Corps finis

(A, +,×) + ×A
(A, +)
×
(x, y, z)∈A3x×(y+z) = x×y+x×z(x+y)×z=x×z+y×z
A1A∈A1A6= 0A
x∈A1A×x=x×1A=x1A
A×
A×A A×={x∈
A|∃y∈A, xy =yx = 1K}
K(+,×)
(K, +,×)
(K\{0K},×)
K
(A, +,×) (B, ⊕,⊗)f:A→B f
∀(a, b)∈A2f(a+b) = f(a)⊕f(b)
∀(a, b)∈A2f(a×b) = f(a)⊗f(b)
f(1A) = 1B

(K, +,×)ω+(1K) 1K
K+ω+(1K)<+∞K
car(K) = ω+(1K)ω+(1K) = +∞
car(K) = 0
car(K)>0
car(K) = min {n∈N∗|n·1K= 0K}=|{n·1K|n∈Z}|
N
K K car(K)
(n·1K)n∈NK(n, n0)∈N2
n6=n0n·1K=n0·1Kn > n0(n−n0)·1K= 0K
ω+(1K)<+∞car(K)6= 0
car(K) = ab a, b ≥2 (ab)·1K= 0
(a·1K)×(b·1K)=0 K a ·1K= 0 b·1K= 0
car(K)
Kcar(K)|K|
K K car(K)
1K(K, +) car(K)||K|
Fp
Fp=Z/pZp
p p
pZ/pZ
p
K p (K, +) p
1K(K, +) a, b ∈K
a=a0·1K= (1K+... + 1K)a0b=b0·1Kab = (a0·1K)×(b0·1K) = (a0b0)·1
K p FpK
(K, +, x)p α ∈N∗
|K|=pα

p= car(K)KFp
K
K α ∈N∗K
Fp
|K|=pα
pαp
n > 0
(A, +,×)I⊂A I A
(I, +) (A, +)
(a, i)∈A×I a ×i∈I I A
(A, +,×) (B, ⊕,⊗)f:A→B
Ker(f)
(A, +,×)A
A{0}
A A {0}x∈A x 6= 0 (x)6={0}
(x) = xA =A y ∈A yx = 1A
A I {0}A x ∈I
y∈A xy = 1Axy ∈I1A∈I
A=I a ∈A a ×1A∈I1A∈I
a∈I
(A, +,×)∼A
π A/ ∼
π I A (x, y)∈A2
x∼y⇔x−y∈I A/ ∼=A/I I = Ker(π)π

I=x−y|(x, y)∈A2x∼y
(A, +, x)I A
A J
A I ⊂J J ∈ {A, I}
(A, +,×)I A A/I
I
A/I J I ⊂J I 6=J
x∈J\I x 6= 0 0A∈I y ∈A π(x)π(y) = π(1A)
π(xy) = π(1A)xy −1A∈I x ∈J x ∈I xy ∈I1A∈I
A= (1A)⊂J J =A
I B =A/I B
J⊂B J {0}B H =π−1[J]
H I ⊂H x ∈I
π(x) = I=π(0K)J+B
π(0K)π(0K)∈J π(x)∈J x ∈H I ⊂H J =I
J={0}I=A J =B B {0}B
B
I1I2A I1∩I2
x∈A x (x)A
x
x∈A x x /∈A×(x)
x∈A x x /∈A×(a, b)∈A2
p|ab p|a p|b
x∈A(x) = xA ={x×a|a∈A}
xA x
(A, +,×)
(a, b)∈A2ab = 0Aa= 0Ab= 0A

(A, +,×)A I
A α ∈I I = (α) = αA
(Z,+,×)K K[X]
I{0}ZK[X]k
I I
I kZkK[X]
(A, +,×) (a, b)∈A2a|b a b u ∈A
b=ua
(A, +,×) (a, b)∈A2a|b bA ⊂aA
(A, +,×)p∈A p
a∈A a|p a ∈A×u∈A×a=up
u∈A×(u) = uA =A
(x) = A aA =bA u ∈A×a=ub
(L, +L,×L) (K, +K,×K) (L, +L,×L)
(K, +K,×K)K⊂L L K
K K L
L K
L·K×L(λ, x)∈K×L
λ·x=λ×Lx(L, +L,·)
[L:K] = dimK(L)L
K[L:K]<+∞