Structures algébriques, feuille 10

A K
A[X]
A a ∈A X −a A[X]
A3X−1 2X+ 1 Z/6Z
C[X]R[X]C
K[X]
K[X]Z[X]
A K
ZA
ZK
φZ−→ K φ nZ
n0n K
Q R Z/pZ
K−→ L K L
K P =
n
P
i=0
aiXi∈K[X]
P P 0n
P
i=1
iaiXi−1K[X]
K[X]−→ K[X]
P7−→ P0
K K[X]K
K0P∈K[X]P0= 0
P
K p p P ∈K[X]
P0= 0 Q∈K[X]P=Q(Xp)
n P P (n)P(0) =P P (n+1) =
(P(n))0K a ∈K r ∈N(X−a)r
P∀i∈ {0, ..., r −1}, P (i)(a)=0

K
K K[X, Y ]K[X][Y]
XiYjP=P
0≤i≤n,0≤j≤m
λi,j XiYj˜ P
K2−→ K
(a, b)7−→ ˜ P(a, b) = P
0≤i≤n,0≤j≤m
λi,j aibj
P(a, b)˜ P(a, b)P(a, b) = 0 (a, b)P
K[X, Y ]
K K[X, Y ]
K2
K˜ P P
K
P∈R(X)P2= (X−1)3
F∈R(X)F2= (X−1)3
C(X)R(X)
1
(X2+ 1)(X2−1)
X7+ 1
(X2+ 1)(X2+X+ 1)
X3+X
(X2+X+ 1)2
P∈C[X]P0/P C(X)
pFppZ/pZ
P2 3 P
2
F2[X]F3[X]
4F2
4K
K2
j K 0 1 j3= 1 K∗
j2+j+ 1 = 0 K={0,1, j, j + 1 = j2}
K[X]
X2+X+ 1

X2+X+j
X2+X+j2
K[X]
X4+X
X4+jX3+j2X2+j2X+j2
X4+X+ 1