épreuve 2

A B B A
A\B={x∈A;x /∈B}
N N∗N\ {0}
Z
Q Q+
Q∗Q\ {0}
R R+
P
pZ(p)
p
x x [x]
k k ≤x < k + 1
Q[X]X
R[X]X
nRn[X]R[X]
n
E F R
P(E, F ) = {P∈R[X] ; P(E)⊂F},
R[X]
E F
CAPES 2002
Epreuve 2

P(Q,Q),P(R,R+),P(Q,Q+).
P(Q,Q)
m i j 0m
δi,j δi,j = 0 i6=j δi,i = 1
q0, q1, . . . , qmm+ 1
j= 0,1, . . . , m LjRm[X]
Lj(qi) = δi,j i= 0,1, . . . , m
(Lj)0≤j≤m
Rm[X]
PRm[X]P
(Lj)0≤j≤m(P(qj))0≤j≤m
P(Q,Q)Q[X]
P(R,R+)
∀(a, b, c, d)∈R4,∃(x, y)∈R2(a2+b2)(c2+d2) = x2+y2.
x y a, b, c, d t z
t2+z2
A
RA
S={z∈A| ∃x∈A, ∃y∈A, z =x2+y2}
S0 1
PP(R,R+)
P
(X−a)α(X2+bX +c)βa, b, c α β
X2+bX +cR[X]
P
α
P
R[X]
P(R,R+)
P(Q,Q+)
P(Q,Q+)P(R,R+)
2X2+4
R[X]
a, b, c, d
2X2+ 4 = (aX +b)2+ (cX +d)2

Ãa
√2
b
2
c
√2
d
2!
a, b, c, d Q
2X2+ 4
Q[X]
P(Z,Z)
nΓn
Γ0(X) = 1 et,pour n > 0,Γn(X) = X(X−1)...(X−n+ 1)
n!.
nΓnP(Z,Z)
kZ0≤k < n k ≥n k < 0
m(Γn)0≤n≤m
Rm[X]
PRm[X]
P=X
0≤n≤m
dnΓnavec d0, d1, . . . , dm∈R.
(P(n))0≤n≤m
(dn)0≤n≤m
P∈ P(Z,Z)
d0, d1, . . . , dm∈Z
P(0), P (1), . . . , P (m)∈Z
m+ 1 P
m= 5
P(X) = X5−15X4+ 85X3−225X2+ 274X−120
(dn)0≤n≤5P=P5
n=0 dnΓnP
Q
m > 0
P=Pm
n=0(−1)nΓn
P
QPΓm

P(E, Z(p))
p
x
k x pka
ba b
p
k vp(x)vp(0) = +∞
vpQ Z∪{+∞}
k+ (+∞) = (+∞) + k= +∞k≤+∞kZ∪ {+∞}
vp
x, y Qvp(xy) = vp(x) + vp(y)
x, y Qvp(x+y)≥min{vp(x), vp(y)}
vp(1) vp(−1) (x, y)Q×Q∗
vp³x
y´vp(x)vp(y)
Z(p)={x∈Q|vp(x)≥0}Z(p)
Q Z(p)vp
(k, n)N∗×N∗
{j∈N|1≤j≤n, vp(j) = k}hn
pki−hn pk+1 i
∀n∈N, vp(n!) = X
k>0·n
pk¸.
EZ
Z=∩l∈PZ(l)
P(E, Z) = ∩l∈PP(E, Z(l)).
(un)n∈NE p
E
∀n∈N∗vpÃn−1
Y
k=0
(un−uk)!= min
x∈EvpÃn−1
Y
k=0
(x−uk)!.
p= 3
E={1}∪{3k|k∈N}(un)n∈NE u0= 0
u1u2
E=Z(n)n∈Np
a E
(un)n∈Np E u0=a

(un)n∈Np
E(Pn)n∈N
P0(X) = 1 et,pour n≥1, Pn(X) =
n−1
Y
k=0
X−uk
un−uk
.
PnP(E, Z(p))
m(Pn)0≤n≤m
Rm[X]
Pn(uk)nN0≤k≤n
m P
Rm[X]
P(X) =
m
X
n=0
cnPn(X) avec c0, c1, . . . , cm∈R.
P∈ P(E, Z(p))
c0, c1, . . . , cm∈Z(p)
P(u0), P (u1), . . . , P (um)∈Z(p)
ω(0) = 0 nN∗ω(n)
vp³Qn−1
k=0 (un−uk)´PP(E, Z(p))
pω(m)PZ(p)P(E, Z(p))
Q[X]
P(N\pN,Z(p))
p
pNpN\pN
p n
ϕp(n) = n+1+·n
p−1¸et ωp(n) = X
k≥0·n
(p−1)pk¸.
p−1
·ϕp(n)
p¸=·n p−1¸et ϕp(n)∈N\pN.
ϕpN N \pN
n vp(ϕp(n)!) = ωp(n)
xRa b N∗£x
ab ¤=·[x
a]
b¸.)