Open Idempotentes em Álgebras de Grupos e Códigos Abelianos Minimais

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4 " 4 , ! !

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%

( 8 < = < > 8 ! FqCn - 8 5 !

Cn $ 5 5 Fq $ 8 8 !

; # # 8 ! - - 8 !

-8 # 8 < 8 - $

- FqCn - 8 < > 8 ! - 8

! <8 8 !> 8 $ 8

#

4 5 6 3 ! ; 5 ; ; *!

• F_q 5 q q / % , ;

• Cn " n;

• FqCn ! ;

• mα(x) 2 α;

• Ci 2 ;

• RG * ;

• ei $ R;

• d(x, y) =_|{i:xi =yi, i= 0, . . . , n−1}| : . ;

• degP $ P.

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8 / " # * / '

/ ) ! ! 9

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"$ 8 #

6 4 5 A q

/ 8 # F A 8 #

% n A / 8 n.

n/ ! 4 $ An_{, n}

∈N:_{C ⊂}An_.

6 / 1 8 L# . #

$ $

, / , # 3 ! 8

- ! ! 1950 M M !

! 8 $

-D ! - ! !

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$ 8 / 6 ( - 4 %

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* 6 - $ $ /

- $ $ /! ! #

! $

Fn

q - ! 8 / 5 Fq q #

C / 8 $ c= (c , c , . . . , cn− )∈ C

$ (cn− , c , . . . , cn− ) ! $/ c i→i+1

n !/ / $ C# F "

' - !

5 5 #

F! $ $ Fn

q n ! Fq / - !

FqCn Cn / " n# 5 !

% " Fn

q FqCn#

" FqCn ' FqCn# * /

! ! / " F_q

$ n# - " / FqCn#

F/ G ! 5 '

! FqG/ ! #

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2

! 8 !4 $ ! #

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! !

6 L ! * ?#E#

5 9 ! ! 6 8=

?#?K ! 9 5 4

8 ! #

G / 5 F / "

F _$ G ! /

!

$ # $

! 8 #

- ! " @

N 6 ! '

# $ " !

FqCn " 5 Cn n "

' ! "

$ #

F " B / ! 8 # ( ,

! ! 5 FG ! 5

! 9 , / " 4 ,

! ! # '

!

* 8 # 3 ! 9

9 : " !

#

#

/

&

1!

" 5& 9 5 9

! ! 5 $ $

! 8 # * 9 "

OH AP#

/(/

#

3 4 p_∈N ,

Fp =GF(p) ={0,1, . . . , p−1}

4 # ϕ:Zp →Fp 5 ϕ(a) =a/ ! 4 # D

F_p 9 ' ϕ / 5 8

p# ϕ/ 5 #

3 4 F 5 K ! F# F ! K

" ! [F:K] / F $ $ ! K# F F /

8 ! K [F:K]/ 5 #

= /(/ " F ! K F# _|K_|=q$ % _|F_|=qm_#

& [F:K] = m$

# F_{/ $ !} K F_{/ 5} [F:K] =m

m_∈N# *

F∼₌Km_.

|F_|=qm_#

! /(/ " F ! $ % _|F_| = pm_{# p F}

[F:Z_p] =m$

3 4 R $ # p_∈R / ! R

9 - +

?# p= 0 p /_{∈ U}(R)#

@# 3 p=bc b _{∈ U}(R) c_{∈ U}(R) / p $ #

# /(/ " K & & f(x) _∈ K[x]$ % f(x) '

K # #

K[x]

f(x)

' $

3 K=F_p ∂(f(x)) =m ,

Fp[x]

f(x) ={r(x) + f(x) :r(x)∈Fp[x] ∂(r(x))< m}

/ , 2 Fp[x] m ! pm#

Fp[x]

f(x) ={a +a x+ +am− x

m− _:a

i ∈Fp} ∼=Fp

7 /(/ >4 ? @ K ' & & f(x)_∈K[x] '

K# L K ( f(x)$

= /(- " K p >0$ % :

)$ pa= 0# a_∈K$ *$ (a_±b)p _=ap

±bp _{# a, b}

∈K k _∈N$

+$ , ϕ:K_→K ! ϕ(a) =ap _{' ! " $}

- #

Im(ϕ) =Kp

' K$

3 4 f(x) _∈ K[x] "$ ! F# 6 Q =

& L K

?# f(x) = (x₋a ) (x₋am) L#

@# L=K[a , . . . , am]#

F L / 8 , f(x) ! K# L /

N I1J / f 9 ! F K _⊆ F _⊆ L

F =L#

7 /(- " p . f(x) = xp _{−x∈ F}

p[x]# n ∈N$

% L ' , f(x) L pn _$

# 3 8 L 4 f(x) ! Fp#

mdc(f(x), f′_{(x)) = 1 "' f(x) L# D L}

pn _{# * ! 4}

F =_{α_∈L:f(α) = 0_}=_{α _∈L:αp _=α

}

L# / - $ 5 F / ! L pn _#

F =L#

3 8 L 8 pn _{# L}• _{/ $}

pn_{−1 6 D α}p − _{= 1 α ∈L}• ₄

αp _{=α α}

∈ L•_{# * α}p _{=α α}

∈ L# f(x)

-L#

f(x) =

α∈L

(x₋α),

/ 4 #

! /(- / & ! # pn _{# $}

! /(9 0 GF(pn_{) =F}

p ' , f(x) = xp −x ∈

F_p[x]$

! /(A 0 Fq # & & . p & & .

n_∈N$

= /(9 " G a, b_∈G# m n$ % :

)$ % k =mmc(m, n)$

*$ G ' N# aN _{= e#}

a_∈G$

7 /(9 " K & & G ! K•_{$ % G '}

$ % # K ' ! # K = Z_p(α)#

α _∈K$

# 3 4 |G_| = n# I2J D ?#B & ,

& N aN _{= 1 a ∈ G# 2 f(x) = x}N _{−1∈ K[x]}

& N "' K n _≤ N#

6 D N _≤ n# n =N G / n

/ G/ " # & α_∈K• _K• _{= α # Z}

p(α) /

& / α Z_p K=Z_p(α)#

7 /(A " K & & $ % xm

−1 xn

−1 K[x] # # m n$ ' #

mdc(xm

−1, xn

−1) =xmdc m,n

−1.

# * $ & q, r_∈Z

n=qm+r, 0_≤r < m.

D

xn

−1 = xqm r

−1 = xr_xqm

−xr_+xr

−1

= xr_(xqm

−1) +xr

−1

= xr q−

i

xiq _(xm

−1) +xr

−1.

xm_{−1 $ x}n_{−1 K[x] m $ n#}

! /(; 1 & & . p#pm

−1 pn

−1 # #m

n$

3 4 F _{5 "} p# α_∈F_/

F=Zp(α)# , $ F /

φ(pn₋1),

φ / - #

7 /(; " p . p m, n_∈N$ % :

)$ Fp ' Fp # # m n$

*$ 1 m n# . F_p F_p $

# I1J 3 8 F_p 4 ! F_p # F_p

! F_{p /} k k _∈N_{# D}

pn = (pm)k=pkm

m $ n# 1 6 ?#C ! xp − ₋₁

$ xp − _{−1# D "' x}p

−x Fp / !/ "' xp −x#

F_{p 4} F_{p / !} F_{p #}

I2J $ m n & , ! Fp Fp

& pm _{"' x}p _−x#

7 /(B " f(x)_∈F_q[x] 2 k# q=pm_{$ %}

F_q ' , f$

# 3 4 α ' f & Fq(α)# 3

f(x) =a +a x+ +akxk,

D ?#@

f(αq_{) =} k

j

aj(αq )j = k

j

ajαjq = k

j

ajαj q

= 0,

/

αq_{, α}q _{, . . . , α}q −

"' f(x) 8 " α# 3 4 l

$

αq _=α.

g(x) =

l−

j

(x₋αq ₎

D ?#@

(g(x))q ₌ l−

j

(x₋αq ₎q ₌ l−

j

(xq

−αq _{) =} l−

j

(xq

−αq _{) =g(x}q_).

*

g(x) =b +b x+ +blxl,

l

j

bq_jxjq _{= (g(x))}q _=g(xq_{) =} l

j

bjxj,

4 bq_j =bj g(x)∈Fq[x]# g(x) $ f(x) f(x)

g(x) $ f(x) Fq[x]# g(x) =f(x) l =k#

F_q / f

1, α, . . . , αk−

#

7 /(8 " f(x)_∈Fq[x] 2 k# q=pm$ %

f(x) xq _{−x F}

q[x] # # k n$

# 3 8 f(x) $ xq _{−x F}

q[x]#

F_q f(x) Fq "' f(x) Fq # *

6 ?#R k $ n# 1 8 k $ n#

6 ?#R F_q Fq # * f(x) $ xq −x Fq [x] f(x)

xq _{−x- ! F}

q # f(x) $ xq −x Fq[x]#

C /(/ 3 & α ' ( 2 f(x) k#

αq_{, α}q _{, . . . , α}q −

( f(x)$ - & mdc(p, q₋1) = 1 &

αq q− = 1,

F•

q ' q−1# " # " α , 4

& & F_q F•

q$ ' # f(x)

( # ( xq _{−x $}

! /(B 0 2 f(x) =xq _−x∈F q[x]

2 2 Fq# " n$

# ! $ g(x) h(x) 2 2 "$

! Fq $ f(x) g(x)h(x) $ f(x) mdc(g(x), h(x)) = 1# *

F! $ #

3 4 α Fq # F 2 α " ! mα(x) /

2 2 "$ ! F_q α '# α αp

2 $ 6 ?#H [mα(x)]p = mα(xp)# * /

k mα(x) / $

αq

≡1 (modqn

−1).

3 4 η_∈F_{q $ #}

F_q =F_q(η) = η =_{ηi _{:i= 0,1, . . . , q}n

−2_}.

D

xq ₋x=m (x) ml(x),

mi(x) =mη (x) =mη (x) i= 1, . . . , l# *

Rm ={ηi, ηqi, ηq i, . . . , η q

− _i

}

/ 4 "' mi(x) ∂(mi(x)) = ki $ n# 4

- F_{q |Rm|} = 1 F_q#

Fq ' 4

Sq − ={0,1, . . . , qn−2},

!

Ci ={i, qi, q i, . . . ,(qk− )i},

qn

−1# 1 - σ : Sq − → Sq −

5

σ(i) = iq (modqn

−1)

! 5 # !

Ci ={i, qi, q i, . . . ,(qk− )i},

Sq − 4 Ci 2 mi(x) ki / $

qki_≡i (modqn₋1) (αq _≡1 (modqn₋1)).

F 4 Ci 8 2 qn−1 Iqn−1J

2 # i " 4 _Ci /

mi(x) = j∈C

(x₋ηj).

# /(/ " p = 2# n= 4 q= 2$ % F ' # 16 #

2 15 :

C = _{0_}

C = _{1,2,4,8_}

C = _{3,6,12,9_}

C = _{5,10_}

C = _{7,14,13,11_}.

/(- %

#

' 5 RG G !

R. * / 9 ! G !

R RG 4 6

8= #

3 4 G R # 5 RG 4

S - T

α =

g∈G

α(g)g, α(g)_∈R.

5

α=

g∈G

α(g)g β =

g∈G

β(g)g _∈RG,

α / β

α(g) =β(g), _∀ g _∈G.

F α, " ! supp(α) / 4

supp(α) =_{g _∈G:α(g) = 0_}.

5 α β RG

α+β =

g∈G

(α(g) +β(g))g I?#?J

λ_∈R

λα=λ

g∈G

α(g)g =

g∈G

(λα(g))g I?#@J

5 α β RG

αβ =

g∈G h∈G

(α(g)β(h))gh=

f∈G

γ(f)f, I?#BJ

γ(f) =

g∈G

α(g)β(g− _f)

α(g)β(h) R gh G# 9

5 I?#?J I?#BJ RG /

1 =

g∈G

α(g)g,

α(1) = 1 α(g) = 0 g _∈ G_{− {}1_}# F RG 8

#

RG I?#?J I?#@J / R # * /

R - $ RG / R ! ! R. 8

RG 3 G ! R#

* - ι:G_→RG 5

ι(g) =

h∈G

x(h)h_∈RG,

x(g) = 1 x(h) = 0 h _∈G_{− {}g_}# / 5 # *

5 G ι(G) RG# 5

G ! RG ! R#

* - ν :R_→RG 5

ν(a) =

g∈G

x(g)g _∈RG,

x(1) =a x(g) = 0 g _∈G_{− {}1_}# / 5 # *

5 R ν(R) RG#

* 5 9 + ag=ga RG a_∈R

g _∈G#

= /(A " σ : R _→ S ! # σ(1) = 1# M S5 4 & $ % M ,

a_∗m=σ(a)m

' R5 4 & $

7 /(D " R G $ %

: & & S R & & , σ:G_→S & σ(gh) =

σ(g)σ(h)# g, h _∈ G# . ! R5 ϕ :

RG _→S & σ =ϕ_◦ι# ι:G_→RG$ % # R_{⊆ Z}(S)# ϕ ' ! R53 $

# * - ϕ:RG _→S 5

ϕ

g∈G

α(g)g =

g∈G

α(g)σ(g)

4 #

! /(8 " R ϕ : G _→ H ! $ % . ! ϕ:RG _→RH & ϕ_|G =ϕ$ % #

R ' # ϕ ' ! R53 $ ' #

ϕ ' " # " " # ϕ ' '$

= /(; " S #R S#

S# G $ %

SG_≃S_⊗RRG.

# ! S_⊗RRG/ 5 ! $

(s_⊗α)(t_⊗β) = st_⊗αβ, _∀ s, t _∈S α, β _∈RG.

* - σ : G _→ S _⊗R RG 5 σ(g) = 1⊗g - ' 9

6 ?#A & , 8 5 R ϕ : SG _→

S_⊗RRG

ϕ(α) = 1_⊗α.

- τ :S_×RG_→SG 5

τ(s, α) =sα

/ 8 5 R ! # * & , 8 5 R

! ψ :S_⊗RRG→SG 5

ψ(s_⊗α) = sα.

(ψ_◦ϕ)(α) =ψ(1_⊗α) = α

(ϕ_◦ψ)(s_⊗α) =ϕ(sα) =s(1_⊗g) = (s_⊗α),

4 ψ_◦ϕ=ISG ϕ◦ψ =IS⊗ RG# SG≃S⊗RRG#

# /(- " R G# H $ %

R(G_×H)_≃(RG)H _≃RG_⊗RRH.

# * - σ:G_×H _→(RG)H 5

ϕ(g, h) = gh

/

σ((g , h ) (g , h )) = σ(g g , h h ) = g g h h =g h g h

= σ(g , h ) σ(g , h ), _∀ (g , h ),(g , h )_∈G_×H.

* 6 ?#A & , 8 5 R !

ϕ:R(G_×H)_→(RG)H

ϕ

 

g,h∈G×H

α(g, h)(g, h)



 =

g,h∈G×H

α(g, h)σ(g, h)

=

h∈H g∈G

α(g, h)g h.

U - $ 5 ϕ/ ! 4 #

D ?#R ! (RG)H _≃RG_⊗RRH RG⊗RRH R(G×H) - #

# /(9 " _{Ri}i∈I R= i∈IRi$ %

RG_≃

i∈I

RiG,

& & G$

F 9 5 ! R G

RG 4 #

7 /(E >7 2? @ " G $ %

RG ' # # ,6 :

)$ R ' $

*$ G ' ! $

+$ _|G_| ' R.

F R =F / / : F/

|G_| / $ "$ F _|G_| = 0 F /

" F _{$ |}G_|#

! /(D " G ! K $ % KG ' #

# K G.

7 6 L ! *

- 9 ! ! #

7 /(/. " G ! F & F |G_|$ % :

)$ FG ' . ! _{Bi} ≤i≤r# 5

FG$ Bi ' $

*$ / & FG' _{Bi} ≤i≤r$

+$ Bi ' (

Mn(Di)# Di F #

!

FG_≃

r

i

Mn (Di)

' ! F53 $ 7$ % ( Mn (Di)# "

Ii =                       

x 0 0

x 0 0

$$

$ $$$ $ $$ 0

xn 0 0        

:x , x , . . . , xn ∈Di               

≃D_in

' 3 & $ 8 x_∈FG#

φ(x) = (α , . . . , αr)∈ r

i

Mn (Di)

! x mi ∈Ii

xmi =αimi.

! , Ii 5 FG5 $

9$ i=j# Ii ≃Ij$

:$ / & FG5 ' Ii# i= 1, . . . , r$

! /(E " G ! F &

F _|G_|$ % :

FG_≃

r

i

Mn(F)

n +n + +nr =|G|.

F , R 8

R.

7 /(// " R=_⊕s

i Ai ,

$ % _{e , . . . , es} R

& :

)$ ei = 0 ' R# i= 1, . . . , s$

*$ i=j, eiej = 0$

+$ 1 =e + +es$

7$ ei ei =e′i+e′′i# & e′i e′′i

& e′

ie′′i = 0# 1≤i≤s$ F e , . . . , es 8

R#

/(9 $

#

%

$ ! ! 5 G !

F " $ /

! 4 # ' - L = OGP#

3 4 G " 5 n+

G= a =_{1, a, a , . . . , an−

}

F " F $ n# - $

ϕ: F[x] _→ FG f(x) _→ f(a)

U - $ 5 ϕ/ 5 6 ( 5

FG_≃ F[x]

kerϕ,

kerϕ=_{f(x)_∈F[x] :f(a) = 0_}.

F[x]/ " kerϕ/ 2

2 "$ f (x) f (a) = 0# f (x) = ma(x)

a / /! ! F# U ! $ ! 5

a /

ϕ(a) =x+ f (x) _∈ F[x]

f (x) .

U xn

−1_∈kerϕ an _{= 1#}

f (x) =

r

i

kixi

/ 2 r < n

f (a) =

r

i

kiai = 0

RG 1, a, a , . . . , ar _{! F#}

kerϕ= xn

−1

FG_≃ F[x] xn₋₁ . 3 4

xn₋1 =f (x)f (x) ft(x)

- xn

−1 - "$ ! F# " F

$ n fi = fj i = j# D 6 8 % 1

!

FG_≃ F[x] xn₋₁ ≃

F[x]

f (x) ⊕

F[x]

f (x) ⊕ ⊕

F[x]

ft(x)

,

fi(x) + fj(x) =F[x].

* a G /

ϕ(a) = (x+ f (x) , . . . , x+ ft(x) ).

3 4 ζi "' fi(x) - 8 /! F#

F[x]

fi(x) ≃

F(ζi).

FG_≃F(ζ )_⊕F(ζ )_{⊕ ⊕}F(ζt).

ζi "' xn−1 FG

-& 9 F# a /

(ζ , ζ , . . . , ζt).

C /(- "

fi(x) =

xn

−1

fi(x)

=

j i

fj, i= 1, . . . , t.

% ' 3 ! & f (x), . . . , ft(x) & xn−1

fi(x)fj(x) i=j$ # g (x), . . . , gk(x)∈F[x] &

f (x)g (x) + +fk(x)gk(x) = 1.

1 ei =fi(a)gi(a)# FG# i= 1, . . . , t#

e + +et= 1 eiej =δijei.

1 #

FG=FGe _{⊕ ⊕}FGet.

; ( ζi =aei#i = 1, . . . , t# & FGei ' 3 ei mζ(x) =

fi(x)$

# /(- " G 7#

G= a =_{1, a, a , a , a , a , a _},

F=Q . $ %

x ₋1 = (x₋1)(x +x +x +x +x +x+ 1)

' , x ₋1 F$ # ζ ' ( '

#

FG_≃F_⊕F(ζ).

- # a G '

(1, ζ).

; #

f (x) = x +x +x +x +x +x+ 1 f (x) = x₋1

&

1

7f (x) + − 1

7 f (x) x + 2x + 3x + 4x + 5x+ 6 = 1.

< #

e = 1

7 a +a +a +a +a +a+ 1 e =− 1

7 a +a +a +a +a +a−6 .

# /(9 " G 25 2m _{F = Q}

. $ & FG' 2m _F$

+ # 7 ! m#

G_≃C _{× ×}Cm,

Ci i= 1, . . . , m " 2# 3 m = 1 & ?#@

FG_≃F_⊕F,

x ₋1 = (x₋1)(x+ 1).

3 8 4 $ /

k " 2 1_≤k < m# ?#@ !

FG_≃F(C _{× ×}Cm− )×Cm)≃(F(C × ×Cm− ))Cm.

D 8

F(C _{× ×}Cm− )≃F ⊕ ⊕Fm− , Fi ≃F⊕F.

FG_≃(F _{⊕ ⊕}Fm− )Cm ≃F Cm⊕ ⊕Fm− Cm FiCm ≃FCm⊕FCm

4 #

* $ - $ !

" ! ' ! 5 #

D ! n n' 2 2

Φn(x) / 5

Φn(x) = mdc n,j

(x₋ζj_),

ζ 'n / $ Φn(x)/ φ(n)#

3 4 ζn ' n/ $ F = Q ,

#

-σ : F[x] Φn →

F(ζn)

5 σ(f+ Φn ) =ζn / 5 #

F(ζn)≃

F[x] Φn

/ & 2 F φ(n)#

3 n = kd ζk _{/ d / ' d/ $}

# * & & d / 2 2

Φd(x) = mdc d,j

(x₋ζjk),

xn₋1 =

≤ ≤

d|n

Φd(x).

d 5& 4

Φd(x) = a

i

fd(x),

- Φd(x) 2 "$ ! F#

FG ! - +

FG_≃

n

d|

a

i

F[x]

fd ≃ n

d|

a

i

F(ζd) .

d5& ζd "' n/ $ #

- F(ζd ) $

FG_≃

n

d|

adF(ζd),

adF(ζd) ad - - F(ζd)#

φ(d) = ad[F(ζd) :F].

ad=

nd

[F(ζd) :F]

,

nd / , d G / φ(d)#

# /(A " G n#

G= a =_{1, a, a , . . . , an− _},

F=Q . $ %

xn

−1 =

≤ ≤

d|n

Φd(x)

' , xn_{−1 F$ # ζ}

d ' ( d5'

#

FG_≃

n

d|

adF(ζd),

ad =

nd

[F(ζd) :F]

= nd

φ(d) = 1.

% # n= 6#

x ₋1 = Φ (x)Φ (x)Φ (x)Φ (x)

= (x₋1)(x+ 1)(x +x+ 1)(x ₋x+ 1).

1 #

FG_≃F_⊕F_⊕F −1 + √

−3 2 ⊕F

1 +√₋3 2 .

* ! !

5 - 4 / - '

6 *! #

7 /(/- > F ? @ " G ! n K

& & & K _|G_|$ %

KG_≃

n

d|

adF(ζd).

! /(/. " G ! n. %

QG_≃

n

d|

adQ(ζd).

! /(// " G ! n F &

F n$ F ' ( n5' #

FG_≃

n

i

F_i, Fi =F#i= 1, . . . , n$

' !

# - 9 , !

#

& / N

! FG & ! ,

FG ' " #

3 g / G o(g) g 4

C(g) =_{h− _gh:h∈G}

/ 4 g G# 3 C(g) / 4 5 5

Ag = h∈C g

h

RG / 8 G ! R#

7 /(/9 " G R $ % 5

" _{Ag}g∈G G R Z(RG)$

7 /(/A " G ! F &

F _|G_|$ % . FG ' . " , G$

C /(9 F F

|G_|# . FG 3 &

. " , G$

3 4 G H ! G# 3 H = _{1_} ?#G

& , 8 5 ǫ:RG_→R

ǫ

g∈g

α(g)g =

g∈g

α(g).

∆(G, H) = kerǫ=

g∈G

α(g)g :

g∈G

α(g) = 0

/ RG 8 #

0 $ e / R e '

R 4

R=Re_⊕R(1₋e),

a =ae+a₋ae =ae+a(1₋e) Re_∩R(1₋e) =_{0_}.

& !

#

RG ! 4 5 H G _|H_|/

$ "$ R $ H

eH =H =

1

|H_|

h∈H

h_∈RG.

= /(B " R H !

G$

)$ _|H_| ' R# e_H =eH RG$

*$ H ' G # #eH' RG$ % #

RGeH ≃eHRG

(RG)eH ≃R

G H .

+$ H ' G#

RG=RGeH⊕RG(1−eH),

RGeH ≃R

G

H RG(1−eH) = ∆(G, H).

# I1J ! $ eH ! 5

eH =

1

|H_|

h∈H

h= 0.

eHy=eH y∈H hy∈H#

eH =eH

1

|H_|

h∈H

h = 1

|H_|

h∈H

eHh=

1

|H_|

h∈H

eH =

1

|H_||H|eH =eH.

I2J 3 H / ! G gHg− _{=H g ∈G# D}

geHg− =g

1

|H_|

h∈H

h g− = 1

|H_|

h∈H

ghg− = 1

|H_|

h∈H

h=eH.

geH =eHg g ∈G 4 eH / # * / RGeH ≃eHRG#

1 eH / RG g ∈G

eH =geHg− ⇒

1

|H_|

h∈H

h= 1

|H_|

h∈H

ghg− .

D gHg− _{=H g ∈G 4 H / ! G#}

g _∈G

geH =g

1

|H_|

h∈H

h =g 1

|H_| |H| −

h∈H

(1₋h) =g+δ,

δ=₋g 1 |H_|

h∈H

(1₋h)_∈∆(G, H).

D - 9

σ:RGeh →

RG

∆(G, H) ϕ:

RG

∆(G, H) →R

G H

5

σ(geH) =g+ ∆(G, H) ϕ(g+ ∆(G, H)) =gH

5 #

(RG)eH ≃R

G H .

I3J U - $ 5

RG=RGeH⊕RG(1−eH).

*

RG(1₋eH) = RG |H| − h∈H

h _⊆

h∈H

RG(1₋h) = ∆(G, H).

(1₋h)(1₋eH) = 1−h−(1−h)eH = 1−h−eH+heH = 1−h

∆(G, H)_⊆RG(1₋eH)# RG(1−eH) = ∆(G, H)#

V H = G eG / 8 RG#

RG / H =G′ _{! $ G ∆(G, G}′_{) /}

$ RG#

= /(8 " R H# K

! G & H _⊆K$ %

dimRG(eH−eK) = dimRGeH−dimRGeK.

# H _⊆K eHeK =eK# D

geH =geK+geH−geK =geK+g(eH−eK) eK(eH−eK) = 0,

4

RGeH =RGeK ⊕RG(eH−eK),

/ 4 #

3 4 G = a " pn _{p , 5& #}

& , ! H = ap _{G i= 0, . . . , n#}

{1_}= ap

≤ ap −

≤ ≤ ap

≤ a =G

/ , / G 4 !

G=G _⊃G _{⊃ ⊃}Gn={1}

! " G Gi = ap #

= /(D " F = Q . # G = a

pn_{# p . ! #}

G=G _⊃G _{⊃ ⊃}Gn={1}

G$ %

e =G ei =Gi −Gi− , i = 1, . . . , n,

" FG$ ' # FGei ≃F(ζp )#

ζp ( pi5' # [FGe:F] =e(1)pn# & &

e FG e(1) ! 1 e$

#

FG_≃

n

j

F(ζp )

FG / & n+ 1 $ #

e ei =G Gi−Gi− =G Gi−G Gi− = 0, i = 1, . . . , n,

G Gi =G , i= 0,1, . . . , n.

* / 1_≤i_≤j

GiGj =Gi.

D

eiej = (Gi−Gi− )(Gj−Gj− ) =Gi−GiGj− −Gi− +Gi− =δij(Gi−GiGj− ).

*

e , e , ei, . . . , em

m+ 1 # $

n

i

ei =G +G −G +G −G + +Gn− −Gn− +Gn−Gn− = 1.

D ?#H !

FG_≃F FGei ≃F

G Gi

ei ≃FL, i= 1, . . . , n,

L " pi_{# * D ?#G}

[FGei :F] = (FG)Gi :F !

− (FG)Gi− :F

!

= pi₋pi−

= φ(pi) ="F(ζp ) :F #

.

FGei ≃F(ζp ) i= 1, . . . , n#

ei(1)pn=

1

pn−i −

1

pn−i− p

n

=pi₋pi− = [FGei :F].

e (1)pn= 1 = [FGe :F].

* [FGe:F] =e(1)pn _{$ e FG#}

C /(A - < 1.8 & 2 2

Q$ # =5 3

# " # < ' # # " G 3 F=F ! $ %

x ₋1 =

≤ ≤

d|n

Φd(x) = Φ (x)Φ (x) = (x−1)(x +x+ 1)

= (x₋1)(x₋2)(x₋4).

< #

FG_≃ F[x] x ₋1 ≃

F[x]

x₋1 ⊕

F[x]

x₋2 ⊕

F[x]

x₋4 .

1 #

QG_≃ Q[x] x ₋1 ≃

Q[x]

x₋1 ⊕

Q[x]

x +x+ 1 ≃Q⊕Q(ζ ),

ζ ( $ # 3 FG ' =

# & 3 QG '

$ 1 # < $

# /(; " G 3 F = Q .

$ %

G = a =_{1, a, a , . . . , a _}

G = a =_{1, a , a , a , a , a , a , a , a _}

G = a =_{1, a , a _}

G = a =_{1_}.

< #

e = G = 1

27(1 +a+a + +a +a )

G = 1

9(1 +a +a +a +a +a +a +a +a )

G = 1

3(1 +a +a )

G = 1.

#

e = G ₋G = 1

27(2−a−a + 2a −a − −a + 2a −a −a )

e = G ₋G = 1

9(2−a −a + 2a −a −a + 2a −a −a )

e = G ₋G = 1

3(2−a −a ).

1 # eiej =δijei# e +e +e +e = 1

ei(1)3 = 3i −3i− .

#

-7

( " ! 6

! '

! ! $

!4 $ ! 8 + "

' / 2

"

Fq[x]

xn₋₁ ,

F_{q / 5} q n/ ,

# * $/ 5

F_q[x]

xn₋₁

! FqCn " 5 Cn n 5 !

% ! "$ 2

! FqCn. $ " !

F_qCn.

6 * / I J

! / I $ J#

" ' ! "

$ ! #

-(/

1!

3 4 A 4 5 q 8 #

F A 8 # % n

A 8 n#

An _{4 $ n ! A, /}

An

=_{(c , c , . . . , cn− ) :ci ∈A, i = 0, . . . , n−1}.

n/ ! 4 $ An _n∈N+

C ⊆ An

.

!4 $ 6 /

$ $/ / !

! # ! $ 9 &

$ 6 #

x= (x , . . . , xn), y = (y , . . . , yn)∈An,

8 > ? , x y

- 4 +

d(x, y) =_|{i:xi =yi, i = 0, . . . , n−1}|

C ⊆An _{8 > C , +}

d = min_{d(x, y) :x, y _{∈ C}, x=y_}.

! - ! A % : - (n, m, d)

$ I , n N

! An _{C J , : " #}

F !4 $ / N

'

-"$ # F N / 8

! , # / 8 ! , #

-(-

=

! 8 4

/! - ! 8 5 Fq q # * /

4 $ _C ! Fq - !

$ $ Fn

q / 8 #

$ x= (x , . . . , xn− ) Fnq, 5 ,

w(x) =_|{i:xi = 0, i= 0, . . . , n−1}|.

$

w(x) =d(x,0),

d : . # F C /

w(_C) = min_{w(x) : x_{∈ C − {}0_}}.

& $ $ _C# *

! 8 +

! 2

{f , . . . , fn}

Fn

q $ v , !

-v=v f +v f + +vnfn, vi ∈Fq i= 1, . . . , n.

3 4 _C k ! Fq# 3

{e , . . . , ek}

/ ! 2 Fk

q

{c , . . . , ck}

/ ! C

-ν :Fk

q −→F n

q 5 ν(ei) =ci, i= 1, . . . , n,

/ 4 Imν = _C# $ ' +

Fk q

ν

−→ Fn q

| |

Fk q

ν|_F

−→ C

7 'G - ν ! 2

Fk

q Fnq $ # $ ! C !

2 Fn

q# _

             

c = b f + b f + + bn fn

c = b f + b f + + bn fn ### ### ### ### ### ### ### ###

ck = b kf + b kf + + bnkfn 5 bij ∈Fq#

ν(ei) =ci=b if +b if + +bnifn, i= 1, . . . , k,

' ν N ! 2 /

G=        

b b b k

b b b k

### ### ###

bn bn bnk         .

'G $ _C

4 ' _C / ! Fn

q ' G I

- ! _CJ# F _C $ w_∈Fn

q

- w=ν(v) v_∈Fk

q# ' G∈Mn×k(Fq) 4

-! _C / 8 ( ! , ( _C#

F $ C / $/

-! 4 π : Fn

q −→ Fnq−k kerπ = C +

!

{c , . . . , ck}

C I $ J !

{c , . . . , ck, v , . . . , vn−k}

Fn

q# * v∈Fnq , !

-v=λ c + +λkck+λk v + +λnvn−k

λi ∈Fq i= 1, . . . , n# * - π :Fnq −→Fnq−k 5

π(v) =λk v + +λnvn−k.

/ kerπ =_C# - +

Fn q

π

−→ Fn−k q

| |

C −→ 0

H _∈ Mn−k×n(Fq) ' (n−k)

- π ! 2 Fn

q Fnq−k $ # kerπ = C

C / 4 $ w_∈Fn

q - '

Hwt_{= 0,}

'H / - $

C# * ' H / 8 ( (

! , _C# _C !/

5 $/ ' #

* $ π ν +

Fk q

ν

−→ Fn q

π

−→ Fn−k q

| | |

Fk

q −→ C −→ 0

C =Imν = kerπ# 3 x_∈Fk q

(π_◦ν)(x) =π(ν(x)) = 0,

ν(x)_∈Imν =_C = kerπ.

!

HG= 0.

-(9

$ Fn

q, 5

+ " # '

2 $ ' "$

-% $ $ - !

9 / ! 5 5

! 9 2 # *

" #

F _{5 $ & " 5}

q 4 F=GL(q)

G= a =_{1, a, . . . , an−

},

" n# 3 i _→ i+ 1

$

c= (c , c , . . . , cn− )

! $

c = (cn− , c , . . . , cn− ),

8 x# 3 x l

9 N ! $

cl = cn−l , . . . , cn−l , . . . , cn, c , . . . , cn−l .

C / 8 "

$

c= (c , c , . . . , cn− )∈ C,

/ !/ $ C# U $ 5 $ c_{∈ C}

2

c(x) =c +c x+ +cn− xn− ∈F[x].

* " $

c= (c , c , . . . , cn− )∈ C,

x c(x),

& / ' n# n

xn_{−1 4}

Rn=

F[x]

xn₋₁ ={r(x) + x n

−1 :r(x)_∈F[x] ∂(r(x))< n_{} ≃}Fn

Rn / F ! / " # 0 $

FG_{≃ R}n.

7 -(/ @ _C n F ' # #

C ' _Rn$

# 3 8 C 4 " ! F# xc(x) _{∈ C}

2 c(x)_∈F[x]#

xic(x)_{∈ C}, _∀ i.

C /

a(x)c(x)_{∈ C}, _∀ a(x)_{∈ R}n.

C / Rn#

1 4

c(x) =c +c x+ +cn− xn−

$ #

xc(x)_{∈ C},

C / Rn# C / " ! F#

F! mdc(q, n) = 1 ?#A ! FG /

4 $ !

! # *

/ !

FG# "

" # $ 5

5 9 - : " ! " #

3 4 I Rn# & , 2 2 g(x)_∈F[x]

g(x) = _I g(x)/ $ xn_{−1# g / 2}

I I / n₋∂(g(x))# 1 $ xn₋₁

Rn# 3 8

g(x) =

n−k

i

cixi =c +c x+ +cn−k− xn−k− +xn−k.

g(x), xg(x), . . . , xk− g(x)

I# I /

-h(x) g(x),

∂(h(x))< k _{I I} / k#

' _I /

-G=        

c c c cn−k− 1 0 0 0

0 c c cn−k− cn−k− 1 0 0

### ### ### ### ### ### ### ###

0 0 0 0 c c cn−k− 1

        .

3 4 _C (n, k) "

g(x) =

n−k

i

cixi h(x) = k

i

bixi.

Rn

h(x) g(x)_≡0 =

n−

i

n−

j

cjbi−j xi

n−

j

cjbi−j = 0, i= 0,1, . . . , n−1.

bi = 0 i <0 i > k# D $

(c , c , . . . , cn−k,0, . . . ,0)

/ $

(bk, bk− , . . . , b ,0, . . . ,0)

" # ' $ 5 C H =        

bk bk− bk− b b 0 0 0

0 bk bk− b b b 0 0

### ### ### ### ### ### ### ###

0 0 0 0 bk bk− b b

        ,

/ In₋kJ

GHt_{= 0.}

* H / '

C⊥ =_{x_∈Fn_:xct _{= 0,}

∀ c_{∈ C}}

C# C⊥ _{/ (n, n−k) " αh}∗_(x)

h∗(x) =xkh 1 x

2 " h(x) α / 8 αh∗_{(x) 4 2 #}

7 ! mdc(q, n) = 1 "' xn

−1 4

Rn#

FG_{≃ R}n ≃

F[x]

g (x) ⊕

F[x]

g (x) ⊕ ⊕

F[x]

gt(x)

,

xn₋1 =g (x)g (x) gt(x)

- xn

−1 - "$ ! F# Mi = gi(x) i= 1, . . . , t

& Rn

Bi = hi(x) = *

xn

−1

gi(x) +

, i = 1, . . . , t.

Rn#

Rn≃FG≃B ⊕B ⊕ ⊕Bt.

* 6 ?#?? & ei(x)

e (x) +e (x) + +en(x) = 1 ei(x)ej(x) =δijei(x).

D

Bi =FGei(x) = ei(x) ,

ei(x) / ! Bi

f(x) =

t

i

fi(x),

fi(x)∈Bi

ei(x)f(x) =fi(x), i= 1, . . . , t.

Bi hi(x) ei(x) Mi gi(x)

1₋ei(x)

Bi⊕Bj = mdc(hi(x), hj(x)) = *

xn

−1

gi(x)gj(x) +

Mi = gi(x) = *

xn₋₁

g (x) gi− (x)gi (x) gt(x) +

= B _{⊕ ⊕}Bi− ⊕Bi ⊕ ⊕Bt.

4 _I g(x)#

mdc(g(x), h(x)) = mdc g(x),x

n₋₁

g(x) = 1.

D & a(x), b(x)_∈F[x]

a(x)g(x) +b(x)h(x) = 1.

e(x) =a(x)g(x)_{∈ I} !

e(x) =e(x) +a(x)b(x)(xn

−1) =e(x)

/ _I = e(x) #

#

9

%

/ " ! 8 # 9 !

"$ #

,

! ! 5 FG 9 ,

4 " 4 , ! !

# 6 !

L Q8W 8 ' 6 # O?KP

' 5 / ' ! 5

/ , !

' 6 '

FG# '

! * 8 #

,

#

9(/ C

G

#

+

#

" F 5 $ & " 5

q 4 F=Fq =GL(q) G ! 5 n

mdc(q, n) = 1# ?#A ! FG / #

* /

FG_≃

r

i

(FG)ei ≃ r

i

Fi,

Fi ≃(FG)ei i= 1, . . . , r & 9 5 F ei $

FG# ' / , r

! # ! 5

! , # 5

A=

r

i

Fei.

F! $ Fei ≃ F $ - , r

/ !/ A _{$ $ !} F_#

= 9(/ " α FG$ % α _∈ A # # αq _{= α$ %}

#

α=

g∈G

α(g)g,

α(g) =α(gq_{) = =α(g}q −

),

g _∈G$

#

α=

r

i

αi ∈A,

αi = αei ∈ Fi i = 1, . . . , r# α / A

αi ∈ Fei i = 1, . . . , r# Fei ≃ F αqi = αi i = 1, . . . , r#

D ?#@ !

αq ₌ r

i

αi q

=

r

i

αqi = r

i

αi =α.

g∈G

α(g)g =α=αq ₌ g∈G

α(g)g

q

=

g∈G

α(g)gq

α(g) =α(gq_)#

3 4 C =_{1_} 8 g /_{∈ C} #

C =_{gq :j = 0, . . . , tg −1}={g , gq, . . . , gq

−

},

tg / $

gq =g qt _≡1 (mod_|g _|),

G / 5 # * 8 g /_{∈ C ∪ C} !

C =_{gq :j = 0, . . . , tg −1},

tg / $

gq =g qt _≡1 (mod_|g _|).

! G q 2

G=_{C ∪ C ∪ ∪ C}s.

G= a / " # g _∈ G !

- g =ai

Ci ={i, qi, q i, . . . , qt− i}.

tg & # - # mdc(q,|gi|) = 1 &

a, b _∈Z

aq+b_|gi| = 1⇔aq≡1 (mod|gi|)⇔q∈ ,

,_U(Z_|g|),,,

U(Z_|g|) =_{r _∈Z_|g|:mdc(r,_|gi|) = 1} ,

,_U(Z_|g|),,=φ(_|gi|). D

gi =gi =g aq b|g|

i =g

aq i .

T =_{g , g , . . . , gs}

/ 4 q 2 #

7 9(/ " F G $ % .

FG' . q5 2 G$

# 3 ! , FG / N A

! F# 7 ! ! ! s #

q 2 Ci 5

ηi = g∈C

g _∈FG, i = 1, . . . , s.

η_iq =

g∈C

g

q

=

g∈C

gq ₌ g∈C

g =ηi

ηi ∈A i= 1, . . . , s#

%: # _B =_{η , . . . , ηs} / ! A ! F s=r#

-s

i

αiηi = 0⇒ s

i g∈C

αig = 0,

αi = 0 i= 1, . . . , s G # D

B / 4 # * $ B A#

α _∈A

α=

g∈G

α(g)g.

α=

g∈G

α(g)g =

g∈G

α(g)g

q

=

g∈G

α(g)q_gq_.

3 α(g) _∈ F α(g)q _{= α(g) D B#? ! α(g) = α(g}q₎

g _∈G#

α=

g∈G

α(g)ηi,

4 _B A#

! 8 $ L = OGP ,

! ! 5 G /

N ! , ! " G , - " #

h _{∈ C}i h=giq j# mdc(q,|gi|) = 1

gi = h .

q 2 Cg / ! 4 4

" g 4

Cg ⊆ Gg ={gr :mdc(r,|g|) = 1}={gr:r ∈ U(Z|g|)}, ∀ g ∈G.

* , ! " G / - , /

Cg =Gg, ∀ g ∈G.

D ! G / $ n gn _{= 1}

g _∈G#

7 9(- " F G e & mdc(q,_|G_|) = 1$ % _Cg = Gg# g ∈ G # # U(Ze) '

q _∈Z_e$ - # q ' ( # " #

qφ e _≡1 (mode).

# 3 8 _Gg =Cg g ∈G# G / & e

& g _∈G e _Gg =Cg # D r∈Z

r _{∈ U}(Ze) gr ∈ Cg & j ∈Z r =qj# q U(Ze)#

1 8 U(Z_e) 4 " q#

g _∈G _|g_| $ e q _∈Z_|g|/ U(Z|g|)# h ∈ Gg & r _∈Z h=gr_{# D r∈ U(Z}

|g|)# * & j ∈Z r =qj

h=gq

∈ Cg# Gg =Cg#

= 9(- _U(Ze) ' # # e = 2# 4# pn 2pn# & p '

. n ' $

! 9(/ " F G e & mdc(q,_|G_|) = 1$ % _Cg =Gg# g ∈G # # ,6 :

)$ e= 2 q ' . $ *$ e= 4 q _≡3 (mode)$

+$ e=pn_{# & p ' . |q|=φ(e) U(Z} e)$

7$ e= 2pn_{# & p ' .}

|q_|=φ(e) _U(Z_e)$

#

G=_{g , . . . , gk},

e=mmc(_|g _|, . . . ,_|gk|).

D 6 B#@ _Cg = Gg g ∈ G U(Ze) /

" q _∈Z_e#

I1J 3 e = 2 G / 2 _U(Ze) / " # 3 q / ,

"

qφ e =q_≡1 (mode).

q / U(Ze)#

I2J 3 e= 4 q _≡3 (mode) _U(Ze) / "

qφ e _=q

≡1 (mode).

q / U(Z_e)#

1 q / _U(Ze)

qφ e =q _≡1 (mode).

D e $ q ₋1 = (q₋1)(q+ 1)# mdc(e, q) = 1 e $ q + 1#

q _≡3 (mode)#

I3J 3 e=pn _{|q|=φ(e) U(Z}

e) U(Ze) / "

qφ e _=q|q|

≡1 (mode).

q / U(Ze)#

1 q / _U(Z_e)

qφ e

≡1 (mode).

|q_|=φ(e) _U(Ze)#

I4J I3J

U(Z _p )_{≃ U}(Z )_{× U}(Z_p )_{≃ U}(Z_p )

φ(2pn_{) = |q| U(Z} p )#

B#? G = Cm / " FCm QCm

, m= 2 4 pn _2pn

q F - ' N # 4

$ FCm 4 8

QCn $ " #

9(-$ QG

$ ! " ! 5

' - #

F & & $

" ' ! "

pn _{p / , #}

= 9(9 " F ! # _|F_|=q#G= a pn_#

p . #

G=G _⊃G _{⊃ ⊃}Gn={1}

G$ %

e =G ei =Gi −Gi− , i= 1, . . . , n,

" FG &

e +e + +en= 1.

# 3 D ?#A#

0 $ F! $ ?#C / D B#B ' 4

$ QG $ ! #

+

! 9(- " F ! _|F_| = q G = a

pn_{# p . $ % " < 3.3'}

" # # , :

)$ p= 2# n= 1 q . n= 2 q _≡3 (mod4)$ *$ p ' . o(q) =φ(pn₎

U(Zp )$

# D B#B & & n+ 1 FG#

& G / pn _{$ p}n _q

B#?#

7 9(9 > 2 % @ " F ! # _|F_|=q# G= a pn_{# p . o(q) =φ(p}n₎

U(Z_p )#

G=G _⊃G _{⊃ ⊃}Gn={1}

G$ % " 5

FG '

e =G = 1

pn g∈G

g ei =Gi−Gi− , i = 1, . . . , n.

# % B#@#

6 B#B 4

FG " pn _{! F#}

8 * O??P

& 2 #

F " 2pn

- #

3 4 G " 2pn _{p , " #}

G=C_×A

A p ! 3> < G C =_{1, t_} 2 ! 3> <# *

FG_≃F(C_×A)_≃(FC)A _≃(F_⊗F)A.

& ?#B $ FC

e = (1 +t)

2 e =

(1₋t) 2 ,

FA 6 B#B# * !

+

7 9(A >% 2 @ " F q G

2pn_{# p . # & o(q) =φ(p}n_{) U(Z} p )$

1 G = C _×A# A p5 A G C = _{1, t_} 25 A$ ei#i= 0,1, . . . , n# FA#

FGp :

1 +t

2 ei

1₋t

2 ei, i= 0,1, . . . , n.

# FC e = (1 +t)

2 e =

(1₋t) 2

FA 6 B#B #

: " "

Ii = (FG)(-Gi−Gi− )

- ,

& ! # F 2

! #

- + 3 ei(X) ∈ Fq[X] / 2

ei(a) =ei 2 Ii /

gi(X) = mdc(ei(X), Xp −1), i= 0,1, . . . , n.

ei(X) =

1

pn−i p −₋

j

Xjp

−_pn−1i

p − ₋

j

Xjp−

= 1

pn−i  p

p −−

j

Xjp ₋

p − −

j

Xjp−

 

= 1

pn−i 

(p₋1)

 

p −₋

j Xjp  ₋ p− j

Xjp−

 

p −₋

j Xjp     = 1

pn−i p− p−

j

Xjp−

 

p −₋

j

Xjp

 

Xp

−1 = (Xp

−1)

p −₋

j

Xjp

= (Xp− ₋1)

 

p− p−

j

Xjp−

 

 

p −₋

j

Xjp

 .

' (Xp−

−1) - 8 ! F / '

p₋

p− p−

j

Xjp−

gi(X) = mdc(ei(X), Xp −1)

= (Xp−

−1)

 

p −₋

j

Xjp  .

D (gi(X))/ pn−pi+pi− #

dim(Ii) =pn−∂(gi(X)) =pi−pi− =ϕ(pi).

9(9

%

$ 3 !

5 # 7 p # 3 4 G p ! #

! H G

G= G

H ={1}

4 " FG# F! $ G/ "

pn _{6 % & , ! H}∗

G H _,

, , ,H ∗ H , , , ,=p.

5 eH =H−H-∗ ! eH = 0 +

= 9(A 0 eH# ! # eG = G " 5

FG# " ' 1$

#

e_H = H₋H-∗ _H−H-∗ _=H−HH-∗_−HH-∗_+H-∗ _=H−H-∗ _=e

H.

3 4 H K - ! G

G

H ={1} G

K ={1}

4 " H∗ _K∗ _{! G H K $}

, , , ,H ∗ H , , , ,=p

, , , ,K ∗ K , , , ,=p.

3 H K $ H _⊂K H∗ _⊆K

eHeK = (H−H-∗)(K −K-∗) =HK−HK-∗−H-∗K+H-∗K-∗ = 0.

3 H K $ H, K _⊂ HK# * H∗_{, K}∗ _{⊂ HK# D}

H∗_K∗ _{⊂HK# HK =H}∗_K∗_#

HK _⊂HK∗ _⊂H∗K∗

HK∗ _{=HK# ! H}∗_{K =HK.}

eHeK = 0.

eHeG = 0 eGeK = 0.

! " C G _G(C) 4

C ! /

G(C) =_{c_∈C :mdc(o(c),_|C_|) = 1_}.

3 _C / - " ! " G

|G_| =

C∈C

|G(C)_|

G / p

|G(C)_|=_|C_{| −}|C| p .

* 4 S 4 ! H G

G= G

H ={1}

4 "

e=

H∈S

eH.

%: # e = 1#

- ! $ (FG)e=FG# 0 $

# D

(FG)e =

H∈S

(FG)eH dim(FG)e= H∈S

dim(FG)eH.

dim(FG)eH = dim(FG)H−dim(FG)H-∗,

H =H-∗_+e

H H-∗eH = 0

(FG)H = (FG)eH⊕(FG)H-∗.

I2J D ?#H !

dim(FG)eH = dimF

G

H −dim(F G

H∗ IB#?J

dimF G H =

, , , ,_HG

, , ,

, dimF _HG∗ =

, , , ,_HG∗

, , , ,.

$ & - ! 4 σ :_{C → S}

|X_|=

, , , ,_σ(G_X)

, , ,

,, ∀ X ∈ C.

3 C _{∈ C} ! G φ(C) = H 6

6 ( 5

dimF G

H =|C| dimF G H∗ =

, , , ,_HG∗

, , , ,= , , , , , G H H∗ H , , , , ,=

|C_| p .

*

|G(C)_|=_|C_{| −} |C|

p = dim(FG)eH.

dim(FG)e=

H∈S

dim(FG)eH = C∈C

|G(C)_|=_|G_|

e= 1#

7 9(; " p . G p5 pr_$

% " < 3.4 ' "

FG # # ,6 :

)$ pr _{= 2 q . $}

*$ pr _{= 4 q≡3 (mod4)$}

# % D B#C B#?#

7 9(B " p G p5 2pn_{$ 1}

G = E _×B# E 25 B p5 $ % #

FG ef# & e '

FE f FB.

F! $ $ FB 6 B#R

& ?#B $ FE - e =

e e em

ei =

1 +ti

2 ei = 1₋ti

2 , i= 0,1, . . . , m.

B#? , $

! ! 5 #

9(A )

)

H

3 8 G / 2m_pn _{p , "}

m _≥0#

G=E_×B E = t _{× ×} tm ,

E 2 ! 2m _{I $ $ J B p}

3> <# 0 $ 6 B#H $ FE

- e=e e em

ei =

1 +ti

2 ei = 1₋ti

2 , i= 0,1, . . . , m.

$ FG - eEeB eE /

$ FE eB $ FB#

& eE FE y∈E

y=tε tε

m , εi ∈ {0,1}, i= 0,1, . . . , m.

*

yeE =tε

1_±t

2 t

ε m

1_±tm

2 =±eE = (−1)

ε _e

E, IB#@J

εy ∈ {0,1}.

$ - eEB#

(FG)eEB ! - γeEB

γ =

y∈E,b∈B

xybyb,

D

γeEB =

y∈E,b∈B

xybyeEbB=

y∈E,b∈B

xyb(−1)ε eEB.

I = (FG)eEB / 1 : " /l(I) =|G|#

* - e = eEeH eE ∈ FE eH =

H₋H-∗ _{H / ! B}

B H

/ " pi _H∗ _{/ , ! B H}

[H∗ :H] =p.

3 4 Ie = (FG)e b∈B B = b, H #

H∗ = bp− , H .

(1₋bp−

)eEH = (1−bp

−

)eE(H-∗+eH) = (1−bp

−

)eEeH ∈Ie.

bp−

∈H

supp((1₋bp− )H) =H_∪bp− H

/ 4 /

w((1₋bp−

)eEH) = 2|E| |H|,

l(Ie) : " Ie l(Ie) ≤

2m _|H|# 3

B =H_∪bH _{∪ ∪}bp− _H,

4 !

G=E_×H_∪b(E_×H)_{∪ ∪}bp− (E_×H)

4 # * α FG !

-α=

p−

j

αjbj, αj ∈F[E×H].

IB#@J - hH =H h_∈H !

αjeEeH =αjeEeH =kjeEeH, kj ∈F, j= 0,1, . . . , pi−1.

(FG)eEeH ⊂(FG)eEH,

0 =γ _∈(F A).eEeH =Ie

!

-γ =αeEH = (k +k b+ +kp− bp− )eEH,

5 kj = 0# 3 γ = kjbjeEH eEH ∈ (FG)eEeH

/ # * 5 - kj kj′

- ' γ#

l(Ie)≥2m |H| l(Ie) = 2m |H|.

$ ! 4

- FGe e $ FG# 3 4 e = eEeH

$ #

FGeEeH =F[E×B]eEeH = ((FE)B)eEeH = (FEeE)BeH.

(FE)eE / - F $ FE,

FGeEeH ≃FBeH.

* IB#?J

dim(FGeEeH) =φ(pi). !

# 9(/ " G 2 F = F = $ %

G = a =_{1, a, a , a , a , a , a , a , a _}

G = a =_{1, a , a , a _}

G = a =_{1, a , a _}

G = a =_{1_}.

1 # FG :

e = G = 2 + 2a+ 2a + 2a + 2a + 2a + 2a + 2a

e = G ₋G = 2 +a+ 2a +a + 2a +a + 2a +a

e = G ₋G = 1 + 2a +a + 2a

e = G ₋G = 2 +a .

& '

1

!:

O?P 8 8 > # # 0 3# Q# 3# 1# # !

< X = 1995#

O@P (# # = 1# # * <

X = 1975#

OBP #L# (# 1 B $ # ( L > (

< X = 1981#

OCP D# 8 Y B == < X = 1971#

ORP 1# ' S3 ! T $ 279

I2004J 191 203#

OHP D 1# .# ; ; $ > - 8 (

* $ # 20 1983#

OGP 3# # L = S*! ! - 5 T $ $ $

# 68 I1950J 420 426#

OAP # 3#Q# 3 8 C B Q < *

8 2002#

OEP # # # 0 # < B 8 .

8# 3 # $ #184 $ * 1996#

O?KP 1# *# ' # # S( ! !

T; ; 13I2007J 382 393#

O??P # 8 3#Q# * S - < 8 T; ; $

3 I1997J 99 113#

O?@P 3#Q# * # 8 S > - 8 2pn_{T ; ; # 5} I1999J 177 187#

O?BP 0#0# 1 C - 8 # 3 7

< X = 1995#