Polynômes irréductibles

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R R
RX8+X4+ 1
X8+X4+ 1 R
Q
1+(X−1)2(X−3)2Q[X]
R
E={P∈R[X]∃Q, R ∈R[X]P=Q2+R2}
E
E={P∈R[X]∀x∈R, P (x)≥0}
P= 65X4−134X3+ 190X2−70X+ 29 A B
Z[X]P=A2+B2
P∈Z[X]P P P
P, Q ∈Z[X] (P) = 1 R=P Q p R
p P p
Q
p P
Q R
(P)6= 1 P Q
R∈Z[X]P, Q ∈Q[X]R=P Q P1, Q1∈Z[X]
P Q R =P1Q1
Q Z
Z
X4+X+ 1 X6+X2+ 1 Z[X]
Z
a1, . . . , an∈Z
(X−a1). . . (X−an)−1Z[X]
(X−a1). . . (X−an)+1 n
P∈Z[X]P=Xn+an−1Xn−1+··· +a0X0p
a0≡0 [p], . . . , an−1≡0 [p], a06≡ 0 [p2].
PZ[X]
Xp−a
K C a∈Kp∈NXp−a
K K Xp−a=P Q P, Q ∈K[X]
PCP(0)

(X2−X+ 1)(X2+X+ 1)(X2−X√3 + 1)(X2+X√3 + 1)
α= 2 + r√2+1
2+ir√2−1 2, α, β = 2 −r√2+1 2−ir√2−1 2, β
PRP= (X2−2 (α)X+|α|2)(X2−2 (β)X+|β|2)
P=|Q+iR|2
P
P=1
65QQ Q = 65X2+ (49i−67)X+ (42 + 11i)Q Q[i]
P=A2+B2= (A+iB)(A−iB)A, B
B−B λ ∈Q[i]A+iB =λQ A −iB =λQ
2A= 65(λ+λ)X2+ ((49i−67)λ−(49i+ 67)λ)X+ ((42 + 11i)λ+ (42 −11i)λ)
2iB = 65(λ−λ)X2+ ((49i−67)λ+ (49i+ 67)λ)X+ ((42 + 11i)λ−(42 −11i)λ)
λλ = 65.
65λ∈Z[i]λ=u+iv
65 u, v ∈Z
A=uX2−67u+ 49v
65 X+42u−11v 65
B=vX2+49u−67v
65 X+11u+ 42v 65
u2+v2= 65.
67u+ 49v65 u≡8v[65]
65 u2+v2= 65 v=±1, u =±8
A=±(8X2−9X+ 5), B =±(X2+ 5X+ 2).
P=QR Q(ai)R(ai) = −1⇒Q(ai) = −R(ai) = ±1Q+R n
P=−Q2x→ ∞
P=Q2Q2−1=(Q−1)(Q+ 1) = (X−a1). . . (X−an)
Q−1Q+ 1 n= 2p
P=QR Q =Xn1+bn1−1Xn1−1+··· +b0X0R=Xn2+cn2−1Xn2−1+··· +c0X0
a0=b0c0p b0c0p
b0, b1, . . . , bk−1ak≡bkc0[p]p bkp
Q

a6= 0 Xp−a=P Q P, Q ∈K[X]n= deg(P)∈
[[1, p −1]] b= (−1)nP(0) ∈Kb p a bp=an
n∧p= 1 nu +pv = 1 bpu =anu =a1−pv a= (bu/av)p
bu/av∈KXp−a