CCINP MP 2019- Math´ematiques 2

CCINP MP 2019- Math´ ematiques 2

Corrig´e propos´e par V. Queffelec.

Lyc´ee Sainte-Anne, Brest.

EXERCICE I

Q1.

Voici un code possible.

import

math#S up p o s e d e f i n i t i v e m e n t i m p o r t e dans l a s u i t e

def

e s t P r e m i e r ( n ) :

””” Donnee : un e n t i e r n a t u r e l n s u p e r i e u r ou e g a l a 2 R e s u l t a t : l e b o o l e e n i n d i q u a n t s i n e s t p r e m i e r ou non ”””

f o r

i

in range

( 2 ,

i n t

( math . s q r t ( n ) ) + 1 ) :

i f

n%i ==0:

return

F a l s e

return

True

Q2.

Voici un code possible.

def

l i s t e p r e m i e r s ( n ) :

””” Donnee : un e n t i e r n a t u r e l n s u p e r i e u r ou e g a l a 2 . R e s u l t a t : une l i s t e co nt ena nt t o u s l e s e n t i e r s p r e m i e r s i n f e r i e u r s ou egaux a n . ”””

L = [ ]

f o r

i

in range

( 2 , n +1):

i f

e s t P r e m i e r ( i ) : L . append ( i )

return

L

Q3.

Voici un code possible.

def

v a l u a t i o n p a d i q u e ( n , p ) :

””” Donnee : deux e n t i e r s n a t u r e l s n e t p s u p e r i e u r s ou egaux a 2 .

R e s u l t a t : l a v a l u a t i o n p−a di que de n obtenue par un a l g o r i t h m e i t e r a t i f ”””

compt=0

while

n%p==0:

compt+=1

return

compt

Q4.

Voici un code possible.

1

def

v a l u a t i o n p a d i q u e ( n , p ) :

””” Donnee : deux e n t i e r s n a t u r e l s n e t p s u p e r i e u r s ou egaux a 2 .

R e s u l t a t : l a v a l u a t i o n p−a di que de n obtenue par un a l g o r i t h m e r e c u r s i f ”””

i f

n%p ! = 0 :

return

0

e l s e

:

return

1+ v a l u a t i o n p a d i q u e ( n//p , p )

Q5.

Voici un code possible.

def

v d e c o m p o s i t i o n f a c t e u r s p r e m i e r s ( n ) :

””” Donnee : un e n t i e r n a t u r e l n s u p e r i e u r ou e g a l a 2 . R e s u l t a t : sa d e c o m p o s i t i o n en f a c t e u r s p r e m i e r s

comme une l i s t e de c o u p l e s , conformement a l ’ enonce . ”””

L = [ ]

f o r

i

in range

( 2 ,

i n t

( math . s q r t ( n ) ) + 1 ) :

i f

e s t P r e m i e r ( i ) :

L . append ( ( i , v a l u a t i o n p a d i q u e ( n , i ) ) )

return

L

EXERCICE II

Q6.

Soit

E

=

R²

et

u

l’endomorphisme canoniquement associ´e `a la matrice

!

0

−1

1 0

"

(rotation d’angle

π

2

). On a bien

u̸= 0L(E)

et

∀x∈R²

,

⟨u(x), x⟩

= 0.

Q7.

Montrons les implications sugg´er´ees par l’´enonc´e. Soit (u, v)

∈

(L(E))

²

d´efinis comme dans l’´enonc´e.

i)⇒ii) On a :

⟨u(x), u(y)⟩

=

⟨x, v(u(y))⟩

par d´efinition de

v

=

⟨x, u(v(y))⟩

d’apr`es l’hypoth`ese i)

=

⟨v(x), v(y)⟩

par d´efinition de

v

et sym´etrie du produit scalaire ii)⇒iii) Est imm´ediat en choisissant

y

=

x.

iii)⇒ii) Employons une identit´e de polarisation.

⟨u(x), u(y)⟩

= 1

2 (∥u(x) +

u(y)∥²− ∥u(x)∥²− ∥u(y)∥²

))

= 1

2 (∥u(x +

y)∥²− ∥u(x)∥²− ∥u(y)∥²

)) car

u

est lin´eaire

= 1

2 (∥v(x +

y)∥²− ∥v(x)∥²− ∥v(y)∥²

)) d’apr`es l’hypoth`ese iii)

= 1

2 (∥v(x) +

v(y)∥²− ∥v(x)∥²− ∥v(y)∥²

)) car

v

est lin´eaire

=

⟨v(x), v(y)⟩

par une identit´e de polarisation

2

.G k ,

úH2K2Mib /2 +Q``2+iBQM

S`Q#HĕK2 ,

S`ûHBKBMB`2b ,

RX aQBi n2N[mB MǶ2bi Tb mM +``û T`7Bi UH2 +``û /ǶmM 2MiB2` Mim`2HVX PM bmTTQb2 T` HǶ#bm`/2 [m2pn=a

b p2+a^b= 1X

X PM T2mi bmTTQb2` [m2 b2bi mM 2MiB2` bi`B+i2K2Mi TQbBiB7 U[mBii2 ¨ T`2M/`2 HǶQTTQbû /2aVX PM M2 T2mi Tb pQB`b= 1+` bBMQM-pn=a2ZQ`nMǶ2bi Tb mM +``û T`7BiX .QM+b>2X S` H2 i?ûQ`ĕK2 /2 /û+QKTQbBiBQM 2M T`Q/mBi /2 MQK#`2b T`2KB2`b-b/K2i /QM+ mM /BpBb2m` pT`2KB2`X m Tbb;2- QM `TT2HH2 [m2 R MǶ2bi Tb T`2KB2`X

#X HQ`bn= a²

b²X _û~2t2 /2 `BbQMM2K2Mi `Bi?KûiB[m2 , QM M2 KMBTmH2 Tb /2b 7`+iBQMb 5 PM

`ûû+`Bi /QM+ nb² = a²X BMbB- np² = a²- /QM+ p/BpBb2a²=a⇥aX *QKK2 p 2bi mM MQK#`2 T`2KB2`- T` H2 H2KK2 /2 :mbb-p/BpBb2aUQmp/BpBb2aXXXVX

BMbB-a2ibbQMi /BpBbB#H2b T`p/QM+ M2 bQMi Tb T`2KB2`b 2Mi`2 2mtX *QMi`/B+iBQM 2i `ûbmHii , pnMǶ2bi Tb `iBQMM2HX

kX aQBipmM MQK#`2 T`2KB2` 2ik2[1, p 1]X X PM `2K`[m2 [m2k!(p k)!

Çp k å

=p! =p⇥(p 1)!X .QM+p/BpBb2k!(p k)!

Çp k å

X

P`p2bi T`2KB2` p2+ iQmi 2MiB2`j2[1, p 1]X .QM+p2bi T`2KB2` p2+ iQmi T`Q/mBi /Ƕ2MiB2`b j2[1, p 1]X BMbB-p^k!(p k)!+`k2[1, p 1]X

S` H2 H2KK2 /2 :mbb- p/BpBb2 /QM+ <sup>p</sup><sub>k</sub> TQm` iQmik2[1, p 1]X ii2MiBQM ,pM2 /BpBb2 Tb ^p₀ MB ^p_p XXX

#X 1M i`pBHHMi /MbZ/pZ- H2 /2`MB2` `ûbmHii b2 i`/mBi 2M <sub>k</sub><sup>p</sup> ⌘0KQ/pTQm` iQmik2[1, p 1]X SQm` iQmi (a, b)2Z<sup>2</sup>- +QKK2Z/pZ 2bi mM MM2m +QKKmiiB7- QM T2mi TTHB[m2` H 7Q`KmH2 /m #BMƬK2 /2 L2riQM 2i QM Q#iB2Mi ,

(a+b)^p⌘ Xp

k=0

Çp k å

a^kb^{p k}= 0 +a^p+b^p KQ/p.

jX X PM T2mi `2K`[m2` [m2 bB x<sup>2</sup> ⌘ dKQ/p- HQ`b ( x)² = dKQ/pX AH bm{i /QM+ /2 +H+mH2`

0²,1²,2², ...,(p 1) 2

2

KQ/mHQp+` HQ`b(p 1)²= 1²- (p 2)²= 2²XXX 2i+XXX

G2b +``ûb /2Z/11ZbQMi /QM+{(0KQ/11)²,(1KQ/11)²,(2KQ/11)²,(3KQ/11)²,(4KQ/11)²,(5KQ/11)²}= {0,1,4,9,5,3}X

*2mt /2Z/17ZbQMi{0,1,4,9, 1, 9,2, 2, 4X

#X 1z2+iBp2K2Mi 5

R

9X aQBi n > 2X Sn =

nX1

k=0

exp(2ik⇡

n ) =

n 1

X

k=0

Å

exp(2i⇡

n ) ãk

2bi mM2 bQKK2 /2 T`2KB2`b i2`K2b /ǶmM2 bmBi2 ;ûQKûi`B[m2X PM `TT2HH2 [m2 b `BbQM exp(2i⇡

n ) 2bi /Bzû`2Mi2 /2 R +` n>2X .QM+ Sn = 1 exp(2in⇡

n ) 1 exp(2i⇡

n

) = 0 +`exp(2in⇡

n ) = 1X

S`iB2 R ,

RX X Z[ ] 2bi mM2 BMi2`b2+iBQM /2 bQmb@MM2mt /2(C,+,⇥)X

1M T`iB+mHB2` (Z[ ],+) 2bi mM2 BMi2`b2+iBQM /2 bQmb ;`QmT2b //BiB7b- /QM+ mM bQmb ;`QmT2 //BiB7 /2(C,+)X

aB (a, b) 2 Z[ ]²- HQ`b TQm` iQmi A bQmb MM2m /2 (C,+,⇥) +QMi2MMi - a2 A 2i B 2 AX

*QKK2 A 2bi mM bQmb MM2m U/QM+ bi#H2 T` T`Q/mBi BMi2`M2V- a⇥b 2 AX 6BMH2K2Mi- a⇥b2T

A2A A=Z[ ]X

.2 KāK2- 12Z[ ] U`û/+iBQM HBbbû2 m H2+i2m` MQM TbbB7VX BMbB-Z[ ] 2bi mM bQmb@MM2m /2(C,+,⇥)X

#X aB B 2bi mM bQmb@MM2m /2 (C,+,⇥) +QMi2MMi - HQ`b B 2 A X 1M T`iB+mHB2`- Z[ ] = BTÄT

A2A \{B}

ä2bi mM2 T`iB2 /2BX .QM+Z[ ]⇢BX

PM pB2Mi /2 KQMi`2` [m2Z[ ] 2bi H2 THmb T2iBi bQmb@MM2m /2(C,+,⇥)+QMi2MMi X

kX X amTTQbQMb 2ZX HQ`b Z2bi mM bQmb MM2m /2 C+QMi2MMi X S` biimi KBMBKH /2 Z[ ]- QM /QM+Z[ ]⇢ZX

JAa- MǶQm#HBQMb Tb /2 `2K`[m2` [m2 12Z[ ]X HQ`b TQm` iQmin2Z- n·12Z[ ] [mB 2bi mM bQmb ;`QmT2 //BiB7 /2CX .QM+Z⇢Z[ ]X

6AMH2K2Mi-

Z[ ] =Z.

#X PM T`Q+ĕ/2 T` /Qm#H2 BM+HmbBQMX

GǶmM2 2bi +HbbB[m2 , bB 2 Z[ ]- HQ`b T` bi#BHBiû /m T`Q/mBi 2i /QM+ /2b TmBbbM+2b 2M T`iB+mHB2`-8k2N, <sup>k</sup> 2Z[ ]X *QKK2 Z[ ] 2bi mM bQmb ;`QmT2 //BiB7 /2 C-8ak 2Z, ak k 2 Z[ ]X 6BMH2K2Mi- TQm` iQmiP 2Z[X]-P( )2Z[ ]X .QM+ ,

{P( )|P 2Z[X]}⇢Z[ ].

_û+BT`Q[m2K2Mi- BH b2`Bi THmb /ûHB+i /2 KQMi`2` [m2 iQmi ûHûK2Mi /2 Z[ ] 2bi mM TQHvMƬK2 2M X AH 7mi /QM+ mM2 mi`2 TT`Q+?2- miBHBbMi H2b T`QT`Bûiûb /2Z[ ]X

S` biimi KBMBKH /2Z[ ]- BH bm{i /2 KQMi`2` [m2{P( )|P 2Z[X]}2bi mM bQmb MM2m /2C +QMi2MMi X .ǶT`ĕb H [m2biBQMRX#- MQmb m`QMb HQ`b HǶBM+HmbBQMZ[ ]⇢{P( )|P2Z[X]}X PM KQMi`2 7+BH2K2Mi [m2{P( )|P 2Z[X]} 2bi mM bQmb ;`QmT2 //BiB7 /2C- [m2 {P( )|P 2 Z[X]} 2bi bi#H2 T` T`Q/mBi 2i [m21 = ⁰2{P( )|P 2Z[X]}XXX

+X LPL , T` 2t2KTH2-i=i⁵ /MbZ[i]X

jX X G `2HiBQM 2bi `û~2tBp2- bvKûi`B[m2 2i i`MbBiBp2X m+mM2 /B{+mHiûX

#X .û}MBbbQMb H2b HQBb+p 2i ⇥^p bmBpMi2 bm`Z[ ]/(p),

SQm` ( ˙a,b)˙ 2(Z[ ]/(p))²- QM TQb2a˙ +pb˙:=a+˙ b 2ia˙⇥^pb˙ :=a⇥˙ bX

AH bm{i /2 KQMi`2` [m2 H2 `ûbmHii M2 /ûT2M/ Tb /m +?QBt /2b `2T`ûb2MiMib /2b +Hbb2ba˙ 2ibX˙ S` +QMbi`m+iBQM- 2bi #B2M mM KQ`T?BbK2 UBKKû/BiV 2i bm`D2+iB7 +` iQmi ûHûK2Mia˙ 2Z[ ]/(p) /K2ia2Z[ ] +QKK2 Miû+û/2MiX

k

S`iB2 k ,

PM bmTTQb2 /Mb +2ii2 T`iB2 [m2 H2 MQK#`2 +QKTH2t2 pû`B}2 H2b +QM/BiBQMb 2/ Z 2i ²2ZX PM MQi2 d= ²X

GǶ2MiB2`p2bi iQmDQm`b T`2KB2` 2i BKTB`X

RX .ǶT`ĕb H [m2biBQMkX#-Z[ ] ={P( )|P 2Z[X]}X aQBiP 2Z[X]X HQ`bP( ) =PN

n=0an n=P^N/2

k=0a2k 2k+P^N/2

k=0a2k+1 2k+1X

P` <sup>2k</sup> = d<sup>k</sup> 2 ZX .QM+ T` bQKK2 2i T`Q/mBib /Ƕ2MiB2`b- P( ) = n0 +m0 2M TQbMi n0 = P^N/2

k=0a2kd^k2Z2i m0=P^N/2

k=0a2k+1d^k2ZX 6BMH2K2Mi-

Z[ ] ={n0+ m0|(n0, m0)2Z²}.

kX R2` +b , bB d <0- HQ`b 2iRX aQB2Mi(n, m)2i (n⁰, m⁰) +QmTH2b /Ƕ2MiB2`b i2Hb [m2a=n+ m = n<sup>0</sup>+ m<sup>0</sup>X S` mMB+Biû /2b T`iB2b `û2HH2b 2i BK;BMB`2b /2b MQK#`2b +QKTH2t2b-n=n⁰ 2i m=m⁰X k2K2 +b , bB d > 0- HQ`b 2 R\Q /ǶT`ĕb H2b T`ûHBKBMB`2bX aQB2Mi (n, m) 2i (n⁰, m⁰) +QmTH2b /Ƕ2MiB2`b i2Hb [m2 a=n+ m=n<sup>0</sup>+ m<sup>0</sup>X aBm 6=m<sup>0</sup>- HQ`b = n n⁰

m m⁰ 2 Q+2 [mB 2bi #bm`/2X .QM+m=m<sup>0</sup> 2i T` bmBi2-n=n⁰X

PM bmTTQb2 iQmDQm`bpT`2KB2` BKTB` 2i QM bmTTQb2 [m2 d^p= 1X

jX aQBi a˙ 2Z[ ]/(p) i2H [m2a˙²= ˙0- +Ƕ2bi ¨ /B`2 (n+˙m)<sup>2</sup>= ˙0 /QM+ BH 2tBbi2(n<sup>0</sup>, m<sup>0</sup>)2MiB2`b i2Hb [m2 (n+ m)²=n²+ 2 nm+dm²=p(n⁰+ m⁰)X

GǶmMB+Biû /2 HǶû+`Bim`2 /ûKQMi`û2 /Mb H [m2biBQM T`û+û/2Mi2 T2`K2i /Ƕû+`B`2 HQ`b ,

2nm =pm⁰ 2i n²+dm² =pn⁰X 1M T`iB+mHB2`- p/BpBb2 2nm 2i n2 +dm²X *QKK2 p 2bi BKTB`- /QM+ T`2KB2` p2+ 2- T` H2 H2KK2 /2 :mbb-p/BpBb2nQmmX

aB p|m- +QKK2p|n²+dm²- HQ`bp|n² 2i }MH2K2Mip|nX

.2 KāK2- bB p|n- HQ`bp|m<sup>2</sup>d2i +QKK2p2bi T`2KB2` p2+d-p|bX BMbB-p|n +m+Ƕ2bi ¨ /B`2(n+˙m)²= ˙0X

SQm` iQmia2Z[ ]- bBa=n+ m- QM MQi2a<sup>c</sup>=n m2i N(a) =a⇥a<sup>c</sup>X 9X aQBia2Z[ ]-N(a) =n<sup>2</sup> dm<sup>2</sup>2ZT` bQKK2 2i T`Q/mBi /Ƕ2MiB2`bX 8X SQm` a<sup>0</sup>=n<sup>0</sup>+ m<sup>0</sup>- QM `2K`[m2 [m2N(aa⁰) =...=N(a)N(a⁰)X

.2 THmb- QM `2K`[m2 [m2 bBa˙ = ˙b- HQ`bN(a)KQ/p=N(b)KQ/p

aBa˙2bi BMp2`bB#H2 /MbZ[ ]/(p)QM /QM+1 =N(1)KQ/p=N(aa <sup>1</sup>)KQ/p= (N(a)KQ/p)(N(a <sup>1</sup> KQ/p)X .QM+N(a)2bi BMp2`bB#H2 KQ/mHQp- +Ƕ2bi ¨ /B`2N(a)^p= 1X

_û+BT`Q[m2K2Mi- bB N(a)^p = 1- HQ`b N(a) 2bi BMp2`bB#H2 KQ/mHQ p 2i HQ`b aa^cN(x) ¹ ⌘ N(x)N(x) ¹⌘1KQ/pX

.QM+a2bi BMp2`bB#H2 /ǶBMp2`b2a^cN(a) ¹X

eX amTTQbQMb [m2 dMǶ2bi Tb mM +``û /Mb Z/pZX aQBia˙ MQM BMp2`bB#H2 /Mb/Z[ ]/(p)X HQ`b /ǶT`ĕb +2 [mB T`û+ĕ/2-p^N(a)6= 1/QM+p|N(a)+`p2bi T`2KB2`X BMbB- bB a=n+ m, p|n² m²- +Ƕ2bi

¨ /B`2 dm²⌘n² KQ/pX

aB pM2 /BpBb2 Tb m- HQ`b mKQ/p2bi BMp2`bB#H2 /Mb Z/pZ U[mB 2bi mM +Q`TbV 2id ⌘a<sup>2</sup>b <sup>2</sup> ⌘ (ab <sup>1</sup>)<sup>2</sup> KQ/p- +2 [mB +QMi`2/Bi HǶ?vTQi?ĕb2 7Bi2 bm` dX

BMbB-p/BpBb2mTmBbp/BpBb2n² 2i /QM+nX .QM+a˙ =n+˙ m= ˙0X

G2 b2mH ûHûK2Mi MQM BMp2`bB#H2 /2 Z[ ]/(p)2bi /QM+ HǶûHûK2Mi MmH /QM+Z[ ]/(p)2bi mM +Q`TbX j

S`iB2 j ,

aQBi ,

q:

® ((Z/pZ)^⇤,⇥) ! ((Z/pZ)^⇤,⇥) (xKQ/p) 7! (xKQ/p)²

RX GǶTTHB+iBQM q 2bi mM KQ`T?BbK2 /2 ;`QmT2b JlGhASGA*hA6a 2i kerq = {x 2 Z/pZ|x² = (1 KQ/p)}X

P` x²= (1KQ/p)bB 2i b2mH2K2Mi bBx² (1KQ/p) = (x (1 KQ/p))(x+ (1KQ/p)) = 0X

*QKK2 Z/pZ 2bi mM +Q`Tb U/QM+ BMiĕ;`2V- QM 2M /û/mBi [m2 x = ±(1 KQ/p)X BMbB- kerq = {(1 modp),( 1 modp)}X

kX .Mb mM 2t2`+B+2- QM KQMi`û H2 i?ûQ`ĕK2 /m `M; TQm` H2b KQ`T?BbK2b /2 ;`QmT2b ,

*`/((Z/pZ)<sup>⇤</sup>) =*`/(E2`q)·*`/(AKq).

jX A+B-Card(kerq) = 2X .2 THmb H2b +``ûb MQM MmHb bQMi 2t+i2K2Mi H2b ûHûK2Mib /Mb HǶBK;2 /2 H 7QM+@

iBQMqX .QM+ H2 MQK#`2 /2 +``ûb MQM MmHb /MbZ/pZ2bi û;H ¨ Card(Im(q)) = Card((Z/pZ)^⇤)

2 =

p 1 2 X

9X S` H2 i?ûQ`ĕK2 /2 G;`M;2- QM KQMi`2 H 7Q`KmH2 /Ƕ1mH2` , 8x2Z, x^p= 1)x^'(p)⌘1KQ/p.

UB+B-'(p) =p 1VX

8X 1M TQbMi y = x^{(p 1)/2}- QM Q#iB2Mi y² = x^{p 1} = (1KQ/p)X .QM+ y 2 kerq 2i }MH2K2Mi-y = x^{(p 1)/2} 2bi +QM;`m ¨ R Qm( 1)KQ/mHQpX

eX AH v 2t+i2K2Mi p 1

2 +``ûb MQM MmHb /MbZ/pZX

*QKK2Z/pZ2bi mM +Q`Tb- /QM+ BMiĕ;`2- HǶû[miBQM TQHvMQKBH2x^{(p 1)/2} 1/2 /2;`û p 1 2 /K2i m THmb p 1

2 bQHmiBQMbX P`- bB x 2bi H2 +``û /2 y- 7Q`+ûK2Mi x^{(p 1)/2} = y^{p 1} = 1- /QM+ x 2bi bQHmiBQMX

BMbB- HǶû[miBQM TQHvMQKBH2x^{(p 1)/2} 1 /K2i 2t+i2K2Mi p 1

2 bQHmiBQMb [mB bQMi 2t+i2K2Mi H2b +``ûb KQ/mHQpX

dX X _2K`[mQMb [mǶûiMi /QMMû [m2p2bi BKTB`- p > 2 2i ( 1)^k = 1bB 2i b2mH2K2Mi bB k 22ZX HQ`b ,

( 1) 2bi mM +``û /Mb Z/pZbB 2i b2mH2K2Mi bB( 1)^{(p 1)/2} = 1 KQ/p bB 2i b2mH2K2Mi bB(p 1)/222ZbB 2i b2mH2K2Mi bBp 124/Z bB 2i b2mH2K2Mi bBp⌘1 KQ/4X

#X PM `2i`Qmp2 [m2 k 2bi mM +``û KQ/mHQ Rd KBb Tb KQ/mHQ RR 2M +H+mHMi2^{(p 1)/2} KQ/mHQp TQm` p= 11TmBbp= 17X

aQB2Mi /2b 2MiB2`bn2i n<sup>0</sup> MQM /BpBbB#H2b T`pX

3X ě bB n2i n⁰ bQMi /2b +``ûb /2 Z/pZ- HQ`bn<sup>(p 1)/2</sup> ⌘1 2i (n<sup>0</sup>)<sup>(p 1)/2</sup>⌘1- /QM+ (nn<sup>0</sup>)<sup>(p 1)/2</sup> = 1⇥1 = 1X BMbB-nn<sup>0</sup> 2bi mM +``û KQ/mHQpX

ě bBn 2bi mM +``û /2Z/pZ 2in<sup>0</sup> MǶ2bi Tb mM +``û /2 Z/pZ-n^{(p 1)/2}⌘1 2i(n⁰)^{(p 1)/2}⌘ 1- /QM+(nn⁰)^{(p 1)/2}= 1⇥( 1) = 1X BMbB-nn⁰ MǶ2bi Tb mM +``û KQ/mHQpX

ě bB n 2i n⁰ M2 bQMi Tb /2b +``ûb /2 Z/pZ- HQ`b n^{(p 1)/2} ⌘ 1 2i (n⁰)^{(p 1)/2} ⌘ 1- /QM+

(nn⁰)^{(p 1)/2}= ( 1)⇥( 1) = 1X BMbB-nn⁰ 2bi mM +``û KQ/mHQpX NX aQBij= exp(2i⇡

3 )X PM bmTTQb2 2M THmbp6= 3X 9

X .Mb C- j 2bi BMp2`bB#H2 2i j <sup>1</sup>=j<sup>2</sup> +`j³ = 1X P` j<sup>2</sup>2{P(j)|P 2Z[X]}=Z[j]X .QM+ j 2bi BMp2`bB#H2 /MbZ[j]/(p)/ǶBMp2`b2j˙²=j²X

PM TQb2b=j j ¹- ûHûK2Mi /2Z[j]/(p)X

#X HQ`bb<sup>2</sup>= (j j <sup>1</sup>)<sup>2</sup>=j<sup>2</sup> 2 +j <sup>2</sup>=j<sup>2</sup> 2 +j<sup>4</sup>=j<sup>2</sup> 2 +j= 3 + (1 +j+j<sup>2</sup>) = 3+`

1 +j+j²= 0X

BMbB-( 3)2bi H2 +``û /2b˙ /MbZ[j]/(p)X ii2MiBQM ,b MǶ2bi Tb mM ûHûK2Mi /2ZXXX

*QKK2p >3-pM2 /BpBb2 Tb( 3)X

HQ`b ( 3) 2bi mM +``û KQ/mHQ p bB 2i b2mH2K2Mi bB( 3)<sup>(p 1)/2</sup> = 1KQ/p bB 2i b2mH2K2Mi bBU/MbZ[j]/(p)V- (b<sup>2</sup>)<sup>(p 1)/2</sup> = ˙1bB 2i b2mH2K2Mi bBb<sup>p 1</sup>= ˙1bB 2i b2mH2K2Mi bBb<sup>p</sup> =b +`b 2bi BMp2`bB#H2 /MbZ[j]/(p)+`b²= ˙32bi BMp2`bB#H2 KQ/mHQpTQm`p >3X

6BMH2K2Mi 3 2bi mM +``û /MbZ/pZbB 2i b2mH2K2Mi bBb^p=b/MbZ[j]/(p)X +X P`-b<sup>p</sup>= (j j <sup>1</sup>)<sup>p</sup>=j<sup>p</sup>+ ( j <sup>1</sup>)<sup>p</sup>=j<sup>p</sup> j <sup>p</sup> +`p2bi BKTB`X

BMbB-( 3)2bi mM +``û KQ/mHQpbB 2i b2mH2K2Mi bBj^p j ^p=j j ¹X PM `2K`[m2 [m2j^p j ^p2bi `2bT2+iBp2K2Mi û;H ¨

• y bB p⌘0[3]-

• j j ¹bB p⌘1[3]-

• j² j ²bBp⌘3[3]X

6BMH2K2Mi- ( 3)2bi mM +``û bB 2i b2mH2K2Mi bBp⌘1[3]X

/X .ǶT`ĕb H [m2biBQM3X-3 2bi mM +``û /2Z/pZsietseulementsi( 3)2i( 1)bQMi iQmb H2b /2mt +``ûb Qm #B2M MB( 3)MB( 1)M2 bQMi /2b +``ûbX

.QM+p⌘1[3]2i p⌘1[4] U+Ƕ2bi ¨ /B`2p⌘1[12]V Qm #B2Mp⌘( 1 [3]2i p⌘( 1)[4] U+Ƕ2bi ¨ /B`2p⌘( 1)[12]V

RyX aQBi!= exp(i⇡

4)- pm +QKK2 ûHûK2Mi /2Z[!]/(p)/ǶBMp2`b2! ¹= ! 2ib=! ! ¹ PM +H+mH2b²=!² 2! ²= 2X

.QM+ T` mM `BbQMM2K2Mi bBKBHB`2- k 2bi mM +``ûsietseulementsi b<sup>p</sup>=!<sup>p</sup>+! <sup>p</sup>=bX PM `2K`[m2 [m2 TQm` /2b pH2m`b /2 pBKTB` ,

• bBp⌘1[8], b^p=b

• bBp⌘3[8], b^p= b

• bBp⌘5[8], b^p= b

• bBp⌘7[8], b^p=b

6BMH2K2Mi- k 2bi mM +``û KQ/mHQpbB 2i b2mH2K2Mi bBp=±1[8]X

S`iB2 9 ,

G2 MQK#`2p2bi iQmDQm`b mM MQK#`2 T`2KB2` BKTB`X PM /û}MBi H 7QM+iBQM /2 G2;2M/`2Lp aQBi⇣= exp(2i⇡

p )X

PM /û}MBi H bQKK2 /2 :mbb `2HiBp2 ¨pT` H2 MQK#`2 +QKTH2t2G= X

a2Z/pZ

Lp(a)⇣^a.

RX X aQBi(a, t)2Z²X HQ`bLp(a<sup>2</sup>t) =Lp(a<sup>2</sup>)⇥Lp(t) =Lp(t)+`a² 2bi mM +``û /QM+Lp(a²) = 1X

#X aQBiS= X

(a,b)2((Z/pZ)^⇤)²

Lp(ab)⇣^a+bX

§ iQmi +QmTH2 (a, b)2(Z/pZ^⇤)²- QM T2mi bbQ+B2` mM mMB[m2 +QmTH2 (a, t)2M TQbMi t=ba <sup>1</sup>X GǶûHûK2Mit2bi BMp2`bB#H2 T` T`Q/mBi /ǶBMp2`bB#H2bX BMbB-

8

S= X

(a,t)2((Z/pZ)^⇤)²

Lp(a(at))⇣^a+(at)

= X

(a,t)2((Z/pZ)^⇤)²

Lp(a²t)⇣^a(1+t)

= X

(a,t)2((Z/pZ)^⇤)²

Lp(t)⇣^a(1+t)

= X

t2(Z/pZ)^⇤

Lp(t)

Ñ X

a2(Z/pZ)^⇤

⇣^a(1+t) é

SQm` t= ( 1)- X

a2(Z/pZ)^⇤

⇣^a(1+t)= (p 1)UMQK#`2 /2 i2`K2b iQmb û;mt ¨ RV SQm` t6= ( 1)- X

a2(Z/pZ)^⇤

⇣^a(1+t)= X

b2(Z/pZ)^⇤

⇣^b T` mM +?M;2K2Mi /ǶBM/B+2b 2i /QM+

X

a2(Z/pZ)^⇤

⇣^a(1+t)= Ñ

X

b2(Z/pZ)

⇣^b é

⇣⁰= 0 1 = ( 1) 2M +QKTHûiMi H bQKK2X 6BMH2K2Mi-

S= (p 1)·Lp( 1) X

t2(Z/pZ)^⇤\{ 1}

Lp(t).

+X _2K`[mQMb [m2P

a2(Z/pZ)^⇤Lp(a) = 0+` BH v miMi /2 +``ûb KQ/mHQp[m2 /2 ǴMQM@+``ûbǴ KQ/mHQpX *2ii2 bQKK2 2bi /QM+ +QMbiBimû2 /ǶmiMi /2 R [m2 /2( 1)X

HQ`bG²=

Ñ X

a2(Z/pZ)

Lp(a)⇣^a é

·

Ñ X

b2(Z/pZ)

Lp(b)⇣^b é

= Ñ

X

a2(Z/pZ)^⇤

Lp(a)⇣^a é

· Ñ

X

b2(Z/pZ)^⇤

Lp(b)⇣^b é

+`Lp(0) = 0

= Ñ

X

(a,b)2((Z/pZ)^⇤)²

Lp(a)Lp(b)⇣^a⇣^b é

=

Ñ X

(a,b)2((Z/pZ)^⇤)²

Lp(ab)⇣^a+b é

=S= (p 1)·Lp( 1) X

t2(Z/pZ)^⇤\{ 1}

Lp(t)/ǶT`ĕb +2 [mB T`û+ĕ/2

=p·Lp( 1) X

t2(Z/pZ)^⇤

Lp(t) =pLp( 1) X

t2(Z/pZ)

Lp(t) =pLp( 1) 0.

6BMH2K2Mi-

G²=p·( 1)^{(p 1)/2}. kX aQB2Mi p2i q/2mt MQK#`2b T`2KB2`b BKTB`b /BbiBM+ibX

X PM /ûD¨ pm [m2 Lp(a) =Lp(q²a) =Lp(q)·Lp(qa)X aB Lp(a) = 0- HQ`bLp(a) = (Lp(a))^qX

aB Lp(a) = 1- HQ`bLp(a) = (Lp(a)p)^q

1M}M- bBLp(a) = ( 1)- HQ`bLp(a) = (Lp(a))<sup>q</sup> +`q2bi BKTB`X

#X SQbQMbt=aq+QKK2 T`û+û/2KK2MiX X

a2(Z/pZ)^⇤

(Lp(a)⇣^a)^q KQ/q= X

a2(Z/pZ)^⇤

((Lp(a))^q⇣^aq) KQ/q

= X

a2(Z/pZ)^⇤

Lp(q)Lp(aq)⇣^aq KQ/q

e

= X

t2(Z/pZ)^⇤

Lp(q)Lp(t)⇣^tKQ/q

=Lp(q)· X

t2(Z/pZ)^⇤

Lp(t)⇣^t KQ/q

=Lp(q)·GKQ/qX

+X 1M miBHBbMi H [m2biBQMkX#/2b T`ûHBKBMB`2b- G^q ⌘

Ñ X

a2(Z/pZ)^⇤

Lp(a)⇣^a éq

KQ/q= X

a2(Z/pZ)^⇤

(Lp(a)⇣^a)^q KQ/q⌘Lp(q)·GKQ/q.

/X .ǶT`ĕb H [m2biBQMRX+-(G²)^{(q 1)/2}·GKQ/q= (p·( 1)^{(p 1)/2})^{(q 1)/2}·GKQ/q

=p^{(q 1)/2}·( 1)^{(p 1)(q 1)/4}·GKQ/q

=Lq(p)·( 1)^{(p 1)(q 1)/4}·GKQ/q jX 6BMH2K2Mi- QM Q#iB2Mi T`kX+2i kX+,

Lp(q)G⌘G^q⌘(G²)^{(q 1)/2}⌘( 1)^{(p 1)(q 1)/4}Lq(p)GKQ/qX BMbB-Lp(q)G⌘( 1)^{(p 1)(q 1)/4}Lq(p)GKQ/qX

1M KmHiBTHBMi T` G- QM Q#iB2Mi , Lp(q)G<sup>2</sup>⌘( 1)<sup>p21·q21</sup>Lq(p)G<sup>2</sup>KQ/q /QM+ T` RX+,

Lp(q)p( 1)^{(p 1)/2}⌘( 1)^p21·q21Lq(p)p( 1)^{(p 1)/2} KQ/q

P` p^q= 1 /QM+p( 1)<sup>(p 1)/2</sup>=±p2bi BMp2`bB#H2 /Mb Z/qZX 6BMH2K2Mi- Lp(q) = ( 1)^p21·q21Lq(p).

d