b mod m = = = = = = = 12 mod = 27 1 mod { a a mod m, b b mod m. { a a = q 1 m, b b = q 2 m. { a = q 1 m + a, b = q 2 m + b. a + b a + b = q 1 m + a + q 2 m + b = q 1 + q 2 m + a + b.
13 12 1 a + b a + b mod m. a b = q 1 m+aq 2 m+b = q 1 q 2 m 2 +bq 1 m+aq 2 m+ab = q 1 q 2 m+bq 1 +aq 2 m+ab. 9 ab a b mod m. i ii a a mod m a a m q 1 a = q 1 m + a a a a = q 1 m m b b mod m q 2 b = q 2 m + b a + b = q 1 + q 2 m + a + b a + b a + b mod m {}}{}{{}}{{} b a {} {{ }} }} {{ {} q 1m a q 2 m b = q 1 q 2 m 2 m = bq 1 m m = aq 2 m m ab 1.1: ab a b mod m a b ab ab a b 3 m m m ab a b mod m
14 k n 0, n 1,, n k 9, n k 0 k + 1 a a = n 0 + n n n k 10 k. a n 0 + n 1 + n n k mod = mod 9. i 10 i 1 i = 1 mod 9. n i 10 i n i 1 mod 9. a n 0 + n 1 + n n k mod = = 8 mod i ii , 2,, a k 0 n 0, n 1,, n k 1 9, n k 1 0 a = n 0 + n n n k 1 10 k 1. a n 0 n 1 + n k 1 n k 1 mod 11.
15 , 1 p a p a p 1 1 mod p 2 p p 1! 1 mod p [ 1 ] a, b n b = na b a a b a b a b a Z b Z n Z s.t b = an. def. 5 d Z a, b Z def. d Z b Z d Z a Z 6 d Z a, b Z d Z def. a, b Z a, b Z a, b gcda, b , = , 120 = , = 3 1 3, 5
16 = = = i , 63 ii , a, b a, b 7 a, b a b a b a b 3 a, b, n Z a, b = a nb, b. a, b = d x, y Z s.t. a = dx, b = dy. a nb = dx ndy = dx ny d a nb, b a nb, b = d x, y Z s.t. a nb = d x, b = d y. a = a nb + nb = d x + nd y = d x + ny d a, b d = d d, d d d d d d = d a, b = d, a nb, b = d d d a, b = a nb, b
17 16 1 a,b Z a = q 1 b + r 1, 0 r 1 < b. a 1 := b, b 1 := r 1. a 1 = q 2 b 1 + r 2, 0 r 2 < b 1. a 2 := b 1, b 2 := r 2. a 2 = q 3 b 2 + r 3, 0 r 3 < b 2... b > r 1 > r 2 > 0 r n+1 = 0 a n 1 = q n b n 1 + r n, 0 r n < b n 1. a n := b n 1, b n := r n. a n = q n+1 b n. 3 a, b = a q 1 b, b = r 1, b = r 2, b 1 = = r n, b n 1 = r n. r n = a, b , = a = 391, b = 221, q 1 = 1, r 1 = = a 1 = b = 221, b 1 = r 1 = 170, r 2 = = a 2 = b 1 = 170, b 2 = r 2 = 51, r 3 = = , 221 = i. 48,36 ii. 1813,777 iii , i ii
18 1.12. Mathematica , Mathematica Mathematica Windows Windows >Wolfram Mathematica Mathematica > >.nb FactorInteger Shift Enter * * Mod[100,7] 2 * 2x mod 7 x=0, 1, 2, 3, 4, 5, 6 * Table[Mod[2*x, 7],{x,0,6}] {0, 2, 4, 6, 1, 3, 5} * a=, 2,, Table[Mod[2*x, 7],{x,0,6}] * Print["a=", 2," ", Table[Mod[2*x, 7],{x,0,6}]] a=2 {0, 2, 4, 6, 1, 3, 5} * * Do[Print["a=",a," ",Table[Mod[a*x, 7],{x,0,6}]], {a,0,6}]
19 18 1 a=0 {0, 0, 0, 0, 0, 0, 0} a=1 {0, 1, 2, 3, 4, 5, 6} a=2 {0, 2, 4, 6, 1, 3, 5} a=3 {0, 3, 6, 2, 5, 1, 4} a=4 {0, 4, 1, 5, 2, 6, 3} a=5 {0, 5, 3, 1, 6, 4, 2} a=6 {0, 6, 5, 4, 3, 2, 1} 1.13 Z/NZ 8 NZ Z N NZ N NZ := {an a Z} = {, 2N, N, 0, N, 2N, }. 9 Z/NZ N N N 0, 1, 2,, N 1 Z NZ = {, kn,, 2N, N, 0, N, 2N,, kn, } N kn kn 0 N a a+nz = {, a kn,, a 2N, a N, a, a+n, a+2n,, a+kn, } a a a + kn a + kn a {0, 1,, N 1, } Z/NZ Z/NZ x x Z/NZ = {0, 1, 2,, N 1}.
20 1.14. Z/NZ Z/NZ Z/NZ x + y := x + y. x y := xy. 15 N = 8 Z/8Z i = = 12 = 4. Z/8Z = ii. 5 5 = 5 5 = 25 = 1. Z/8Z = Z/13Z 0 12 i ii iii. 8 5 iv Z/NZ 16 Z/5Z 1.2:
21 : Z/6Z 18 Z/5Z Z/6Z 19 Z/NZ Z/5Z 20 Z/NZ Z/6Z
22 1.15. Z/NZ , Z/NZ Z/NZ N = 0 N = 8 8 = 0 Z/8Z 26 = 2 8 = 0 26 = = = 2 Z/NZ Z/8Z 26 = 2, 3 = = 2 + 5, 26 3 = 2 5 Z/NZ a = a, b = b a + b = a + b, ab = a b Z/NZ Z/13Z 2 < 7, 3 < < 7 6, 6 < 42 Z/13Z 42 = = 3 6 < 42 = 3 3 < 6
23 22 1 Z/NZ x,y 3x + 4y = 5 x 2 + y 2 = x, y = 2, 3 5x + 3y = 1 x, y = 1, 4 y A B x 2 y A 1 x y 1.17 ax + by = c , 221 = x + 221y = 17. x, y Z
24 1.17. ax + by = c = = = = = = = = { 170 = = = = , 221, 170, 51, , 170, 51, =
25 = = = = = , 221 = x + 221y, x, y Z 391, 221 = 221, 170 = 170, 51 = 51, = { 391 = , 221 = = { 221 = ,. 170 = = { 170 = , 51 =
26 1.17. ax + by = c 25 ax + by = c ax + by = c i. a, b = d ii. c d iii. c d d a = da, b = db, c = dc a x + b y = c iv. a x 0 + b y 0 = 1 x 0, y 0 v. x, y = x 0 c, y 0 c vi. x, y = x 0 c + b t, y 0 c a t. t ax + by = c ax + by = c i. a, b a, b ii. ax + by = c iii. ax + by = c 4 a, b d a b ax + by = 0 a = da, b = db { x = b t y = a t, t Z
27 26 1 ax + by = 0 da x + db y = 0 d a x + b y = 0 d a b a b a x = b y b a, b = 1 x b x = b t, t Z a x = b y a b t = b y y = a t. x = b t, y = a t ax + by = c x = x 0, y = y 0 ax + by = c x 0, y 0 ax + by = c ax 0 + by 0 = c ax x 0 + by y 0 = 0 x x 0 = b t, y y 0 = a t x = x 0 + b t, y = y 0 a t a, b d ax + by = c d c d x + 221y = , 221 = x + 13y = 2
28 1.17. ax + by = c = = = = = = = = = = x + 13y 23x + 13y = = = 2 x = 8, y = 14
29 28 1 a b c d 20 a a b c d 1 = a = 391x + 221y = 35 ad bc 0 1 ad bc d b c a 1 a a = a 391, 221 = x + 13y = ax + by = da x + db y = da x + b y ax + by d c d x, y = x 0, y 0 x, y = x 1, y 1 { ax1 + by 1 = c ax 0 c + by 0 c = c. ax 1 x 0 c + by 1 y 0 c = 0.
30 1.17. ax + by = c 29 a, b a, b = d a = da, b = db da x 1 x 0 c + db y 1 y 0 c = 0. d a x 1 x 0 c + b y 1 y 0 c = 0. a x 1 x 0 c = b y 1 y 0 c. a, b = 1 1 d a,b x 1 x 0 c b y 1 y 0 c a x 1 x 0 c = b t y 1 y 0 c = a t x 1, y 1 x 1, y 1 = x 0 c + b t, y 0 c a t. x, y = x 0 c + b t, y 0 c a t, x + 221y = , 221 = = = 17 2 x, y = 8, 14. t Z, t :. 391 = 17 23, 221 = t x, y = t, 14 23t.
31 i. 3x + 2y = 0 ii. 18x + 15y = 0 iii. 7x + 5y = 1. iv. 7x + 5y = 3 v. 18x + 15y = 3 vi. 18x + 15y = 2 vii. 209x + 57y = 19. viii. 209x + 57y = 17. ix. 209x + 57y = 76.
32 , ax + by = 0 d = a, b, a = da, b = db { x = b t y = a t t Z { x = bt y = at t Z ax + by = 0 d > x + 221y = = 17 23, 221 = 17 13, 34 = a = 391, b = 221, d = 17, a = 23, b = 13. { x = 221t y = 391t t Z 391x + 221y = t t = t = 0 { x = 13t y = 23t t Z t = 1 { x = 221t y = 391t t Z { x = 13 y = 23
33 32 1 { x = x0 ax + by = c y = y 0 ax + by = c x 0, y 0 d = a, b, a = da, b = db 23 { x = x1 y = y 1 ax 1 + by 1 = c ax 0 + by 0 = c ax 1 x 0 + by 1 y 0 = 0 X = x 1 x 0, Y = y 1 y 0 ax + by = 0 X = b t, Y = a t x 1 x 0 = b t, y 1 y 0 = a t { x = x0 + b t y = y 0 a t t Z 391x + 221y = = 17 23, 221 = 17 13, 34 = d = 17, a = 23, b = 13, c = 2. 23x + 13y = 1 x 0 = 4, y 0 = 7 { x = t y = 14 23t t Z
34 i. 1204, 817. ii. 2747, i. 1204x + 817y = 83. ii. 2747x + 804y = a, b d = a, b, a = da, b = db a, b, d i. a = 1204, b = 817. ii. a = 2747, b = i. 28x + 19y = 1. ii. 2747x + 804y = i. 28x + 19y = 3. ii. 2747x + 804y = i. 28x + 19y = 0. ii. 2747x + 804y = i. 28x + 19y = 3. ii. 2747x + 804y = i. 11x + 9y = 4. ii. 1909x y = 498. iii x y =
35 , Z/NZ 1 Z/15Z x = 1. Z/15Z 13 x 7 13 = 91 = = 1. Z/15Z 1 7 = Z/15Z Z/15Z = {0, 1, 2,, 14} i. 18 ii. 1 2 iii. 4 5 N Z/NZ Z/NZ Z 1 N 0 1 = 1, = 2, = 3,, N {}}{ = N = 0. Z/NZ = {0, 1, 2, 3,, N 1} Z/NZ N N 0
36 1.21. Mathematica 35 Z/NZ Z/NZ Z/NZ a,b b a ax = b. a = 0, b 0 b a Z/NZ a 0 b a 24 Z/15Z 1 7 7x = 1. Z/15Z Z/15Z 13 x 7 13 = 91 = = 1. Z/15Z 1 7 = Z/7Z Z/7Z = {0, 1, 2, 3, 4, 5, 6} Mathematica Mathematica Windows > > Mathematica
37 Mathematica file>open new Shift Enter * * Mod[100,7] 2 * 2x mod 7 x=0, 1, 2, 3, 4, 5, 6 * Table[Mod[2*x, 7],{x,0,6}] {0, 2, 4, 6, 1, 3, 5} * a=, 2,, Table[Mod[2*x, 7],{x,0,6}] * Print["a=", 2," ", Table[Mod[2*x, 7],{x,0,6}]] a=2 {0, 2, 4, 6, 1, 3, 5} * * Do[Print["a=",a," ",Table[Mod[a*x, 7],{x,0,6}]], {a,0,6}] a=0 {0, 0, 0, 0, 0, 0, 0} a=1 {0, 1, 2, 3, 4, 5, 6} a=2 {0, 2, 4, 6, 1, 3, 5} a=3 {0, 3, 6, 2, 5, 1, 4} a=4 {0, 4, 1, 5, 2, 6, 3} a=5 {0, 5, 3, 1, 6, 4, 2} a=6 {0, 6, 5, 4, 3, 2, 1}
38 1.22. Z/NZ Z/NZ a Z/NZ, a 0 1 a ax = 1. Z/NZ a a 25 Z/NZ a Z/NZ, a 0 1 a Z/15Z ax = 1 Z/NZ ax 1 mod N ax 1 N ax + Ny = 1 26 Z/13Z 1 7 7x + 13y = 1
39 = = = = = = = = Z/21Z Z/NZ N
40 , Mathematica Print Print[ ], Yn Print : : :Print["HelloYn, 2+3, " end."] : Hello 5 end. Table Table[,{n, n_min, n_max}] : n_min n_max : : Table[2^n,{n,1,5}] : {2, 4, 8, 16, 32} Do Do[,{n, n_min, n_max}] : n_min n_max : : Do[Print[n," =",Prime[n]],{n,1,5}] : 1 =2 2 =3 3 =5 4 =7 5 = x Z/NZ x {0, 1, 2,, N 1} {0, 1, 2,, N 1}
41 40 1 a, b Z Z/NZ b a ax = b Z ax + Ny = b a, N = d d = 1 ax + Ny = b b Z/NZ a d > 1 d b b Z/NZ a d b Z/NZ b a b Z/NZ a 27 Z/21Z, b a = 3 11, 21 = x + 21y = 3 x = 6, y = = 6 d > 1 b a = 2 3, 21 = 3 > 1, Z/21Z 3 b a = 3 6, 21 = 3 > 1, x + 21y = 3 x = 4, 11, 18 Z/21Z Z/21Z 6
42 , a, b Z n Z b = na b a a, b, n 9, 18 Z 2 Z 18 = a b q 0 r < b r a = qb + r. r a b q a r
43 = i. ii. iii. iv m, a, b q 1, q 2 0 r 1, r 2 < m { a = q1 m + r 1, b = q 2 m + r 2. r 1 = r 2 a b mod m. a b m m a b mod m a b m
44 1.27. Z/N Z a,a, b, b m { a a mod m, b b mod m. i. a + b a + b mod m. ii. ab a b mod m k 0 n 0, n 1,, n k 1 9, n k 1 0 k a a = n 0 + n n n k 1 10 k 1. a n 0 + n 1 + n n k 1 mod Z/N Z N Z/NZ := {0, 1, 2,, N 2, N 1} Z/NZ 1.28 Z/NZ ax + by = c.
45 a,b Z a = bq 1 + r 1, 0 r 1 < b. a 1 := b, b 1 := r 1. a 1 = b 1 q 2 + r 2, 0 r 2 < b 1. a 2 := b 1, b 2 := r 2. a 2 = b 2 q 3 + r 3, 0 r 3 < b 2... b > r 1 > r 2 > 0 r n+1 = 0 a n 1 = b n 1 q n + r n, 0 r n < b n 1. a n := b n 1, b n := r n. a n = b n q n+1. 3 a, b = a bq 1, b = r 1, b = r 2, b 1 = = r n, b n 1 = r n. r n = a, b , = = = = , 221 = 17.
46 ax + by = c = { 391 = , 221 = = { 221 = ,. 170 = = { 170 = , 51 = , 221 = 221, 170 = 170, 51 = 51, = = = = =
47 = , 221 = x+221y, x, y Z ax + by = c ax + by = c i. a, b = d ii. c d iii. c d d a x + b y = c a, b = 1 iv. a x 0 + b y 0 = 1 x 0, y 0 v. c = dc x, y = x 0 c, y 0 c vi. a = da, b = db x, y = x 0 c + b t, y 0 c a t. t ax + by = da x + db y = da x + b y ax + by d c d x + 221y = , 221 = x + 13y = 2
48 1.29. Z/NZ 47 23x + 13y = = 1 x, y = 4, 7 23x + 13y = 2 34 = x + 13y = 1 x, y = 4, 7 2 x, y = 8, = 17 23, 221 = t x, y = t, 14 23t, t Z 1.29 Z/NZ 1 Z/15Z 7 7x = 1. Z/15Z 13 x 7 13 = 91 = = 1. Z/15Z 1 7 = 13
49 48 1 Z/15Z 1 7 Z/15Z 7x = 1 7x 1 mod 15 7x + 15y = 1 Z/NZ 1 a Z/NZ ax = 1 ax 1 mod N ax + Ny = 1 Z/NZ Z/NZ a Z/NZ, a 0 1 a ax = 1. Z/NZ a a 33 Z/13Z 1 7 7x + 13y = 1 13 = = = 1 6.
50 = = = = = 2. Z/NZ a, N > 1 b a 1 a a b a b q, r0 r < b a = qb + r a, b a a = qb + r i. a = 13, b = 5 ii. a = 26, b = 7 35 i. 0 x 10 ii. 12 x 22 iii. 50 x , 5, 22, 287, 36472
51 a = , b = i. x a mod 17 x 0 x < 17 ii. y b mod 17 y 0 y < 17 iii. z a + b mod 17 z 0 z < 17 iv. w ab mod 17 w 0 w < i. 65, 26 ii. 774, Z/7Z Z/7Z = {0, 1, 2, 3, 4, 5, 6} i ii. 2 6 iii i , 21 ii. { 119 = = = iii = = =
52 iv = v = vi. 119x + 21y = x + 7y = x + 7y = x + 7y = 3, 44 13x + 7y = 3, 45 48x + 14y = x + 14y = x + 14y = x + 48y = Z/5Z {0, 1, 2, 3, 4} i. 38 ii. 1 2
53
54 , A B C D i. A B 90 ii. A C iii. A D 1/2
55 : A A 2. A B B A :A B B A A B 3. A B B C A C :A B B C A C B A 7 C
56 G i. a, b G ab G. ii. e G s.t. a G ea = ae = a. iii. a G b G s.t. ab = ba = e. iv. a, b, c G abc = abc
57 :
58 ,
59 , ,
60 n n σ, τ n S n n 1 n σ τ σ = , τ = σ σ τ σ σ τ τσ τσ = σ
61 60 2 σ σ 1 σ σ = σ1 = 3, σ2 = 1, σ3 = 2, σ4 = 4 {1, 2, 3, 4} σ τ A B C D B C A D σ = τ = τσ = σ = τ = τσ = D B C A σ, τ τσ A B C D σ, τ, τσ B C A D σ τσ A B C D B D C A τσ A B C D = τ σ A B C D = τ B C A D = D B C A
62 a, b c a b c a a b c a b c = b c a = c a b a b c a c b n n , , = = σ = σ3 = 1 3 1
63 σ2 = σ3 = σ5 = 3 σ4 = 4 σ
64 σ = , τ = i. τσ ii. στ iii. σ 1 iv. σ 1 2 v. τ , , , , ,
65 n S n n 1 2, 2 3,..., i i + 1,..., n 1 n 58 n σ i < j σi > σj 1 i, j n i, j σ lσ σ = i, j 1, 2, 1, 3, 1, 4, 1, 5, 2, 3, 4, 5 σi > σj lσ = , 2 3,..., i i + 1,..., n 1 n τ n σ τσ lτσ lσ ± 1 60 σ lσ 61 σ lσ
66 , S 4 σ = τσ , τ = σ τ A B C D B C A D σ = τ = τσ = σ = τ = τσ = D B A C S 4 σ = i. στ. ii. τσ. iii. σ 2. iv. τ , τ =
67 = = σ, τ S n στ = τσ
68 k τ τk τ τk σ τk σ στk = τk στk = στk = τk = τσk = τσk l σ l τ τl = l. στl = στl = σl = τσl = τσl m σ, τ σm = τm = m στm = σm = m = σm = τσm = τσm x στx = τσx στ = τσ 7 σ, τ S n στ r = τσ r = σ r τ r 36 σ = σ σ , σ 2 = σ 3 = σ 4 = 1 4 σ 3 = σ σ 1 =
69 i ii iii iv n n 1 2, 2 3,, k 1 k,, n 1 n Joker K, Q, J, A Joker K Q J A K Q J A K Q J A K Q J A
70 Joker K Q J A K Q J A K Q J A K Q J A
71 , a, b 1 ax = b a x = b a a 0 a 0 a a = 0 a 0 [a 0 ] x = b a. [a = 0 ] a = 0 0x = b 0 = b b b = 0 b 0 [b = 0 ] ax = b 0 = 0 x x = t, t :. [b 0 ] ax = b 0 = b 0
72 G i. a, b G ab G. ii. e G s.t. a G ea = ae = a. iii. a G b G s.t. ab = ba = e. iv. a, b, c G abc = abc. G G 17 A a B b S a b a, b a b + 38 A = B = S = Z a, b Z a + b + a, b 39 σ, τ S n τσ S n S n σ, τ τσ G a, b G a b G G, a, b G ab G G, G 40 0 R := R \ {0} = {x R x 0} R R, 41 Z Z + Z, +
73 G a, b G ab = ba. a b G a, b G, ab = ba. G G G R 2 M 2 R M 2 R, + M 2 R, + 68 M 2 R, + 43 R 2 GL 2 R 19 n n G a G, n N a n a n := n {}}{ aa a. a n a n := a 1 n. G a G, n N a n na := n {}}{ a + a + + a. a n na := n a. n n
74 G e e G, 1, 1 G G G e, e G ii e = ee = e. e ii e ii 69 iii G G G G #G G G G #G = G =. G G a 1, a 2,, a n G G a 1, a 2,, a n a G, a = b e1 1 be2 2 ber r,
75 74 2 b 1, b 2, b r {a 1, a 2,, a n }, e 1, e 2,, e r Z. G a 1, a 2,, a n a 1 a 5 2a 3 1 G = a 1, a 2,, a n. {a 1, a 2,, a n } G 46 Z, + 1 Z Z = 1. n Z n = n 1 + e i e i 1 47 n S n S n = 1 2, 2 3,..., i i + 1,..., n 1 n. n 1 2, 2 3,..., i i+1,..., n 1 n 48 n 1 n σ = 12 k n, τ = 2 n3 n 1 k n k + 2. σ τ 1 n 2 D n D n = σ, τ.
76 G G G G G G G 22 H G i. H G ii. H G 49 G G G {e G } G 50 G a G a G 51 n S n A n A n S n A n n 23 G a G G a 52 k k. 8 G e G G a G n a n = e G N S = {a 1, a 2, a 3,, a k,, a N+1 } i, j a i = a j
77 76 2 S G S G S N + 1 G a a 1 a i a 1 j = a j a 1 j a i j = e n = i j σ, τ S n στ σ τ στ r = 1 n r στ r = σ r τ r σ r = τ r = 1 n r σ τ 53 σ σ = σ σ = , 3 σ M 2 R 2 M 2 R, + 72 M 2 R GL 2 R 73 n S n
78 n 1 n σ = 12 k n, τ = 2 n3 n 1 k n k + 2. e i. σ n = e. ii. τ 2 = e. iii. τστ = σ n 2 D n D n 76 τ n σ τσ lτσ lσ ± 1 77 n σ σ sgn σ. { 1 σ sgn σ = 1 σ 78 n A n n S n 79 G G 80 n n 1 µ n µ n τ =
79
80 , [ ] Z Z 54 Z Z 55 N N 2, 5 N, 2 5 = 3 N. 56 Q[x] Q[x] 25 F field i. F ii. 0 F 57 Q 58 R 59 C
81 a N x + a = 0. x = a a Z a 0, b Z ax = b. b a a n, a n 1,, a 0 Q, a n 0 a n x n + a n 1 x n a 0 = 0. a x + a x x + a a ax x ax a 0, a 1,, a n a n x n + a n 1 x n a 0 x 3.2.2
82 x + 3 = 5. x x = 2 x + 3 = 0. x = 3 a x + a = 0. x = a Z Z
83 Z {0} := {a Z a 0}. a Z {0}, b Z ax = b. b a S T S S S T := {s S s T }. S T S T S = Z, T = {0} S T = Z {0} = 6 10 = Q a, b Z ax = b a Q a 0 a a 1 a Q Q
84 Q Q Q a n, a n 1,, a 0 Q, a n 0 a n x n + a n 1 x n a 0 = 0. Q Q 60 x 2 2 = 0. x = ± x = 0. i = 1 C := {a + bi a, b R} Z/N Z Z/NZ N 0 p Z/pZ F p N 0
85 , R R [ ] R + [ ] x, y R xy R [ ] x, y, z R xyz = xyz. [ ] x, y, z R xy + z = xy + xz, x + yz = xz + yz. R [ ] x, y R xy = yx. R + R + x, y R x + y = y + x Z Z Z 1 2 Z
86 R x R[x] := {a 0 + a 1 x + a 2 x a k x k + + a n x n a k R, n N}. R 1 R[x] 63 R { } a11 a M 2 R := 12 a 21 a a 11, a 12, a 21, a 22 R 22 R 2 M 2 R R = Z A = AB = BA = AB BA., B = = = M 2 Z Z/NZ Z/NZ N = 6 2 Z/6Z R R R R + R R R R
87 86 3 R + R R + R R + R 65 F F + = F. F F ax = 1. F a F F a 0 F F = {a F a 0}. 66 Z Z ax = 1. Z a Z a > 1 a x = 1 0 < x < 1 x Z a = ±1 Z Z = {±1}. 67 Z/pZ p Z/pZ a x = 1. Z/pZ a Z/pZ ax 1 mod p. ax + py = 1. a, p = 1.
88 p a 0. Z/pZ = {a Z/pZ a 0} = {1, 2,, p 1}. p Z/pZ 82 Z/6Z Z/6Z 68 Z/pZ Z/pZ Z/7Z Z/7Z = {1, 2, 3, 4, 5, 6} = 3, 3 2 = 6, 3 3 = 2, 3 4 = 5, 3 5 = 1, 3 6 = S = Z/7Z 3 Z/7Z = 3. p a Z/pZ Z/pZ = a. Z/pZ G n a G G 68 3 Z/7Z S 6 6 Z/7Z 6 S 6 n S n n
89 x + 8 = 3 x x = x = x = ax + b = c b a a, b R b a R R R R R 0 R R R
90 , G e G a G a k = e k a n G G n a G a n = e n k a k = e R R R R := {a R a } 70 Q = {a Q a 0}. R = {a R a 0}. C = {a C a 0}. Z = {±1}. N Z/NZ := {a Z/NZ x Z/NZ s.t. ax 1 mod N}. Q, R, C 0
91 A P P A i. A ii. P iii. A P iv. P P ε-δ 71 R R := R \ {0} = {x R x 0}. R
92 i. A R ii. P. P G i. a, b G ab G. ii. e G s.t. a G ea = ae = a. iii. a G b G s.t. ab = ba = e. iv. a, b, c G abc = abc. R R R R R 0 R iv i R 0 ii e G s.t. e e e R iii a R a 1 a 1 R i R a, b R = ab R. a, b R, a 0, b 0 ab 0 a, b R = ab R. i ii 1 0, 1 R 1 R. 1 R a R, a 1 = 1 a = a. ii
93 92 3 iii a R a 0 1 a 1 a R 1 a a a R, a 1 a = 1 a a = 1. iii iv R R iv R R 3.9 A P F n n F F S P S P A S A P 83 F F := F \ {0} = {x F x 0}. F 84 N NZ Z 85 Z/NZ = {a Z/NZ ax = 1 }. Z/NZ
94 p Z/pZ 87 Z/17Z Z/17Z {0, 1, 2, 3,, 15, 16} 88 Z/15Z Z/15Z {0, 1, 2, 3,, 13, 14} Z/NZ Z/NZ a, b Z/NZ a 1, b 1 Z/NZ ab b 1 a 1 Z/NZ abb 1 a 1 = abb 1 a 1 = a 1 a 1 = aa 1 = 1 ab Z/NZ. 1 Z/NZ 1 1 = 1 1 Z/NZ a Z/NZ a 1 Z/NZ aa 1 = a 1 a = 1 a 1 Z/NZ a 1 Z/NZ. Z/NZ Z/NZ a Z/NZ ax + Ny = 1 a a, N = p a p a p 1 1 mod p.
95 = mod 11. p a p a p 1 1 mod p. p 1 G G G e G a G 3.14 a G = e. G = Z/pZ mod mod = = 2 8 = mod = 16 1 mod 17
96 mod 131 {0, 1, 2,, 130} 3.13 G G H H G G G 5 G p G p G = Sylow G H G a, b G H h H s.t. b = ah. H a ah ah := {x G x = ah h H}. ah H a 90 3 S 3 H = {1 3, 1 2 3, 1 3 2} H
97 G H G a, b G { ah = bh ah bh =. x bh x ah bh h H x = bh ah h H x = ah bh = ah. H h h 1 bhh 1 = b = ah h 1 H h h 1 H b ah bh = ahh = ah. bh ah ah = bh ah bh ah bh = 11 G H G a G H ah H H f : H ah h ah h, h H ah = ah a 1 h = a 1 ah = a 1 ah = h
98 f ah ah fh = ah f f H, ah H ah 6 G H G G : H G H G = H G : H. a 1 H, a 2 H,, a g:h H G a i H a j H =, i j, G = a 1 H a 2 H a g:h H. a i H H G = H G : H. 1 G H G H H G G H G 91 G a G a f e G a f = e.
99 G a G a G = e. G a G a a G a G G = k a. a G = a k a = a a k = e k = e. 2 p G = Z/pZ a G a p 1 = 1. Z/pZ p Z/pqZ 2 p, q Z/pqZ p 1q 1. Z/pqZ = {a Z/pqZ x Z/pqZ s.t. ax = 1} a Z/pqZ a, pq = 1 a, p = 1 a, q = 1 {0, 1, 2,, pq 1} p q Z/pqZ p 0 p = 0, 1 p = p, 2 p = 2p,, q 1 p = q 1p
100 q q 0 q = 0, 1 q = q, 2 q = 2q,, p 1 q = p 1q p 0 Z/pqZ 74 pq p q + 1 = p 1q mod = Z/35Z Z/35Z = = mod = = 2 4 = 16 mod Z/ Z 93 p n Z/p n Z p n p n
101 , x 2 + ax + b = 0 x = α, β x 2 + ax + b = x αx β = x 2 α + βx + αβ = 0 { α + β = a αβ = b α + β, αβ 2 α β x 2 +ax+b = x 2 α+βx+αβ = x αx β = x βx α = x 2 β+αx+βα 2 α + β = a, αβ = b α, β x = α, β, γ 3 x 3 +ax 2 +bx+c = x αx βx γ = x 3 α+β+γx 2 +αβ+αγ+βγx αβγ = 0 α + β + γ = a αβ + αγ + βγ = b αβγ = c α, β γ
102 x, y fx, y fx, y x y y x fy, x fx, y = fy, x fx, y fx, y = x + y fx, y = xy fx, y = x y 2 fx, y = x y fy, x = y x = fx, y 76 2 x 2 + ax + b = x αx β = 0 { α + β = a, αβ = b. a, b α, β i. x 2 + y 2 ii. x + y 2 iii. x 2 y 2 iv. x + 2y 3 + 2x + y 3
103 n x 1, x 1,, x n fx 1, x 2,, x n n S n σ σfx 1, x 2,, x n := fx σ1, x σ2,, x σn. 77 fx 1, x 2, x 3 = x x 1 x 3 4x 2 x 3 σ = σfx 1, x 2, x 3 = fx σ1, x σ2, x σ3 = fx 2, x 3, x 1 = x 2 2+5x 2 x 1 4x 3 x 1. n S n S n 30 n fx 1, x 2,, x n n σ S n σfx 1, x 2,, x n = fx 1, x 2,, x n. n S n fx 1, x 2,, x n S n i i + 1 S n i = 1, 2,, n 1 S n S n n! i i + 1 n i. x + y + z ii. xy + xz + yz iii. xyz iv. x 2 y + y 2 z + z 2 x
104 n x 1, x 2,, x n σ k k σ 1 = x 1 + x x n. σ 2 = x 1 x 2 + x 1 x x i x j + + x n 1 x n, i < j. σ k = {x 1, x 2,, x n k } σ n = x 1 x 2 x n σ 1 = x 1 + x 2 + x 3. σ 2 = x 1 x 2 + x 1 x 3 + x 2 x 3. σ 3 = x 1 x 2 x 3. σ k 3 σ 1 = x + y + z 4 σ 1 = x + y + z + w n x + x 1 x + x 2 x + x n = x n + σ 1 x n 1 + σ 2 x n σ n. x x 1 x x 2 x x n = x n σ 1 x n 1 + σ 2 x n n σ n. 96 x, y, z, w 4
105 , x 2 + ax + b = x αx β = 0 { α + β = a, αβ = b. a, b α, β 2 α, β x 2 + 5x + 3 = 0 α 2, β 2 2 x 2 α 2 + β 2 x + α 2 β 2 = 0. α, β α, β α 2 + β 2 = α + β 2 2αβ = = 25 6 = 19. α 2 β 2 = αβ 2 = 3 2 = 9 x 2 19x + 9 = 0.
106 α 2, β 2 α, β = 5 ± = 5 ± α = = β == = α 2 + β 2 = = α 2 β = = = = n 97 2 x 2 3x 1 = 0 α, β 1 α, 1 β 2 3 n n 79 x 3 + y 3 = x + y 3 3x + yxy = σ 3 1 3σ 1 σ 2. x x x 3 3 = x 1 + x 2 + x 3 3 3x 1 + x 2 + x 3 x 1 x 2 + x 2 x 3 + x 3 x 1 + 3x 1 x 2 x 3 = σ 3 1 3σ 1 σ 2 + 3σ x 2 + 5x + 3 = 0 α, β α β 2 σ 1 = α + β = 5, σ 2 = αβ = 3 α β 2 = α + β 2 4αβ = = 13
107 x, y x i y j i+j 81 i. x 3 y = 5 ii. x 4, x 3 y, x 2 y 2, xy 3, y fx, y = x 3 y 2 + x 2 y 3 + x 3 + y 3 + 2x 2 y + 2xy 2 + 3xy. x 3 y 2, x 2 y 3 2 x 3 y 2 x 3 y xy 2 x + y = = 3 3xy x + y = = 2 x 3 y 2, x 2 y x + y 5, x + y 3 xy, x + yxy p, q, r x 3 y 2 + x 2 y 3 = px + y 5 + qx + y 3 xy + rx + yxy 2 p = 0, q = 0, r = 1 x + y 5 x 5 x + y 3 xy x 4 y fx, y x + yxy 2 = x 3 + y 3 + 2x 2 y + 2xy 2 + 3xy. 3 3 x + y 3, x + yxy
108 s, t x 3 + y 3 + 2x 2 y + 2xy 2 = sx + y 3 + tx + yxy s = 1, t = 1. 3 fx, y x + yxy 2 x + y 3 + x + yxy = 3xy. 1 0 fx, y x + yxy 2 x + y 3 + x + yxy 3xy = 0. fx, y fx, y = x+yxy 2 +x+y 3 x+yxy+3xy = σ 1 σ 2 2 +σ 3 1 σ 1 σ 2 +3σ 2. i. fx, y ii. iii. iv. v. 83 fx 1, x 2, x 3 = x x x 2 3 σ 1 = x 1 + x 2 + x 3 σ 2 = x 1 x 2 + x 2 x 3 + x 3 x 1 σ 3 = x 1 x 2 x 3 x x x 2 3 = aσ bσ 2. x 1, x 2, x 3 = 1, 0, 0 1 = f1, 0, 0 = aσ 1 1, 0, bσ 2 1, 0, 0 = a. x 1, x 2, x 3 = 1, 1, 0 2 = f1, 1, 0 = aσ 1 1, 1, bσ 2 1, 1, 0 = 4a + b. a = 1, b = 2 x x x 2 3 = σ 2 1 2σ 2.
109 x, y, z σ 1, σ 2, σ 3 x, y, z 99 x, y, z σ 1, σ 2, σ 3 i. x 2 + y 2 + z 2 ii. x 2 y + y 2 z + z 2 x + x 2 z + y 2 x + z 2 y iii. x y 2 + y z 2 + z x i. fx, y ii. iii. iv. v. n n = 3 3= =1+2 3=3 3 3
110 σ 1 = x + y + z 3 fx, y, z = x 3 +y 3 +z 3 σ 2 = xy + xz + yz σ 3 = xyz fx, y, z 3 σ 1, σ 2, σ σ 1, σ 2, σ σ σ 1 σ 2 3 σ 3 a, b, c fx, y, z = x 3 + y 3 + z 3 = aσ bσ 1 σ 2 + cσ 3 [ ] aσ bσ 1 σ 2 + cσ 3 x, y, z a, b, c a, b, c x 3 + y 3 + z 3 =aσ bσ 1 σ 2 + cσ 3 =ax + y + z 3 + bx + y + zxy + xz + yz + cxyz =ax 3 + y 3 + z 3 + 3x 2 y + 3x 2 z + 3y 2 z + 3xy 2 + 3xz 2 + 3yz 2 + 6xyz + bx 2 y + x 2 z + y 2 z + xy 2 + xz 2 + yz 2 + 3xyz + cxyz =ax 3 + y 3 + z 3 + 3a + bx 2 y + x 2 z + y 2 z + xy 2 + xz 2 + yz 2 + 6a + 3b + cxyz a = 1 3a + b = 0 6a + 3b + c = 0
111 110 3 a = 1 b = 3 c = 3 x 3 + y 3 + z 3 = σ 3 1 3σ 1 σ 2 + 3σ 3 [ ] x 3 + y 3 + z 3 = aσ bσ 1 σ 2 + cσ 3 x, y, z a, b, c σ 1, σ 2, σ 3 0 x, y, z x, y, z = 1, 0, 0. σ 2 = σ 3 = 0 σ 1 = x + y + z = = 1 σ 2 = xy + xz + yz = = 0 σ 3 = xyz = = 0 f1, 0, 0 = = 1 aσ1 3 + bσ 1 σ 2 + cσ 3 1, 0, 0 = a b c 0 = a a = 1 x, y, z = 1, 1, 0. σ 3 = 0 σ 1 = = 2 σ 2 = = 1 σ 3 = = 0 f1, 1, 0 = = 2 a = 1. σ1 3 + bσ 1 σ 2 + cσ 3 1, 1, 0 = b c 0 = 8 + 2b 8 + 2b = 2 b = 3
112 x, y, z = 1, 1, 1. σ 1, σ 2, σ 3 0 σ 1 = = 1 σ 2 = = 1 σ 3 = = 1 f1, 1, 1 = = 1 a = 1, b = 3. σ1 3 3σ 1 σ 2 + cσ 3 1, 1, 1 = c 1 = 4 c 4 c = 1 c = 3 x 3 + y 3 + z 3 = σ1 3 3σ 1 σ 2 + 3σ 3. x, y, z σ 1, σ 2, σ 3 0
113 , σ 1 = x + y, σ 2 = xy σ 1 σ 2 i. x 4 + y 4 ii. x 5 + y α, β 1 { α + β = A α β = B α = A + B 2 2 β = A B 2 x 2 + bx + c = 0 α, β α + β = a α β α β a, b 2 α β 2 α β 2 = α + β 2 4αβ = b 2 4c
114 α β = ± b 2 4c α, β = b ± b 2 4c 2 α β 1 2 α β 2 α β n fx 1, x 2,, x n i j fx 1, x 2,, iˆx j,, jˆx i,, x n = fx 1, x 2,, x n x 1 x 2 3 x 1 x 2 x 1 x 3 x 2 x 3 n i x j i<jx 12
115 a 1 a 2 a n a 2 1 a 2 2 a 2 n.... a n 1 1 a n 1 2 a n 1 n 102 ω 2 + ω + 1 = 0 x 1 + ωx 2 + ω 2 x x 2 + ωx 1 + ω 2 x 3 3
116 , σ 1 = x + y, σ 2 = xy σ 1, σ 2 i. x 2 + y 2 + 4xy ii. x y 4 iii. xx + 2y 2 + y2x + y σ 1 = x + y + z, σ 2 = xy + xz + yz, σ 3 = xyz σ 1, σ 2, σ 3 i. x 2 + y 2 + z 2 ii. x 3 + y 3 + z 3 iii. x y 4 + y z 4 + z x x 2 = x 2 = 9 x = ±3. 9 = 3 2 a a 2 x 2 = 5 x = ± x + 2 x = 5 x = 7 ± 5. x 2 = x 1 x 2 + 4x 1 = 0. x = 5 x = 2 ± 5.
117 ω 1 3 ω ω 3 = 1, ω ω ω ω 3 = 1 ω 3 1 = ω 1ω 2 + ω + 1 = 0 ω 1 x 2 + x + 1 = 0 2 ω, ω 2 = 1 ± 3i 2 ω ω x 3 = a x = 3 a 1 3 ω x = 3 a, ω 3 a, ω 2 3 a ω 3 = x 3 + ax 2 + bx + c = 0. x = t a 3 t 3 + pt + q = 0.
118 t = α, β, γ x 3 + ax 2 + bx + c = 0. x = α a 3, β a 3, γ a x 3 + ax 2 + bx + c = 0 x = t a 3 t3 + pt + q = 0 t 3 + pt + q = t = 0 q = a 3 + a a 3 3 t 3t 2 + p = 3 t a 3 t = 0 p = 3 a 3 t a 3 + a t a 2 + b t a + c a a b a 3 + c 2 + 2a t a + b 3 + b
119 x 3 6x 2 + x + 5 = 0. t 3 + pt + q = 0 x 3 6x 2 + x + 5 = 0.a = 6 6 x = t = t t 3 + pt + q = t t t t = 0 q = = = 9. t 3t 2 + p = 3t t t = 0 p = = = 11. t 3 11t 9 = x 3 + 3x 2 7x + 1 = 0 t 3 + pt + q = x 3 6x + 4 = 0 α = i i, i = 1
120 α = u + v x = u + v u + v 3 6u + v + 4 = 0. u 3 + v u + vuv 2 = 0. u + v = α uv uv = β t 2 αt + β = 0 t = u, v uv 2 = 0 u 3 + v = 0 u 3 v 3 = uv 3 = 2 3 = 8 u 3 + v 3 = 4, u 3 v 3 = 8 u 3, v 3 2 T 2 + 4T + 8 = 0. T = 2 ± = 2 ± 2i. u, v 3 uv = 2 u = i, v = 3 2 2i. u = ω i, v = ω i. u = ω i, v = ω 3 2 2i. ω 3 = 1, ω 1. x = i i, x = ω i + ω i. x = ω i + ω 3 2 2i.
121 , x 3 + px + q = 0. 3 x 3 + ax 2 + bx + c = 0. 3 x 3 + ax 2 + bx + c = 0. x = t 1 3 a t a + a t a + b t 13 a + c = 0. t 3 + a2 2a b t + 27 ab 3 + c = 0. 2 p = a2 3 + b q = 2a3 27 ab 3 + c t 3 + pt + q = 0. α, β, γ x 3 + ax 2 + bx + c = 0.
122 α 1 3 a, β 1 3 a, γ 1 3 a t = x a. x 3 + px + q = 0 u, v u + v u + v = α { uv uv = β u + v = α u, v uv = β t 2 αt + β = 0 u, v 3 x 3 + px + q = 0. x = u + v uv uv x 3 + px + q = 0 x = u + v u + v 3 + pu + v + q = 0. u 3 + v 3 + q + u + v3uv + p = 0. 3uv + p 0 uv uv 3uv = p. uv u 3 + v 3 + q = 0 u 3 v 3 u 3 v 3 = p3 27 u 3 + v 3 = q.
123 u 3, v 3 2 T 2 + qt p3 27 = 0. T = q ± 2 q 3 + s = q 3 t = q p3. q q p3 27 p3,. u, v 3 3uv = p u = s, v = t. u = ωs, v = ω 2 t. u = ω 2 s, v = ωt. ω 3 = 1, ω 1. x = s + t, x = ωs + ω 2 t. x = ω 2 s + ωt. 3 x 3 + px + q = 0. T 2 + qt p3 27 = n x 3 + 3x 2 + 4x + 7 = 0.
124 , x 2 + ax + b = 0 2 x = α, β { α + β = a αβ = b x 2 + ax + b = x αx β = x 2 α + βx + αβ i. 3 x 3 + px + q = 0 x = u + v ii. u v iii. u 3 =, v 3 = ω u, v iv. u 3 + v 3 u 3 v 3 u 3 v 3 2 v. u 3 v 3 uv 3 vi. x 3 + px + q = 0 x = u + v u 3 + v 3 + q + u + v3uv + p = 0 uv = p 3 u3 + v 3 = q
125 i. x 3 + ax 2 + bx + c = 0 x = t a 3 t 3 + pt + q = 0 a = 0 ii. t 3 + pt + q = 0 t = u + v u 3 + v 3 = q u 3 v 3 = p3 27 u, v iii. u 3 v 3 2 T 2 + qt p3 27 = 0 { m = 3 α α, β n = 3 β iv. ω 1 3 x = m + n a 3 x = mω + nω 2 a 3 x = mω 2 + nω a ω 3 = 1, ω 1 ω x 3 6x 2 + 6x 2 = i. x 3 + 3x 6 = 0 ii. x 3 + 3x 2 6 = 0
126 , ω 2 + ω + 1 = 0 x 1 + x 2 + x 3 = p x 1 + ωx 2 + ω 2 x 3 = q x 1 + ω 2 x 2 + ωx 3 = r. x 1, x 2, x 3 p, q, r 110 ω 2 + ω + 1 = 0 x 1 + ωx 2 + ω 2 x x 2 + ωx 1 + ω 2 x 3 3 S 3 = 1 2, i. 3 x 3 + ax 2 + bx + c = 0 ii. x = t a 3 t3 + pt + q = 0 a = 0 iii. t = t 1 x = t 1 a 3 iv. t = u + v u + v 3 + pu + v + q = 0. u 3 + v 3 + q + u + v3uv + p = 0. { u 3 + v 3 = q, v. 3uv + p = 0 uv = p 3, u 3 + v 3 = q, u 3 v 3 = p3 27,.
127 vi. u 3, v 3 T 2 + qt p3 27 = 0 α, β u 3 = α, v 3 = β vii. u, v = { 3 α 3 β, { ω 3 α ω 2 3 β, { ω 2 3 α ω 3 β viii. t = 3 α + 3 β, ω 3 α + ω 2 3 β, ω 2 3 α + ω 3 β. ω ω 3 = 1, ω 1 ix. x = 3 α + 3 β a 3, ω 3 α + ω 2 3 β a 3, ω2 3 α + ω 3 β a a 2 x 2 = a 0 a a i. 3 ii. n, x 3 + px + q = 0.
128 x 1, x 2, x 3 x 1 + x 2 + x 3 = 0, x 1 x 2 + x 2 x 3 + x 1 x 3 = p, x 1 x 2 x 3 = q. x 1 + ωx 2 + ω 2 x 3 ω 3 = 1, ω 1. ω 2 + ω + 1 = 0. σ S n f σ x 1, x 2,, x n = fx 1, x 2,, x n fx 1, x 2,, x n S n S n S 3 = 1 2, { f 1 2 x 1, x 2, x 3 = fx 2, x 1, x 3 = fx 1, x 2, x 3, f x 1, x 2, x 3 = fx 2, x 3, x 1 = fx 1, x 2, x 3. fx 1, x 2, x 3 1 2x 1 + ωx 2 + ω 2 x 3 = x 2 + ωx 1 + ω 2 x x 1 + ωx 2 + ω 2 x 3 = x 2 + ωx 3 + ω 2 x 1 = ω 2 x 1 + ωx 2 + ω 2 x ω 2 ω 3 = x 1 + ωx 2 + ω 2 x 3 3 = ω 6 x 1 + ωx 2 + ω 2 x 3 3 = x 1 + ωx 2 + ω 2 x 3 3 x 1 + ωx 2 + ω 2 x x 2 + ωx 1 + ω 2 x 3 3
129 x 1 + ωx 2 + ω 2 x 3 3 x 2 + ωx 1 + ω 2 x x 1 + ωx 2 + ω 2 x 3 3 x 2 + ωx 1 + ω 2 x 3 3 x 1 + ωx 2 + ω 2 x 3 3, x 2 + ωx 1 + ω 2 x T 2 + kt + l = 0. k, l l 3 x 1 + ωx 2 + ω 2 x 3, x 2 + ωx 1 + ω 2 x 3. 1 x1 + x 2 + x 3 + x 1 + ωx 2 + ω 2 x 3 + ω 2 x 2 + ωx 1 + ω 2 x 3 3 = x ω + ω2 x 2 + x 3 = x 1. 1 x1 + x 2 + x 3 + ω 2 x 1 + ωx 2 + ω 2 x 3 + ωx 2 + ωx 1 + ω 2 x 3 3 = x ω + ω2 x 1 + x 3 = x 2. 1 x1 + x 2 + x 3 + ωx 1 + ωx 2 + ω 2 x 3 + ωx 2 + ωx 1 + ω 2 x 3 3 = x ω + ω2 x 1 + x 2 = x 3. 3 x 1 + ωx 2 + ω 2 x 3 3
130 S ζ 5 = 1, ζ 1 5 x 1, x 2, x 3, x 4, x 5 x 1 + ζx 2 + ζ 2 x 3 + ζ 3 x 4 + ζ 4 x 5 5 S S n S n S 5 A 5 n x 3 + ax 2 + bx + c = 0 x = x 1, x 2, x 3 a, b, c 3 x 1, x 2, x 3 1 x 1, x 2, x x 1, x 2, x 3 S x 1 + ωx 2 + ω 2 x 3 3
131 i σ σ = ii τ τ τ = iii. Z/18Z Z/18Z {0, 1, 2,, 17} Z/18Z i. σ = a σ b σ 5 c σ
132 ii i a mod 71 a 0 a ii. σ 1 = x + y, σ 2 = xy 2x + y 4 + x + 2y 4 σ 1 σ ω x 3 3x 2 3x 7 = 0
ad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(
I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A
ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4
20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d
16 B
16 B (1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i 1 1. 1.1..
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6
1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
2012 A, N, Z, Q, R, C
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yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2004 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 7 7 1 8 8 1 9 9 1 10 10 1 E-mail:hsuzuki@icu.ac.jp 0 0 1 1.1 G G1 G a, b,
