5 Division euclidienne

0A,1A0 1
(A,+,·),0A̸= 1A.
A,A∗=A\{0A},A×
A.
a, b A,
u∈A×b=ua.
pA, p ̸= 0A,
p /∈A×
(p=uv)⇒u∈A×v∈A×
pAp
pA, p ̸= 0A, p
(p uv)⇒(p u p v)
A
φ:A∗→N.
A
φ(a, b)Ab̸= 0A,(q, r)
A2a=bq +r r = 0Ar̸= 0Aφ(r)< φ (b).
q r a b.
(A, φ)A,
A A
I=a0·A.(a0)
(A, φ)I
A{0A}, a0I\ {0A}φ(a0) = min
a∈I\{0A}φ(a)
I=a0A.

(A, φ)IA.
I={0},0.
I̸={0},
n0= min φ(I\ {0A}) = min
a∈I\{0A}φ(a)
φ(I\ {0A})
N, a0I\ {0A}n0=φ(a0) = min φ(I\ {0A}).
a I a0a=qa0+r r = 0A
a0I\ {0A}r̸= 0A, r =a−qa0I\ {0A}
I φ (r)< φ (a0) = n0, n0a=qa0
I⊂a0A. a0A⊂I I I =a0A.
A
Aa0∈A A =a0A.
a∈A, q ∈Aa=qa0.
e∈Aa0=ea0a=qa0∈A, ae =qa0e=qa0=a,
eA
Aa=bq q =ab−1(a, b)∈A×A∗, r = 0A
φ:A∗→N
(A, φ)φA
φ φ (r)< φ (b)
a∈Ab∈A∗a=bq. 1A=bq
b∈A∗,A
Z[X]
I= (2, X) = 2Z+XZ2XZ[X]
P∈Z[X]I= (2, X) = (P).
2∈I, Q ∈Z[X] 2 = P Q, P 2, P =±1
P=±2.
P∈I, P = 2A+XB A, B Z[X], P =P(0) = 2A(0)
P=±2.
X∈I, X =QP Q ∈Z[X] 1 1 = P(1) Q(1) =
2a a ∈Z,
A
A[X]

Z[X]A[X]
A
Aλ∈A∗I= (λ, X) =
λA+XAλ X A[X].
P∈A[X]I= (P).
λ∈I, Q ∈A[X]λ=P Q, P λ.
P∈I, P =λA +XB A, B A[X], P =P(0A) = λA (0A) = λa
a∈A.
X∈I, X =QP Q ∈A[X] 1A1A=
P(1A)Q(1A) = λb b ∈A, λ
(A, φ)φ
∀(a, b)∈A∗×A∗, φ (ab)≥φ(a)
a, c A∗a c, φ (a)≤φ(c).
(A, φ)
(A, φ), φ :A∗→N
∀a∈A∗, φ (a) = min
x∈A∗φ(ax)
(A, φ)
a∈A∗,{φ(ax)|x∈A∗}
N, φ (a).
a, b A∗,
φ(ab) = min
x∈A∗φ(abx) = φ(abx0)≥φ(a)
φ
a, b Ab̸= 0 x0∈A∗φ(b) = min
x∈A∗φ(bx) = φ(bx0).
(A, φ)a bx0, a =bqx0+r r = 0 r̸= 0
φ(r)≤φ(r)< φ (bx0) = φ(b)
(A, φ).
(A, φ)φ
(a, b)∈A∗×A∗, φ (ab)≥φ(a),
b
φ(−a) = φ(a)φ(ab)> φ (a)a, b b
a∈A∗,
φ(a) = min
x∈A∗φ(ax)
φ(1A) = min
x∈A∗φ(x)

A×={a∈A∗|φ(a) = φ(1A)}
a∈A∗.
φ φ (a)≤φ(ab)b∈A∗.
b∈A×,
φ(ab)≥φ(a)φ(a) = φ(ab)b−1≥φ(ab)
φ(ab) = φ(a).
b∈A∗φ(a) = φ(ab), a ab, a =q(ab) + r
r= 0, r =a(1 −qb)∈A∗
φ(a)≤φ(a(1A−qb)) = φ(r)< φ (ab)
φ(a) = φ(ab). r =a(1A−qb) = 0 a∈A∗,
qb = 1Ab
x∈A∗, φ (a)≤φ(ax)φ(a) = φ(a·1A) 1A∈A∗, φ (a) =
min
x∈A∗φ(ax).
a∈A∗, φ (a) = φ(1A·a) = φ(1A)a∈A×,A×=
{a∈A∗|φ(a) = φ(1A)}.
AaA
a∈A
AI= (a, b) = aA+bA
δ δ pgcd a b AI=δA.
a∈I, δ a, δ a, a
δ δA=A,1A∈I, 1A=au +bv u, v
Aa c =acu +bcvw.
δ a, δA=aA, b ∈aAa b.
a∈A
a=bc, bc, b c. b, b =ua,
a=uac 1 = uc, c a
a
A,
r
k=1
bk, bk
(A, φ)a∈A∗
a∈A∗\A×
a=
r
k=1
pk=
s
j=1
qj

pgcd
pkqjA, r =s, pk
qj
φ φ
φ
φ(a)∈N, a ∈A∗.
φ(a) = φ(1A)φ a
Aa∈A∗
bA∗φ(b)< φ (a).
a a =bc b c
φ(b)< φ (bc) = φ(a), φ (c)< φ (bc) = φ(a),
b c a.
a=
r
k=1
pk=
s
j=1
qj1≤r≤s, pkqjA, p1
A
s
j=1
qjqi
p1qiA,
p1
r
k=2
pk=u
s
j=1
j̸=i
qj, u A.
r1A=v
j∈J
qj, v AJ
{1,··· , s}qj
r=s pk, qj
pgcd
(A, φ)
r(a1,··· , ar)A, r ∈N\ {0,1},
(a1,··· , ar) = x∈A| ∃(u1,··· , ur)∈Ar;x=
r
k=1
ukak
A.A
{a1,··· , ar}.
r∈N\ {0,1}a1,··· , arA. δ
A
(a1,··· , ar) = (δ)
δ=
r
k=1
ukak
∀k∈ {1,··· , r}, δ ak
a1,··· , arδ