Algèbre 4 - Examen Questions de cours [5 pts] Soit A un anneau

a b A
ha, bia b
A
S⊂A
I0⊆I1⊆I2⊆I3⊆. . . A
n∈NIn=In+1 =In+2 =. . .
A
A
A[t]
Z[t1, . . . , tn]
K[t1, . . . , tn]K
n p
f(t) = tn−pQ
f(t)Q Q
Qn n ∈Nn≥1
f(t) = t4−3
Q
A a A
A→A
x7→ ax
A
p p
p2
Fp2Fp×FpZ/p2Z
Fp×Fp6∼
=Z/p2Z

p
P(t)∈Fp[t] 2 Fp2
P(t)∈Fp[t] 3 Fp3
P(t)∈Fp[t] 4 Fp4
P(t)∈Fp[t] 5 Fp5
F×
9
θ∈F×
9θ4∈ {1,−1}θ4=−1θ
F×
9
P(t) = t2+t−1∈F3[t]F3
P(t)F9θ θ4
θF×
9
Q(t)∈F3[t] 2 F3
Q(t)F×
9
n
n
x1, . . . , xn
P1(x1, x2) = x3
1x2 2+x3 2x2 1(n= 2);
P2(x1, x2, x3) = x3
1x2 2+x3 2x2 3+x3 3x2 1(n= 3);
P3(x1, x2, x3) = x2
1x2 2+x2 2x2 3+x2 3x2 1+x1x2x3(n= 3).
P3(x1, x2, x3)Q(σ1, σ2, σ3)Q
σ1, σ2, σ33
a
f(t) = t3−3at2+ 4 ∈C[t]
a
S a f(t) 3
a∈S α1, α2, α3f(t)P3(α1, α2, α3)