Énoncé Partie 1. Exemples. Partie 2. Lemme de Gauss

Q[X]Z[X]
Q Z
+×
ΦQ[X]Z[X]Q
ZΦ = P Q P Q Q[X]
Z[X] deg(P)≥1 deg(Q)≥1
X2−X−1X3−X−1Z
n∈N∗a1, . . . , anΦ
Φ=(X−a1). . . (X−an)−1
Φ∈Z[X]
ZP Q Z[X] 1
Φ = P Q
a1, . . . , anP+Q
Φ = −P2
n a1, . . . , an
(X−a1). . . (X−an)+1 Z
P=Pn
k=0 akXkZ[X]P
c(P)
c(P) = (a0, . . . , an)
P c(P)=1
P Q Z[X]
P Q
P=
n
P
k=0
akXkQ=
m
P
k=0
bkXkP Q =
r
P
k=0
ckXk
p
k∈ {0, . . . , n}p ak
k0
k1k∈ {0, . . . , m}p bk
p ck0+k1
c(P Q) = 1
c(P Q) = c(P)c(Q)
Φ∈Z[X]P Q
1 Φ = P Q
P0Q0Z[X]P Q Φ = P0Q0
Φ∈Z[X] Φ Q
Z
Φ =
n
P
k=0
akXkZ[X]
p
•p an
•p aii∈ {0, . . . , n −1}
•p2a0
ΦQ

a, b, a0, b0Z
P1=X2−X−1=(aX +b)(a0X+b0)
aa0= 1 bb0= 1 a=±1b=±1P1
1−1P1(−1) = 1 P1(1) = −1
P1
a, b, a0, b0, c0Z
P2=X3−X−1=(aX +b)(a0X2+b0X+c0)
a0X2+b0X+c0Zaa0= 1
bc0= 1 a=±1b=±1 1 −1P2
P2(−1) = −1P2(1) = −1P2
k1nΦ(ak) = P(ak)Q(ak) = −1P(ak) = −Q(ak) = ±1
P(ak) + Q(ak)=0 akP+Q
deg(Φ) = ndeg(P)≥1 deg(Q)≥1
deg(P)≤n−1 deg(Q)≤n−1P+Q
n−1n Q =−P
Φ = −P2
x∈RΦ(x) = −(P(x))2≤0 lim+∞Φ=+∞
ΦZ
Φ2= (X−a1). . . (X−an)+1 Z
Φ2=P Q P Q Z[X]
1P Q
n−1
k1n
Φ(ak) = P(ak)Q(ak)=1⇒P(ak) = Q(ak) = ±1
P(ak)−Q(ak)=0 akP−Q
n−1 Φ = P2
n
Y
k=1
(X−ak) = P2−1=(P−1)(P+ 1)
Pdeg(P−1) = deg(P+ 1) = deg(P)
n= deg(P2−1) = 2 deg P n
(X−a1). . . (X−an)+1 Z
P p (a0, . . . , an)
k∈ {0, . . . , n}p akk
0n p akN
k0k p akk1k
p bkk0+k1
ck0+k1=
k0−1
X
k=0
akbk0+k1−k+ak0kk1+
k0+k1
X
k=k0+1
akbk0+k1−k
0≤k≤k0−1p akk0p
k0+ 1 ≤k≤k0+k1k0+k1−k < k1p bk
k1p
p ak0p bk1
p ck0+k1
c(P Q)
1
P Q Z[X]a0,· · · , anP
c(P)
∀k∈ {0, . . . , n}, ak=c(P)a0
kP1=
n
X
k=0
a0
kXk
a0
kZ(a0
0, . . . , a0
n)=1 P=c(P)P1P1∈Z[X]
Q=c(Q)Q1Q1∈Z[X]
c(P Q) = c(P)c(Q)c(P1Q1)
c(P Q) = c(P)c(Q)

a b P
Q P1=aP ∈Z[X]
Q1=bQ ∈Z[X]abΦ = P1Q1
ab|c(abΦ) = c(P1)c(Q1)m
c(P1)c(Q1) = abm
P2Q2Z[X]P1=c(P1)P2
Q1=c(Q1)Q2Φ = P0Q0P0=mP2Q0=Q2
ΦZ
Φ Φ Φ = P Q P Q
Z[X]m= deg(P)≥1r= deg(Q)≥1
P=Pm
k=0 bkXkQ=Pr k=0 ckXk
an6= 0 p6 |ann= deg(Φ) n=m+r
p|a0=b0c0p26 |a0=b0c0p
p|b0k{1, . . . , m}p6 |bk
p6 |an=bmcrp6 |bm
ak=bkc0+Pk
i=1 bk−icik≤m r ≥1k < n =m+r
p|ak
p|Pk
i=1 bk−icik p 6 |bkp6 |c0Φ
Z Q