@tsujimotter

AQ(⇣691)' Z/691Z� Z/691Z

⇣(�11)

tsujimotter

• 

• 

• 

tsujimotter

1810 – 1893

⇣(s) =1X

n=1

1

ns (Re s > 1)

s

s

(s)

⇣(1� s) = 21�s⇡�s sin

✓⇡(1� s)

2

◆�(s)⇣(s)

⇣(1� r) = �Br

r

r

Br r

13 + 23 + 33 + · · ·+ n3

12 + 22 + 32 + · · ·+ n2

=1

2n2 +

1

2n12 + 22 + 32 + · · ·+ n2

=1

3n3 +

1

2n2 +

1

6n

=1

4n4 +

1

2n3 +

1

4n2 + 0 · n

14 + 24 + 34 + · · ·+ n4 =1

5n5 +

1

2n4 +

1

3n3 + 0 · n2 -

1

30n

n1

B0 = 1, B1 =1

2, B2 =

1

6, B3 = 0, B4 = -

1

30, · · ·

B0 = 1, B1 =1

2, B2 =

1

6, B3 = 0, B4 = -

1

30, · · ·

B0 = 1, B1 =1

2, B2 =

1

6, B3 = 0, B4 = -

1

30, · · ·

B0 = 1, B1 =1

2, B2 =

1

6, B3 = 0, B4 = -

1

30, · · ·

B0 = 1, B1 =1

2, B2 =

1

6, B3 = 0, B4 = -

1

30, · · ·

x

e

x � 1=

1X

n=0

B

n

n!x

n

= 1� 1

2· x+

1

6 · 2! · x2 + 0 · x3� 1

30 · 4! · x4 + 0 · x5 + · · ·

⇣(�1) = � 1

12= � 1

22 · 3

⇣(�3) =1

120=

1

23 · 3 · 5

⇣(�5) = � 1

252= � 1

22 · 32 · 7

⇣(�7) =1

240=

1

24 · 3 · 5

⇣(�9) = � 1

132= � 1

22 · 3 · 11

⇣(�11) =691

32760=

691

23 · 32 · 5 · 7 · 13

⇣(�11) =691

32760=

691

23 · 32 · 5 · 7 · 13

von-Staudt Clausen

von Staudt–Clausen

Dm =Y

(p�1)|m

p

Bm

1, 2, 3, 4, 6, 8, 12, 2424 �!

⇣(1� 24) = �B24

24

2, 3, 4, 5, 7, 9, 13, 25

D24 =Y

(p�1)|24

p = 2 · 3 · 5 · 7 · 13

= 24 · 2 · 3 · 5 · 7 · 13⇣(1� 24) = �B24

24

• 

⇣(�23) =236364091

65520=

103 · 229479724 · 32 · 5 · 7 · 13

⇣(1� r1) ⌘ ⇣(1� r2) (mod p)

r1 ⌘ r2 (mod p� 1)

p r1, r2 r1 - p� 1

A ⌘ B (mod p) A�B

p

1� 321� 68

⇣(�31) = 37 · 683 · 305065927/26 · 3 · 5 · 17

⇣(�67) = 37 · 101 · 123143 · 1822329343 · 5525473366510930028227481/23 · 3 · 5

r2 r1

r2 � r1 ⌘ 0 (mod p� 1)

p = 37

⇣(1� r2)� ⇣(1� r1) ⌘ 0 (mod p)

p = 37

36363636

⇣(�31)⇣(�67)⇣(�103)⇣(�139)⇣(�175)

36 37

-1 -199 37

⇣(�23)� ⇣(�11) =103 · 2294797

24 · 32 · 5 · 7 · 13 � �1

22 · 3

=103 · 2294797 + 22 · 3 · 5 · 7 · 13

24 · 32 · 5 · 7 · 13

=11 · 2148814124 · 32 · 5 · 7 · 13

⇣(�23)� ⇣(�11) =103 · 2294797

24 · 32 · 5 · 7 · 13 � �1

22 · 3

=103 · 2294797 + 22 · 3 · 5 · 7 · 13

24 · 32 · 5 · 7 · 13

=11 · 2148814124 · 32 · 5 · 7 · 13

⇣(�23)� ⇣(�11) =103 · 2294797

24 · 32 · 5 · 7 · 13 � �1

22 · 3

=103 · 2294797 + 22 · 3 · 5 · 7 · 13

24 · 32 · 5 · 7 · 13

=11 · 2148814124 · 32 · 5 · 7 · 13

p = 11

10

⇣(�13)

mod p

mod pn n = 1

p r1, r2 r1 - p� 1

(1� 1/p1�r1)⇣(1� r1) ⌘ (1� 1/p1�r2

)⇣(1� r2) (mod pn)

r1 ⌘ r2 (mod (p� 1)pn�1)

n � 1

p

⇣p(1� r) := (1� 1/p1�r)⇣(1� r)

p r1, r2 r1 - p� 1

n � 1

⇣p(1� r1) ⌘ ⇣p(1� r2) (mod pn)

r1 ⌘ r2 (mod (p� 1)pn�1)

r1 ⌘ r2 (mod (p� 1)pn�1)

⇣p(1� r1) ⌘ ⇣p(1� r2) (mod pn)

(p� 1)pn�1

pn

y = f(x)

x

y

pn-1 pn

p

|x|p

:= p

�vp(x) p

x p

–1 –11 5 –1 –251 5

|� 11� (�1)|5 = |� 10|5 =1

5

=1

5

|� 251� (�1)|5 = |� 250|5

=1

53

–1

–251

–11

–1 –11 5 –1 –251 5

|� 11� (�1)|5 = |� 10|5 =1

5

=1

5

|� 251� (�1)|5 = |� 250|5

=1

53

=1

53

=1

5

p

r0

r1

r2

⇣p(1� r0)

⇣p(1� r1)

⇣p(1� r2)

• 

•  p

• 

• 

• 

x

n + y

n = z

n

n � 3

(xyz 6= 0)

(x, y, z)

FLT(n)

FLT(4)

FLT(3)

FLT(5)

FLT(7)

FLT(14)

p

FLT(p)

FLT(p)

FLT(3) FLT(5) FLT(7) FLT(11)

FLT(13) FLT(17) FLT(19) FLT(23) FLT(29)

FLT(31) FLT(37) FLT(41) FLT(43) FLT(47)

FLT(53) FLT(59) FLT(61) FLT(67) FLT(71)

FLT(73) FLT(79) FLT(83) FLT(89) FLT(97)

Q(⇣p)

Q(⇣p)

Z[⇣p]

Z[⇣p]

aq

q0 + q1⇣p + · · ·+ qp�2⇣p�2p a0 + a1⇣p + · · ·+ ap�2⇣

p�2p

⇣p

FLT

x

p + y

p = z

p

(x+ y)(x+ ⇣py)(x+ ⇣

2py) · · · (x+ ⇣

p�1p y) = z

p

Q(⇣p)

z = ✏pe11 · · · pegg

= (✏pe11 · · · pegg )p

x+ ⇣

kp y = ✏

0↵

p

(x� ⇣

kp y), (x� ⇣

k0

p y)

( )( ) = z2

= (P1P2)2

z = P1P2

= (P1P2)(P1P2)

= P 21P

22

( )( ) = z2

z = P1P2 = Q1Q2

= (P1P2)(Q1Q2)

= (P1Q1)(P2Q2)

p = 23

6 = 2 · 3 = ⇠1 · ⇠2

⇠1 = ⇣23 + ⇣423 + ⇣923 + ⇣1623 + ⇣223 + ⇣1323 + ⇣323 + ⇣1823 + ⇣1223 + ⇣823 + ⇣623

⇠2 = ⇣2223 + ⇣1923 + ⇣1423 + ⇣723 + ⇣2123 + ⇣1023 + ⇣2023 + ⇣523 + ⇣1123 + ⇣1523 + ⇣1723

6 = 2 · 3 = ⇠1 · ⇠2

A, B, C, D

Z“ ”

3Z = (3)

3Z+ 5Z = (3, 5)

“ ” “ ”

Z[⇣p]

2Z[⇣p] = (2)

3Z[⇣p] = (3)

⇠1Z[⇣p] = (⇠1)

⇠2Z[⇣p] = (⇠2) 3Z[⇣p] + ⇠2Z[⇣p] = (3, ⇠2)

3Z[⇣p] + ⇠1Z[⇣p] = (3, ⇠1)

2Z[⇣p] + ⇠1Z[⇣p] = (2, ⇠1)

2Z[⇣p] + ⇠2Z[⇣p] = (2, ⇠2)

(2)(3) = (6) (⇠1)(⇠2) = (⇠1⇠2)

(2, ⇠1)(3, ⇠2) = (2 · 3, 2⇠2, 3⇠1, ⇠1⇠2)(3)(2, ⇠1) = (6, 3⇠1)

積

(2, ⇠1)(2, ⇠2) = (22, 2⇠1, 2⇠2, ⇠1⇠2)

= (22, 2⇠1, 2⇠2, 6)

= (2)(1)2, 3 = 1

= (2)(2, ⇠1, ⇠2, 3) 2

= (2)(1)

(2) = (2, ⇠1)(2, ⇠2)

(6) = (2)(3) = (⇠1)(⇠2)

(6) = (2, ⇠1) (2, ⇠2) (3, ⇠1) (3, ⇠2)

(↵)↵

Q(⇣p)

Q(⇣p) Q(⇣p)

A

A2

⇥AA3

⇥A

• 

• 

• 

• 

JK PK

Cl(K) := JK�PK

PK ⇢ JK

#Cl(K) K

Cl(Q(⇣p))

Cl(Q(⇣p))

x

p + y

p = z

p

(x+ y)(x+ ⇣py)(x+ ⇣

2py) · · · (x+ ⇣

p�1p y) = z

p

Q(⇣p)

(z) = pe11 · · · pegg

=�pe11 · · · pegg

�p

(x+ y)(x+ ⇣py)(x+ ⇣

2py) · · · (x+ ⇣

p�1p y) = (z)p

(x+ ⇣

kp y) = A

p

(x� ⇣

kp y), (x� ⇣

k0

p y)

A = (↵)

(x+ ⇣

kp y) = (↵p) x+ ⇣

kp y = ✏

0↵

p

A = (↵)

(x+ ⇣

kp y) = A

p

A

(x+ ⇣

kp y) = A

p

#Cl (Q(⇣p))

pA = (↵)

p

A = (↵)

A = (↵)

p

p 1

1

#Cl(Q(⇣p)) p

#Cl(Q(⇣p)) p

FLT p

FLT p

#Cl (Q(⇣23)) = 3

#Cl(Q(⇣7)) = 1

#Cl(Q(⇣11)) = 1

#Cl(Q(⇣13)) = 1

#Cl(Q(⇣17)) = 1

#Cl(Q(⇣19)) = 1

#Cl(Q(⇣29)) = 8

#Cl(Q(⇣p)) = h�p h

+p

p p

h�7 = 1

h�11 = 1

h�13 = 1

h�17 = 1

h�19 = 1

h�23 = 3

h�29 = 8

h�31 = 9

h�37 = 37

37

100

h�37 = 371

h�59 = 591 · 699

h�67 = 671 · 12739

Remark

200

h�37 = 371

h�59 = 591 · 699

h�67 = 671 · 12739

h�101 = 1011 · 35122815625

h�103 = 1031 · 88049462555

h�131 = 1311 · 217529616253985775

h�149 = 1491 · 4616697044880367249149

h�157 = 1572 · 2281404020463379154005

• 

• 

ex. ( ) hp p

• 

• 

• 

• 

p p

h37 37

⇣(�31) 37

h103 103

103 ⇣(�23)

p p

(1) p hp p

(2) p

⇣(�1), ⇣(�3), ⇣(�5), ⇣(�7), . . .

691

⇣(�1), ⇣(�3), . . . , ⇣(1� (p� 3))

(2) p

(1) p hp p

31

–1 –27

31

⇣(�1), ⇣(�3), . . . , ⇣(1� (p� 3))

(2) p

(1) p hp p

⇣(�1), ⇣(�3), . . . , ⇣(1� (p� 3))

(2) p

p-

p- p

Cl(Q(⇣p)) = AQ(⇣p) �A0Q(⇣p)

A

!i

Q(⇣p)= {x 2 AQ(⇣p) | 8� 2 �,�(x) = x

!(�)i}

� = Gal(Q(⇣p)/Q) ! : �⇠�! (Z/pZ)⇥

AQ(⇣p) =p�2M

i=0

A!i

Q(⇣p)

(1) (2)

(1)

(2)

p ⇣(1� r)

� 2 ��(x) = x

!(�)1�r

pxCl(Q(⇣p))

A!1�r

Q(⇣p)6= {e}

p = 37

AQ(⇣37) ' Z/37Z

⇣(�31) =7709321041217

16320=

37 · 683 · 30506592726 · 3 · 5 · 17

37

�(x) = x

!(�)�31

p

x

k ⇠�! k. mod 37

p = 691

AQ(⇣691) ' Z/691Z� Z/691Z

�(x) = x

!(�)�11

�(y) = y!(�)�199

x

k · yl ⇠�! (k. mod 691, l. mod 691)

p

691 ⇣(�11), ⇣(�199)

691

• 

•  p ó p

• 

[1]

[2]

[3]

[1] [2] [3]

>zeta199.numerator=>498384049428333414764928632140399662108495887457206674968055822617263669621523687568865802302210999132601412697613279391058654527145340515840099290478026350382802884371712359337984274122861159800280019110197888555893671151>zeta199.numerator%691=>0

⇣(�199) ⌘ 0 (mod 691)

↵A

↵(1) = (↵)

↵ 2 K⇥ A ⇢ OK

AB = (n)

A

B n

↵A⇥ 1

↵nB =

↵

n↵AB =

1

n(n) = (1)

Sagemath

x=k.ideal(6)x.factor()(Fractionalideal(2,z^11+z^10+z^6+z^5+z^4+z^2+1))*(Fractionalideal(2,z^11+z^9+z^7+z^6+z^5+z+1))*(Fractionalideal(3,z^11+z^10+z^9-z^8-z^7+z^5+z^3-1))*(Fractionalideal(3,z^11-z^8-z^6+z^4+z^3-z^2-z-1))a=k.ideal(2,z^11+z^10+z^6+z^5+z^4+z^2+1)b=k.ideal(2,z^11+z^9+z^7+z^6+z^5+z+1)c=k.ideal(3,z^11+z^10+z^9-z^8-z^7+z^5+z^3-1)d=k.ideal(3,z^11-z^8-z^6+z^4+z^3-z^2-z–1)x=k.ideal(23)x.factor()z1=z+z^4+z^9+z^16+z^2+z^13+z^3+z^18+z^12+z^8+z^6z2=z^22+z^19+z^14+z^7+z^21+z^10+z^20+z^5+z^11+z^15+z^17k.ideal(z1)k.ideal(z1).reduce_equiv()k.ideal(z2)k.ideal(z2).reduce_equiv()z1*z2