¯
k
φ(ps)k1≤k < ps
psE={1,2, . . . , ps}F={k∈E:p|k}φ(ps) = card(E)−card(F) =
ps−card(F)F={p, 2p, 3p, . . . , ps=ps−1p}(F) = ps−1φ(ps) = ps−ps−1
m n
f:Z/mnZ→Z/mZ×Z/nZ
¯x[mn]7→ (¯x[m],¯x[n])
f¯x[mn] = ¯y[mn]mn |x−y m n
m|x−y n |x−y¯x[m] = ¯y[m] ¯x[n] = ¯y[n]
f f(¯x¯y[mn]) = (xy[m], xy[n]) == (x[m], x[n])(y[m], y[n]) =
f(¯x[mn])f(¯y[mn])
f(¯x[m],¯x[n]) = (¯y[m],¯y[n]) m|x−y n |x−y m n
mn |x−y¯x[mn] = ¯y[mn]f
¯x∈Umn ⇔f(¯x)∈U(Z/mZ×Z/nZ=Um×Unφ(mn) = card(Umn =card(Um×Un) =
card(Um)(Un) = φ(m)φ(n)
n=pk1
1···pks
sφ(n) = Qs
i=1 φ(pki
i) = Qs
i=1(pki
i−pki−1
i)
Gn={z∈C|zn= 1}
Gn(C∗,·)
G n (C∗,·)G=Gn
G
Gn(C∗,×) 1 ∈Gnu, v ∈Gn(uv−1)n=
un(vn)−1= 1 uv−1∈Gn
Gnz=exp(2kπi
n) = exp(2πi
n)kGn=grhξi={1, ξ, ξ2, . . . , ξn−1}
ξ=exp(2πi
n)
G n (C∗,×)∀z∈G zn= 1
G⊂Gn|G|=|Gn|G=Gn
z=a+ib ∈Ca, b ∈R(z) = ea(cos b+isin b)
f: (C,+) →(C∗,×)f(z) = exp(z)
u, v ∈Cu=a+bi v =c+di a, b, c, d ∈R
exp(u+v) = exp(a+c)(cos(b+d) + isin(b+d)) = exp(a) exp(c)[cos(b) cos(d)−sin(b) sin(d) + isin(b) cos(d) +
isin(d) cos(b)]
exp(u+v) = exp(a)(cos(b) + isin(b)) exp(c)(cos(d) + isin(d)) = exp(u) exp(v)
exp (C,+) →(C∗,×)
z=a+bi ∈Cu∈Kerf ⇔exp(z) = ea(cos(b) + isin(b)=1 ea(cos(b)=1 ea(sin(b)=0
ea>0 sin(b) = 0 cos(b) = 1 a= 0 b= 2kπ k ∈Z
Kerf ⊂2πiZz= 2kπi k ∈Zexp(z) = 1
Kerf = 2πiZ
z=a+bi ∈Imf z 6= 0 z=ρeiθ u=c+di exp(u) = ecedi =ρeiθ
c= ln(ρ)d=θ2π z ∀u∈C∗f
f=C∗
G x 7→ x−1G G
∀x, y ∈G(xy)−1=y−1x−1= (yx)−1xy =yx G
2(R)A=0 1
−1−1B=0−1
1 0 A, B
AB