Fiche 2

X2−2 = 0
1 + X−5X2+ 8X3
s7→ 1 + s−5s2+ 8s3
z7→ 1 + z−5z2+ 8z3
F(,)f7→ Id +f−5f◦2+ 8f◦3
X♥
♥0+♥1−5♥2+ 8♥3
X
A
A, ,
P X A
X
P=a0X0+a1X1+· · · +anXn
n
X
k=0
akXk, ak∈ A an6= 0
akP
a0X0a0P
A
A[X]
15X6
P P P
deg(P) = max {k∈ | ak6= 0}
a6= 0 deg(aX0) = 0 0X0−∞
adeg(P)P

1A
2−5X+X33
3−5X+X3−2X55−2
A[X]
P=
n
X
k=0
λkXk∈ A[X]e PA ∀a∈ A e P(a) =
n
X
k=0
λkak
deg(P)<1Pe P
A[X]
n
X
k=0
akXk+
p
X
k=0
bkXk=
max(n,p)
X
k=0
(ak+bk)Xk
ak= 0 n<k6p bk= 0 p<k6n
∀P, Q ∈ A[X] deg(P+Q)6max(deg(P),deg(Q))
P Q
n
X
k=0
akXk×
p
X
k=0
bkXk=
n+p
X
k=0 
X
i+j=k
aibj
Xk
X0A1
∀P, Q ∈ A[X] deg(P Q) = deg(P) + deg(Q)
−∞
P, Q ∈ A[X]P Q Q P Q|P
T∈ A[X]P=QT
a∈ A P⇐⇒ (X−aX0)Pe P(a)=0
a P m ⇐⇒ m(X−aX0)mP
P∈ A[X]n n
P
P∈ A[X]
αX0αP α AP
P
A[X]
0A[X]
P Q (P|Q Q|P)α∈ A P=αQ
D|(P Q ∀U, V ∈ A[X]D|(P U +QV )
∀a∈ A ∀P∈ A[X]aP ∈ A[X]aP =aX0×P
a= 0 aP = 0 a6= 0 deg(aP ) = deg(P)
∀a, b ∈ A ∀P, Q ∈ A[X]
(a+b)P=aP +bP
(ab)P=a(bP )
a(P+Q) = aP +aQ
a(P×Q) = (aP )×Q=P×(aQ)

P, Q ∈ A[X]Q X P A[X]P◦Q=P(Q)
P◦Q6=Q◦P
P Q deg(P◦Q) = deg(P)×deg(Q)
∀P1, P2, Q ∈ A[X]∀a1, a2∈ A (a1P1+a2P2)◦Q=a1P1◦Q+a2P2◦Q
(P1P2)◦Q= (P1◦Q)×(P2◦Q)
∀a∈ A P◦(aX0) = e P(a)X0
n
X
k=0
akXk!0 =
n−1
X
k=0
(k+ 1)ak+1Xk
P0= 0 ⇐⇒ P P deg(P0) = deg(P)−1
∀P, Q ∈ A[X]∀a∈ A ,((P+Q)0=P0+Q0
(aP )0=a P 0
∀P, Q ∈ A[X] (P Q)0=P0Q+P Q0
∀P, Q ∈ A[X] (P◦Q)0= (P0◦Q)×Q0
P∈ A[X]k∈(P(0) =P
P(k+1) =P(k)0
s∈s P P (s)
Xk(s)=
k!
(k−s)! Xk−ss<k
s!X0s=k
0s>k
P=
n
X
k=0
akXkP(s)=
0s>n
n
X
k=sk!
(k−s)!akXk−ss6n
∀k>s, k!
(k−s)! =s!Cs
kCs ks k
s6nP(s)=s!
n
X
k=sCs
kakXk−s
P, Q ∈ A[X]m>1
QmP Qm−1P0∀k∈ {1,· · · , m −1}, Qm−kP(k)
QmP P =QmP1P0=mQm−1Q0P1+QmP0
1=Qm−1×(mQ0P1+QP 0 1)
Qm−1P0
A= 1 2
A

=
[X]\ {0}
[X] 0
P6= 0 [X] [X]Pu
P=Pu=1
λP λ =adeg(P)P.
∀A, B ∈[X]B6= 0 (Q, R)∈( [X])2
A=BQ +Rdeg(R)<deg(B)
Q R A B
A
B|A⇐⇒ R= 0
D|(A B) deg(D)6min(deg(A),deg(B)) A B
A B
A B A B
RdA B
A B Rd
=A B A ∧B
A B A ∧B
[X]
A B ∈[X]⇐⇒ ∃ U, V ∈[X]AU +BV =X0
∀A, B, C ∈[X]AU +BV =C( [X])2(U, V )
C A ∧B
A, B, C ∈[X]A BC A B A C
B C A BC A
P A ∈[X]P A A
P A
P BC B C
1 [X]
∀P∈[X]∀α∈α P ⇐⇒ (X−αX0)P⇐⇒ e P(α)=0
α P m ⇐⇒ (X−αX0)mP(X−αX0)m+1 P
n>1n

[X]
[X]n>1n
1 [X]
[X]
P∈[X]
P=λ
p
Y
k=1 X−αkX0mk
λ P α1,· · · , αpP m1,· · · , mp
P∈[X]
P=
n
X
k=0
akXkn α1,· · · , αnn
k1n
(−1)kan−k
an
=X
16i1<···<ik6n
αi1· · · αikk k P
P∈[X]
⊂P∈[X]β P µ
¯
β P µ
[X] 1 2
[X]
P∈[X]
P=λ
p
Y
k=1 X−αkX0mk×
s
Y
`=1 X2−2Re(β`)X+|β`|2X0µ`
λ P α1,· · · , αpP m1,· · · , mp
β1,· · · , βsP
µ1,· · · , µs
m1+· · · +mp
P∈[X]P
n P1,· · · , Pn∈[X]
P1,· · · , Pn
P1,· · · , PnP1∧P2∧ · · · ∧ Pn
P1∧P2∧ · · · ∧ Pn
∧P1∧P2∧P3=P1∧(P2∧P3)=(P1∧P2)∧P3
∧
P1∧P2∧P3=P1∧(P3∧P2)=(P2∧P3)∧P1=· · ·
P1∧P2∧ · · · ∧ Pn