Giáo trình đại số giao hoán

f(x, y)=0 f(x, y)
r r f(x, y)
r r
r
x =0
(0,α) α f(0,y)=0 f(0,y)
r f(0,y)=0
r
f(x, y)=0
x =0
A x
f c
0
+ c
1
x + ···+ c
r
x
r
c
0
,c
1
, , c
r
∈ A x

A c
r
=0 r deg f c
r
f f =0 deg f = −∞ A[x]
x A
A[x]
g ∈ A[x]
f ∈ A[x] f = gh + v deg v<deg g
r =degf s =degg r<s h =0
v = f r ≥ s
f
1
:= f − cx
r−s
g,
c f deg f
1
<r r
f
1
= gh
1
+ v deg v<s
f = f
1
+ cx
r−s
g = g(h
1

+ cx
r−s
)+v.
n A
A[x
1
, , x
n
]:=A[x
1
, , x
n−1
][x
n
],
A[x
1
, , x
n
] x
n
A[x
1
, , x
n−1
]
A[X]=A[x
1
, , x
n

]
A[X] n k[X]
n f
f(X)=

r
1
+···+r
n
≤t
c
r
1
, ,r
n
x
r
1
1
···x
r
n
n
t c
r
1
, ,r
n
∈ A c
r

1
, ,r
n
=0
x
r
1
1
···x
r
n
n
f x
r
1
1
···x
r
n
n
r
1
+ ···+ r
n
f =0 f
f deg f f =0 deg f = −∞ deg f ≤ 0
f ∈ A f deg f =1
x
r
1

1
···x
r
n
n
>x
s
1
1
···x
s
n
n
r
1
+ ···+ r
n
>s
1
+ ···+ s
n
r
1
+ ···+ r
n
= s
1
+ ···+ s
n
(r

1
− s
1
, , r
n
− s
n
) f,g,h
f>g fh > gh
x
m
1
>x
m−1
1
x
2
> ···>x
m
2
> ···>x
1
>x
2
> 1
m
f = c
d,0
x
m

1
+ ···+ c
m−1,1
x
m−1
1
x
2
+ ···+ c
0,m
x
m
2
+ c
1,0
x
1
+ c
0,1
x
2
+ c
0,0
.
A A
cd =0 c, d =0 A
Z
A deg fg =degf +degg
f,g ∈ A[X]
fg uv u

f v g u
max
,v
max
f,g u = u
max
v = v
max
uv < u
max
v
max
uv = u
max
v
max
c, d ∈ A
u
max
,v
max
c, d =0 cd =0 cdu
max
v
max
fg
deg uv ≤ deg u
max
v
max

=degu
max
+degv
max
=degf +degg.
deg fg =degf +degg
A A[X]
A[X] A
f,g A[X] deg f,deg g ≥ 0
deg fg ≥ 0 fg =0 A[X]
fg =1 deg fg =0 deg f =degg =0 f,g ∈ A
f,g A
k k k[X]=
k[x
1
, , x
n
]
f ∈ k[X]
a =(α
1
, , α
n
) ∈ k
n
f(a):=

r
1
+···+r

n
≤m
c
r
1
, ,r
n
α
r
1
1
···α
r
n
n
.
f k
n
k a f
f(a)=0 f a
k
k
n
k f(a)=0 a ∈ k
n
f =0
n =1
n>1
f =0 f x
n

f
f = f
0
+ f
1
x
n
+ + f
m
x
m
n
f
0
, , f
m
x
1
, , x
n−1
f
m
=0
(α
1
, , α
n−1
) ∈ k
n−1
f

m
(α
1
, , α
n−1
) =0
f(α
1
, , α
n−1
,x
n
)=
f
0
(α
1
, , α
n−1
)+f
1
(α
1
, , α
n−1
)x
n
+ + f
m
(α

1
, , α
n−1
)x
m
n
x
n
f(a)=0 a ∈ k
n
k f,g k[X]
f(a)=g(a) a ∈ k
n
f = g
f −g
k
k = {α
1
, , α
s
} f =(x −α
1
) ···(x −α
s
)
f k f =0
k
k
• ∅ f =0 f ∈ k, f =0
• a =(α

1
, , α
n
) x
i
−α
i
=0
i =1, , n.
• k
n
0=0
(α
1
, , α
n
) (β
1
, , β
n
)
α
1
= c
10
+ c
11
β
1
+ ···+ c

1n
β
n
,
·····················
α
n
= c
n0
+ c
n1
β
1
+ ···+ c
nn
β
n
,
c
ij
∈ k i =1, , n j =0,1, , n V
S V
f(c
10
+ c
11
x
1
+ ···+ c
1n

x
n
, , c
n0
+ c
n1
y
1
+ ···+ c
nn
y
n
),f∈ S.
S
V = ∅
f
1
, , f
d
1 ≤ d ≤ n c
ij
f
i
(c
10
+c
11
x
1
+···+c

1n
x
n
, , c
n0
+c
n1
y
1
+···+c
nn
y
n
)=x
i
,i=1, , d.
V = {(0, , 0,α
d+1
, , α
n
)| α
d+1
, , α
n
∈ k}
V k
n−d
V
f Z(f) deg f =0 f = c
Z(f) k

n
c =0 ∅ c =0
deg f>0 Z(f) deg f =1 Z(f)
k
2
k
3
k
n
n ≥ 4
Z(f)
k[x, y]
• f = x
2
−y Z(f)={

α, α
2
)| α ∈ k}.
x
2
−y =0
• f = x
3
− y
2
Z(f)={(α
2
,α
3

)| α ∈ k}. (α
2
,α
3
) ∈
Z(f) α ∈ k (α
1
,α
2
) ∈ Z(f) α
3
1
= α
2
2
α
1
= α
2
α
2
= α
3
α ∈ k
α
1
=0 α
2
=0 α =0 α
1

=0 α = α
2
/α
1
α
1
= α
3
1
/α
2
1
= α
2
2
/α
2
1
= α
2
,
α
2
= α
3
2
/α
2
2
= α

3
2
/α
3
1
= α
3
.
x
3
− y
2
=0
S Z(S)
Z(S)=

f∈S
Z(f).
Z(f) k k
k
k
S → Z(S)
S
1
⊇ S
2
Z(S
1
) ⊆ Z(S
2

)
S
1
S
2
k[X]
Z(S
1
) ∪ Z(S
2
)=Z(S)
S = {fg | f ∈ S
1
,g ∈ S
2
}.
S
1
,S
2
S Z(S
1
) ∪
Z(S
2
) ⊆ Z(S). a S a
S
1
f ∈ S
1

f(a) =0 g ∈ S
2
fg ∈ S f(a)g(a)=0 g(a)=0 a ∈ Z(S
2
)
Z(S
1
) ∪Z(S
2
) ⊇ Z(S)
{S
i
} k[X]

Z(S
i
)=Z


S
i

.
a S
i
a

S
i


Z(S
i
)=Z


S
i

k
n
k
m
k[Y ]=k[y
1
, , y
m
]
k
m
k[X, Y ]=k[x
1
, , x
n
,y
1
, , y
m
]
k
n+m

S ⊆ k[X] T ⊆ k[Y ] S ∪T
k[X, Y ]
Z(S) × Z(T )=Z(S ∪ T ).
(a, b) ∈ k
n
×k
m
S ∪T
a S b T
k
n
k
n
n A
n
k
A
n
A
n
Z(f)
A
n
D(f):=k
n
\Z(f )={a ∈ k
n
| f(a) =0}.
D(f) D(f)
A

n
A
n
D(f)∩D(g) = ∅ D(f) D(g)
D(f) ∩ D(g)=k
n
\ Z(f) ∪ Z(g)=k
n
\ Z(fg).
D(f),D(g) = ∅ Z(f),Z(g) = k
n
f,g =0
fg =0 Z(fg) = k
n
k
n
\ Z(fg) = ∅
A
n
V ⊆ A
n
V
V
k
n
p, q
Z(x
p
−y
q

)={(α
q
,α
p
)| α ∈ k}.
f g k[x, y] f,g
f =0 g =0
k
2
R
R
n
A I A 0 ∈ I I
• f + g ∈ I f,g ∈ I
• hf ∈ I h ∈ A f ∈ I
A
0 A A 1 ∈ I
I = A A
f A
(f):={hf| h ∈ A}.
(f) f
S A
S
(S):={h
1
f
1
+ + h
r

f
r
| h
1
, , h
r
∈ A, f
1
, , f
r
∈ S, r ≥ 1}.
(S) S (S)
S SA (S)
I J A
I ∪ J I J I + J
fg f ∈ I g ∈ J I,J IJ
I + J = {f + g| f ∈ I, g ∈ J},
IJ = {f
1
g
1
+ ···+ f
r
g
r
| f
1
, , f
r
∈ I,g

1
, , g
r
∈ J, r ≥ 1}.
IJ ⊆ I ∩ J
I =(x, y
2
) J =(y) k[x, y]
I + J =(x, y) IJ =(xy, y
3
) I ∩ J =(xy, y
2
)
I + J = J + I,
IJ = JI,
(I + J)+K = I +(J + K),
(IJ)K = I(JK),
(I + J)K = IK + JK
I,J,K A
I r ≥ 0
I
r
:= I ···I r
I
0
:= A I
r
r I
(x, y)
r

=(x
r
,x
r−1
y, , y
r
)
S ⊂ A
I : S := {f ∈ A| fg ∈ I g ∈ S}.
S g I : g
I =(xy, y
3
) I : x =(y) I : y =(x, y
2
)
S k[X] I =(S)
Z(I)=Z(S).
S ⊆ I Z(I) ⊆ Z(S) a ∈ Z(S)
f ∈ I f = h
1
f
1
+ ···+ h
r
f
r
f
1
, ,f
r

∈ S
f
1
(a)= = f
r
(a)=0 f(a)=0. a ∈ Z(I)
Z(S) ⊆ Z(I)
I J k[X]
Z(I) ∪ Z(J)=Z(I ∩J)=Z(IJ)
Z(I) ∩ Z(J)=Z(I + J)
S = {fg| f ∈ I, g ∈ J} I,J ⊇ I ∩J ⊇ IJ ⊇ S
Z(I) ∪ Z(J) ⊆ Z(I ∩ J) ⊆ Z(IJ) ⊆ Z(S).
Z(I) ∪ Z(J)=Z(S)
Z(I) ∪ Z(J)=Z(I ∩J)=Z(IJ).
Z(I) ∩ Z(J)=Z(I ∪ J) I + J
I ∪ J Z(I ∪J)=Z(I + J)
V A
n
I
V
:= {f ∈ k[X]| f(a)=0 a ∈ V }.
I
V
V
I
V
V k[X] V a
I
a
I

{a}
• I
∅
= k[X]
• I
a
=(x
1
− α
1
, , x
n
− α
n
) a =(α
1
, , α
n
) ∈ A
n
f ∈ k[X]
f = h
1
(x
1
− α
1
)+ + h
n
(x

n
− α
n
)+α
α ∈ k f(a)=0 α =0
f = h
1
(x
1
−α
1
)+ + h
n
(x
n
− α
n
) ∈ (x
1
− α
1
, , x
n
− α
n
).
• V ⊂ A
2
x
2

− y =0
I
V
=(x
2
−y) I
V
⊆ (x
2
−y)
f ∈ k[x, y] y k[x]
f = h(x
2
− y)+v
v ∈ k[x] V ⊆ Z(x
2
− y)={(α, α
2
)| α ∈ k} f ∈ I
V
f(α, α
2
)=v(α)=0 α k
v =0 f = h(x
2
−y) ∈ (x
2
− y)
• V ⊂ A
2

x
3
−y
2
=0
I
V
=(x
3
−y
2
) I
V
⊆ (x
3
−y
2
)
f ∈ k[x, y] y k[x]
f = h(x
3
− y
2
)+uy + v
u, v ∈ k[x] V ⊆ Z(x
3
−y
2
)={(α
2

,α
3
)| α ∈ k} f ∈ I
V
f(α
2
,α
3
)=u(α
2
)α
3
+ v(α
2
)=0 α
k u(x
2
)x
3
+ v(x
2
)=0 u(x
2
)x
3
v(x
2
)
u(x
2

)x
3
=0 v(x
2
)=0 u =0 v =0
f = h(x
2
− y) ∈ (x
2
− y)
• V = {(0, , 0,α
d+1
, , α
n
)|α
d+1
, , α
n
∈ k} I
V
=(x
1
, ,x
d
)
f ∈ k[X]
f = h
1
x
1

+ ···+ h
d
x
d
+ v
v ∈ k[x
d+1
, , x
n
] f ∈ I
V
v(α
d+1
, , α
n
)=0
α
d+1
, , α
n
∈ k v =0
f = h
1
x
1
+ ···+ h
d
x
d
∈ (x

1
, , x
d
).
• I
A
n
=0 0=0 A
n
V W A
n
I
V
⊇ I
W
V ⊆ W
I
V
∩I
W
= I
V ∪W
.
I
V
+ I
W
⊆ I
V ∩W
I

V
+ I
W
= I
V ∩W
.
V = Z(x
2
− y) W = Z(y) I
V
=(x
2
− y) I
W
=(y)
I
V
+ I
W
=(x
2
−y,y)=(x
2
,y). V ∩W =(0, 0) I
V ∩W
=(x, y)
I
V
+ I
W

= I
V ∩W
V
V V V
V
A
1
A
1
A
1
A
1
V I
V
V A
n
V = Z(I
V
)
I
V
= I
V
V ⊆ Z(I
V
) V = Z(S)
S V S ⊆ I
V
Z(S) ⊇ Z(I

V
)
V ⊆ V I
V
⊆ I
V
f ∈ I
V
V = Z(I
V
)
f(a)=0 a ∈ V f ∈ I
V
I
V
⊆ I
V
V ⊆ A
n
W ⊆ A
m
V ×W = V ×W.
V ×W V ×W = Z(I
V ×W
)
I
V ×W
= I
V ×W
V × W = Z(I

V ×W
)=Z(I
V ×W
)=V ×W.
I
V ×W
⊆ I
V ×W
f ∈ I
V ×W
a ∈ V,b ∈ W f(a, b)=0 f(a, Y ) ∈ I
W
f(a, b

)=0
b

∈ Z(I
W
)=W f(X, b

) ∈ I
V
f(a

,b

)=0
a


∈ Z(I
V
)=V f ∈ I
V ×W
V V = V
V = Z(I
V
).
V I
V
I
V
V
I → Z(I) V → I
V
k[X] k
n
I
V
V → I
V
→ Z(I
V
)=V.
I
V
I
V
I A f ∈ A
f

r
∈ I I
√
I
f ∈ A f
r
=0
√
I
f,g ∈
√
I f
r
,g
s
∈ I
(f + g)
r+s
=
r+s

i=1

r + s
i

f
r+s−i
g
i

.
r + s − i ≥ r i ≥ s f
r+s−i
g
i
f
r
g
s
i =1, , r + s (f + g)
r+s
∈ I f + g ∈
√
I
hf ∈
√
I h ∈ A (hf)
r
= h
r
f
r
∈ I
√
I
I ⊆
√
I =

√

I. I =
√
I I
A
A
I
V
f
r
∈ I
V
f
r
(a)=0 a ∈ V
f(a)=0 α ∈ V f ∈ I
V
k[X]
k[X] I = k[X]
√
I
√
I = I
V
V
V = Z(I
V
)=∅
√
I = I
∅

= k[X] 1 ∈ I
I = k[X]
x
2
1
+1 R[X]
R
n
k[X]
k[X]
I A I I
A I
(I,f)=A f ∈ I
I
1
⊆ I
2
⊆···⊆ I
j
⊆···

I
j
I
a
=(x
1
− α
1
, , x

n
− α
n
) k[X]
f ∈ I
a
f = h
1
(x
1
− α
1
)+···+ h
n
(x
n
− α
n
)+α
α ∈ k, α =0 α ∈ (I
a
,f) (I
a
,f)=k[X]
k[X]
I x
2
1
+1 R[X]
I = I

a
a I x
2
1
+1
I,J I ∪ J
I,J I + J = A I ∩J = IJ
I,J
√
IJ =
√
I ∩J =
√
I ∩
√
J.
V = Z(xy − 1) I
V
=(xy − 1)
V Z(x
2
− y)
V = Z(x
2
− y)
I A I
fg ∈ I f ∈ I g ∈ I
Z
A
I = A I

I
I I = J
1
∩ J
2
J
1
J
2
⊆ I
J
1
J
2
I
J
1
⊆ I J
2
⊆ I
I f,g ∈ I
fg ∈ I J
1
=

(I,f) J
2
=

(I,g) I ⊆ J

1
∩J
2
h ∈ J
1
∩ J
2
h
r
∈ (I,f) h
s
∈∩(I,g) ,s > 0
h
r+s
∈ (I,f)(I,g)=I
2
+ I(f)+I(g)+(fg) ⊆ I
h ∈ I I = J
1
∩ J
2
J
1
,J
2
I
I = A I
J I
f ∈ J f ∈ I
I

0
⊆ I
1
⊆···⊆I
j
⊆··· f ∪I
j
f Q I
f Q f Q
Q
J ⊆ Q f ∈ Q
A f ∈ A
f = gh g h
Z
A
f ∈ A
f = g
1
···g
r
= h
1
···h
s
r = s g
i
= c
i
h
i

c
i
i =1, , r
f
Z
A f ∈ A (f)
f
f gh ∈ (f)
gh f