Feuille d`exercices n 2

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2α5βa α, β, a ∈Za10
I N ={a∈Z|∃α∈Z,∃β∈Z,2α5βa∈I}Z
Z[i] = {a+ib, a ∈Z, b ∈Z}
N:Z[i]→NN(a+ib) = a2+b2NZ[i]
N
Z[i]
z∈Z[i]N(z)zZ[i]
2,3,5,7, . . .
Z[i]
7+3i2−i9−i5−11i
p
pZ[i]
z∈Z[i]p=N(z)
a, b ∈Zp=a2+b2p
−1p
(c)⇒(a)−1≡x2(mod p)p/x2+ 1 x2+ 1
p
1
p
p p = 2 p≡1 (mod 4)
A=Z[i√5] = {a+bi√5, a ∈Z, b ∈Z}
AC
N:A→NN(a+bi√5) = a2+ 5b2
A z ∈A N(z)=1
2 3 1+ i√5 1 −i√5A A