Nombre de polynômes irréductibles de degré n sur un corps fini

n
Q d Xpn−X d|n
Q d Fp/(Q)K Fpd
pd−1y y
ypd=y y =R(¯ X)R d −1¯ X
pd−1pd−1pn−1d|n n =dq +r
pn−1 = pdq+r−1 + pr−pr=pr(pq−d−1) + pr−1qr−1
r= 0 d|n¯ X pn−1
Q|Xpn−X Q|Xpn−X Xpn=X
R(¯ X)pn=R(¯ X)ypn=y pd−1|pn−1
d|n
Xpn−X=Qd|nQQ∈Kd
pQ
Xpn−1
Fpn
Qd|nQQ∈Kd
pQ|Xpn−X
Kp
d
pn=Pd|ndIp
d
Id
n= 1/n Pd|nµ(n/d)qd
f
R+x < 1F(x) = P∞
n=0 f(x/n)
F(x)µ(1) = f(x) + f(x/2) + . . .
F(x/2)µ(2) = −f(x/2) −f(x/4) −. . .
F(x/3)µ(3) = −f(x/6) −f(x/6) + . . .
F(x/4) = 0 ∗F(x/4) + . . .
f(x) = P∞
1µ(n)F(x/n)
f(x) = xIp
xx0F(x) = Px|ndIp xx
qx0f(x) = P∞
1µ(n)F(x/n) = Pd|nµ(d)qd/n =
Pd|nµ(n/d)qdd7→ n/d n