TD n°10 : Anneaux noethériens et polynômes

P(X) =
n
X
k=0
akXkZp
an
p P (X)Z/pZ[X]
P(X)Z[X]
p anp
P(X)Z[X]
A K P (X) =
n
X
k=0
akXk∈A[X]
A p ∈A
i)p an
ii)k∈ {0, . . . , n −1}p ak
iii)p2a0
P K[X]A[X]
P
pZ[X] Φp(X) =
p−1
X
k=0
Xk
K6= 2 PX2
i−1
K[X1, . . . , Xn]
K6= 3 X3+Y3−1
K[X, Y ]
Z[X]I
I∩Z Z
I∩Z= 0 ˜ I=IQ[X]IQ[X]
˜
I∩Z[X] = I I
I∩Z=pZI/pI Fp[X]
I p p
p
C[X, Y ]
(X−a, Y −b)
A A
A[X]A
A

A⊂C(X)|z|= 1
P(0) ∈Z
A
I A p1,...,nnI
p1p2. . . pn⊂I
A
A
I I =I1∩I2⇒I=I1I=I2
A I √I
√I={x∈A, ∃n > 0, xn∈I}n > 0 (√I)n⊂I
A A
m1, ..., mnAm1m2. . . mn= 0
A A
A
A A
A a, b ∈A a b
u, v au +bv = pgcd(a, b)
A
A
I a ∈I
(a)⊂(a1)⊂(a2)⊂. . . (an)⊂I
I
E f :Z→Qf(n)∈Zn
P=E∩Q[X]
P6=Z[X]
∆ : E→E(∆f)(n) = f(n)−f(n−1)
f∈P
∆f∈P
∃n∈N,∆nf= 0
d∆df6= 0
Pi=1
i!X(X−1) . . . (X−i+1) (Pi)Q Q[X]
ZP f ∈PPniPi
ni∈Z
f∈P d d!f∈Z[X]f(n)∈Zn∈Z