Polynômes à une indéterminée

KPK[X]n n ∈N∗
n
KPK[X]P(x) = 0
x∈KP= 0
K=Z/pZp
Xp−1Xq−1R[X]p q
P=X5−13X4+ 67X3−171X2+ 216X−108 P P 0
P P 05P
A, B, C C[X]A2+B2=C2
B+C B −C
A, B C
K
K[X]P Q m n
P Q
R, S ∈K[X]
deg(R)≤n−1
deg(S)≤m−1
P R =QS
m= 3 n= 2
P=a3X3+a2X2+a1X+a0Q=b2X2+b1X+b0
≥1
a30b20 0
a2a3b1b20
a1a2b0b1b2
a0a10b0b1
0a00 0 b0
= 0
P=X3+pX +qK[X]
m n

R[X]C[X]
P n ≥1
P=
n
X
i=0
aiXi
R∈R+|P(z)|>|P(0)|z|z|> R
z0|P(z0)|= infz∈C|P(z)|.
P(z0) = 0 P(z0)6= 0 h(X) = P(X+z0)
P(z0)∈C[X]h
n h(X) = 1 + Pn
i=kbiXik=min{i∈N∗/bi6= 0}
ck, ..., cnck6= 0
s
h(s
ck
)=1−sk+
n
X
i=k+1
cisi.
x∈]0; 1[,
|
n
X
i=k+1
cixi−k| ≥ 1.
R[X]
R[X]
P1=X5−1
P2=X4+X2−2
P3=X4−X2+ 1
P= 48X4−32X3+ 1 P
R[X]
Q= 48X5+ 32X4+ 20X3−X−1
Q aj a
j=e2iπ
3
QR[X]
Q[X]
X3+ 2X2+X−1Q
R[X]
Q[X]
X4−X2+ 1
X4−2
P(X) = anXn+an−1Xn−1+... +a1X+a0Z[X]n n ≥1p
p an
p aii∈ {0, ..., n −1}
p2a0
PQ[X]

P
Z[X]
PQ[X]uvP =Q1.R1u, v ∈ZQ1, R1∈Z[X]
p uv p Q1R1
Xn−2Q[X]
nUnnC
Unn
U∗
nn
n
Φn=Y
ζ∈U∗
n
(X−ζ).
n≥1Xn−1 = Qd|nΦd(X)
Φ1, , Φ2, ..., Φ8.
n≥1 Φn∈Z[X]n
pΦpQ[X]
Φp(X+ 1)
Φn(X)Q[X]n≥1
L K L a∈LaK
P∈K[X]P(a) = 0 Ia={P∈
K[X]/P (a) = 0}K[a] = {P(a)/P ∈K[X]}
√2iQX2−2X2+ 1
p2 + √3Q
αQ[X]
αQPmin α
3
p2−√3
L K L a∈L
aK
K[a]
K[a]K
L K
M x ∈M x 6= 0 −x x−1
x y
K[x;y] = K[x][y]K
K[x]
x+y x ×y M
5
√3−3 √2
E E N

E Enn∈N
Z Q
[0; 1] R
(xn)n xn
y xn
Q
Qn[X]n n ≥1
α n Pmin α
(p;q)∈Z×N∗|Pmin p
q| ≥ 1 qn
k
∀(p;q)∈Z×N∗,0<|α−p
q|<1⇒ |α−p q|>k qn
α=
+∞
P
i=1
1
10i!