C4 - Jean-Romain HEU

Q
n∈N∗P=anXn+an−1Xn−1+· · · +a1X+a0n
p
p an
∀k∈ {0,1, . . . , n −1}p ak
p2a0
PQ[X]
Q[X]
P1= 2X4−3X3+ 12X2+ 9X−6, P2=X10 −5, P3=X2−4X+ 4.
pQ[X]
X−1Xp−1
P=Xp−1
X−1Xp−1X−1
PQ[X]
P
PC[X]
QQ[X]P X + 1 Q(X) = P(X+ 1)
P X + 1
P=Xp−1
X−1Q
QQ[X]
p1≤k≤p−1p
p
k
PQ[X]

M z
z
Q[X] 2kk∈N
p
p
e
2iπ
p
p
2k+ 1 k
Q
n n ≥3n
n
n= 2jp1p2· · · pm,
j m pi
pi= 2ki+ 1 ki∈N
n n

1,2,3,4 6
u v R2u v
C={nu +mv ∈R2|n∈Z, m ∈Z}.
θRRθO θ R
Rθ(R,◦)
Id
G={Rθ∈ R | ∀w∈C, Rθ(w)∈C}
G∀Rθ∈G, ∀Rθ0∈G, Rθ◦Rθ0∈G
Id ∈G
G
(u, v)C
u= (1,0)
v= (0,1) u= (2,0) v= (0,1) u= (1,0)
v= (1
2,√3 2)
G={Id, Rπ
2, Rπ, R3π
2}
G
u v C θ ∈R
Rθ∈G
Rθ
Mθ=cos(θ)−sin(θ)
sin(θ) cos(θ)∈ M2(R).
R2
w=w1
w2R2Rθ(w) = Mθw

u=u1
u2v=v1 v2
n, m, p q Mθu=nu+mv Mθv=pu+qv
P=u1v1
u2v2
MθP=nu1+mv1pu1+qv1
nu2+mv2pu2+qv2
u1v2−u2v16= 0 P
P−1=1
u1v2−u2v1v2−v1
−u2u1.
N=P−1MθP N
A=a b
c d∈ M2(R)ATr(A) = a+d
A B M2(R) Tr(AB) = Tr(BA)
Tr(N) = Tr(Mθ)
2 cos(θ)
θ
R3R4R5
R3R4