Fiche résumée du cours d`algèbre 1 1 Rappel sur les

(In)n∈N
X
k
Ik={X
k
qk∈Ik}
A⇒A[X]
p∈A
p /∈A∗
p=ab ⇒p|a p |b
∀a∈A, a 6= 0,∃p1...pr∈A∃u∈A∗
a=up1...pn
∀p p |ab ⇒p|a p |b
⇔(p)
a|bc a ∧b= 1 a|c
⇒
⇒A[X]
a∧b=Qpmin(vp(a),vp(b))
a∨b=Qpmax(vp(a),vp(b))
v:A− {0} → N∀a∈A, ∀b∈A− {0}
∃q, r ∈A, a =bq +r v(r)< v(b)

K=F rac(A)
A∗
A∗
c(P Q) = c(P)c(Q)
P∈A[X]
K[X]
P=anXn+... +a0∈A[X]
p∈A
p-an
∀i∈ {0...n −1}p|ai
p2-a0
PK[X]A[X]P
B=A/I
L=F rac(B)P=anXn+... +a0∈A[X]P=anXn+... +a0∈B[X]
an6= 0 P
K
K→L K
L L K−
L
K−[L:K] = dimK(L)
K⊂L⊂M
(ei)iK−L(fj)jL M
(eifj)i,j K−L
[M:L][L:K] = [M:K].
K⊂LS⊂L
L
L K L =K(S)L(x1...xn)
S={x1...xn}
(Ki →L)→(K0i0 →L0)
Ki →L
f↓ ↓ g
K0i0 →L0
.
K⊂L
α∈Lφ:K[X]→L
P(X)7→ P(α)
α Kerφ ={0}
K
K[α] = Imφ
αK⇒K[X]
P∈K[X]Kerφ = (P)
α
K⊂Lα∈L
αK
K[α] = K(α)

DimKK[α]<∞
P∈K[X]
K⊂K(α)
α
K⊂K(α)K⊂K(β)α β
K
α β
K
i
∼
→L K(α)L(β)
i(Pα) = Pβ
Ki →L
↓ ↓
K(α)
α
i0
→
→L(β)
β
.
P∈K[X]K⊂L
PL=K(α)P(α) = 0
K
K−
P∈K[X]
K K ⊂L
L[X]P=Q(X−αi)
L L =K(α1...αn)
P∈K[X]deg(P) = n
K K−αi
K
K K0i:K∼ →K0P∈K[X]P0=i(P)∈
K0[X]LPK L0
P0K0L→L0
Ki →K0
∩ ∩
L
ϕ
∼
→L0
.
K
X−α α ∈K
K⊂K
K
K⊂K

K
∃p∃n≥0|K|=pn
|K|=pnKXpn−XZ/pZ
Fq
K
G⊂K∗
F∗
q=Z/(q−1)Z
Zrr≥0
Z/nZn > 0
Z/nZ×Z/mZ'Z/(m∧n)Z
KAutKK⊂L
K⊂M
HomK(L,M) = {K− } ={ }
={σ:L→M, σ|K=IdK, σ }
Gal(L|K)L|K=HomK(L,L)
={L|K}
KGal(L|K) = Aut(L)
K⊂M,K⊂L
HomK(L,M)
MLinK(L,M)
[L:K]<∞#HomK(L,M)≤[L:K]
K⊂M K−
HomK(A, M)M
LinK(A, M)
K,K0i:K∼ →K0
i:K[X]∼ →K0[X]P∈K[X]P0=i(P)∈K0[X]
L L0P P 0K K0
L→L0K
q≤[L:K]
q= [L:K]PL
P∈K[X]deg(P) = nL
K L #Gal(L|K) = [L,K]
x, y ∈K
K⇔ ∃σ∈Gal(K|K)|σ(x) = y
K⊂L
P∈K[X]K
L L
K⊂L⊂K
K⊂L

∀x∈L K L
K⊂L
L|K
∃P∈K[X]LPK
(Pi)i∈I
K[X]
(Pi)i∈IK L K
PiLPiL
K⊂L
L|K
L(Pi)i∈I
K⊂L K ⊂M
K⊂L
L⊂M
K⊂M
K⊂M
L⊂M0K⊂M0M0⊂M M =M0
K⊂L⊂M M|KM|L
K⊂L
P∈K[X]α∈K
α P ⇔P(α) = P0(α)=0
P∈K[X]K
P∧P0= 1 K[X]
K
P∈K[X]
P
P06= 0
car(K)=0 car(K) = p > 0P /∈K[Xp]
K⊂Lα∈L
αKα
αK
K⊂L L
K
K⊂Kx∈K K
(∀σ∈Gal(K,K), σ(x) = x)⇔x∈K