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Nullstellensätze and Applications ∗
Kriti Goel1†, Dilip P. Patil2and Jugal Verma3
1,3Department of Mathematics, Indian Institute of Technology Bombay
2Department of Mathematics, Indian Institute of Science Bangalore
E-mail : 1kriti@math.iitb.ac.in ,2patil@iisc.ac.in and 3jkv@math.iitb.ac.in
Dedicated to the memory of Prof. Dr. Uwe Storch
Abstract
In this expository paper, we present simple proofs of the Classical, Real, Projective and
Combinatorial Nullstellensätze. Several applications are also presented such as a classical
theorem of Stickelberger for solutions of polynomial equations in terms of eigenvalues of
commuting operators, construction of a principal ideal domain which is not Euclidean, Hilbert’s
17th
problem, the Borsuk-Ulam theorem in topology and solutions of the conjectures of Dyson,
Erdös and Heilbronn.
§ 1 Introduction
The objective of this paper is to present an exposition of four variations of the Hilbert’s Nullstellen-
satz, namely, the classical, real, projective and combinatorial. Each of these versions has given rise
to new techniques and insights into the basic problem of understanding the common solutions of
polynomial equations.
Analogues of the HNS have been investigated for non-algebraically closed fields. Notable among
them are the real Nullstellensatz [
24
], [
29
] and the combinatorial Nullstellensatz [
1
]. There is a
Nullstellensatz for partial differential equations [
40
] and most recently a tropical Nullstellensatz
[
41
] has also been proved. We have selected simple and short proofs and a few striking applications
for each of these versions which are accessible to students with basic background in algebra. The
paper by Sudhir R. Ghorpade in these proceedings presents the Nullstellensatz for finite fields.
Hilbert’s Nullstellensatz (HNS) is one of the fundamental results of Hilbert which paved the way
for a systematic introduction of algebraic techniques in algebraic geometry. It was proved in the
third section of his landmark paper on invariant theory [
20
]. The proof runs into five pages. In
fact, Hilbert proves it for homogeneous polynomials. He applies induction on the number of
indeterminates and uses elimination theory and resultants. Since the appearance of this proof,
several new proofs have appeared in the literature. Notable among them are: (1) proof by A.
Rabinowitsch [
37
], (2) Krull’s proof based on dimension theory of algebraic varieties, Noether
normalization lemma and the concept of integral dependence [
28
], (3) proof by Krull and Van der
Waerden for uncountable fields [
6
], (4) proof by E. Artin and J. Tate based on the Artin-Tate lemma
[
5
] (5) O. Zariski’s proof based on field theory [
46
], (6) proof by R. Munshi [
33
] and its exposition
by P. May [
31
] and (7) a remarkably simple proof by Arrondo [
3
] using resultants. (8) Krull [
27
]
and independently Goldman [
18
] introduced the notion of Jacobson ring, a ring in which every
∗
This article grew out of discussions with late Prof. Dr.Uwe Storch (1940-2017) and lectures delivered by the second
and the third author in various workshops and conferences. Prof. Uwe Storch was known for his work in commutative
algebra, analytic and algebraic geometry, in particular derivations, divisor class group and resultants.
†The first author is supported by UGC-JRF Fellowship of Government of India.
1
arXiv:1809.02818v1 [math.AC] 8 Sep 2018
Page 2 §2 Nullstellensätze Goel-Patil-Verma
prime ideal is an intersection of maximal ideals. They proved that a finitely generated algebra over
a Jacobson ring is a Jacobson ring which implies the HNS.
We now describe the contents of various sections. In section 2, we discuss the classical version
of the Nullstellensatz over algebraically closed fields. We present a proof due to E. Arrondo [
3
]
which uses two lemmas about polynomials and their resultants. This proof is very much in the
spirit of Hilbert’s original proof. We gather six versions of Classical Nullstellensatz and show that
they are all equivalent to the weak Nullstellensatz. As an application, we present a theorem of
Stickelberger [
42
] about systems of polynomial equations which have finitely many solutions. This
theorem converts the problem of construction of the solutions to the problem of finding common
eigenvectors of commuting linear operators acting on a finite dimensional vector space. We also
discuss a general construction of a Principal ideal domain that is not a Euclidean domain.
We present the Real Nullstellensatz in section 3. It answers the question about existence of a real
solution of a system of real polynomial equations. The Real Nullstellensatz has a weak version and a
strong version which are similar to the corresponding versions of the classical HNS for algebraically
closed fields. The central concepts here are those of real fields, real closed fields and real radical of
an ideal. We shall present the proofs of both the versions assuming the Artin-Lang homomorphism
theorem. We present a modern solution of Hilbert’s
17th
problem. The Real Nullstellensatz was
proved only in the 1970’s. A systematic study of real algebraic varieties was started soon after.
We shall discuss the Projective Nullstellensatz in section 4. This answers the question of existence
of a nontrivial solution of a system of homogeneous polynomial equations. We shall prove that if
f1,f2,..., fn
are homogeneous polynomials in
K[X0,X1,...,Xn]
where
K
is a
2
-field then there is
a nontrivial solution to the system
f1=f2=··· =fn=0.
We follow the approach given in [
35
]
which uses Hilbert functions and multiplicity of a graded ring. As an application, we prove the the
Borsuk-Ulam Theorem in topology.
Section 5 is devoted to the most recent version of the Nullstellensatz, namely the Combinatorial
Nullstellensatz. We present a proof of Noga Alon’s formulation [
1
] using the Classical Nullstellen-
satz. We describe two striking applications of the Combinatorial Nullstellensatz: a proof of Dyson’s
conjecture about the constant term of a Laurent polynomial and a solution of a conjecture of Erdös
and Heilbronn about a lower bound on the cardinality of
A+B
where
A
and
B
are subsets of a finite
field.
This expository article is not intended to be a survey paper on the Nullstellensatz. There are
important works which we do not discuss, for example, the Tropical Nullstellensatz and the Nullstel-
lensatz for partial differential equations, the Eisenbud-Hochster’s paper about Nullstellensatz with
nilpotents [
16
], role of Gröbner bases in computation of radical ideals and testing whether an ideal
is the unit ideal of a polynomial ring and works of many authors about Effective Nullstellensatz[
25
],
[22], [8],[14],[26] etc.
§ 2 Nullstellensätze
Hilbert’s Nullstellensatz is the starting point of the classical algebraic geometry, it provides a
bijective correspondence between affine algebraic sets which are geometric objects and radical
ideals in a polynomial algebra (over a field) which are algebraic objects.
In this section we formulate several versions of Nullstellensatz and prove their equivalence. First
we recall some standard notation, definitions and preliminary results. For other undefined terms and
notions we recommend the reader to see the books [7] and [34].
2.1 Notation and Preliminaries
All rings considered in this article are commutative rings with
unity. The letter
K
will always denote a field and the letters
A
,
B
,
C
,
R
will be generally used for
gpv2018-ToJournal.tex September 11, 2018 ; 12:38 a.m. 2/35
Goel-Patil-Verma §2 Nullstellensätze Page 3
rings. As usual we use
N
,
Z
,
Q
,
R
and
C
to denote the set of non-negative integers, the ring of
integers, the fields of rational, real and complex numbers respectively.
(1) Algebras over a ring
Let
A
be a ring. An
A
-algebra
B
is a ring together with a ring homomorphism
ϕ:A→B
called the structure homomorphism of the
A
-algebra
B
. Overrings and residue class rings of
A
are
considered
A
-algebras with natural inclusions and surjections as the structure homomorphisms, respectively.
The polynomial ring
A[Xi|i∈I]
in the indeterminates
Xi
,
i∈I
, is an
A
-algebra with the natural inclusion
A→A[Xi|i∈I]as the structure homomorphism.
Let
B
and
C
be
A
-algebras. An
A
-algebra homomorphism from
B
to
C
is a ring homomorphism
θ:B→C
such that the diagram
Bθ//C
A
ϕ
__
ψ
??
is commutative, that is, θ◦ϕ=ψ, or equivalently θis A-linear.
The set of all A-algebra homomorphisms from Bto Cis denoted by HomA-alg(B,C).
(2) Polynomial algebras
Polynomial algebras are the free objects (in the language of categories) in the
category of (commutative) algebras over a ring Awith the following universal property :
Universal property of polynomial algebras
Let
B
be an
A
-algebra and let
x= (xi)i∈I
, be a family of
elements of
B
. Then there exists a unique
A
-algebra homomorphism
εx:A[Xi|i∈I]→B
such that
Xi7→ xi
for every
i∈I
.In particular, we can identify
HomA-alg(A[Xi|i∈I],B)
with
BI
. For
I={1,...,n}
, we
can identify
HomA-alg(A[X1,...,Xn],B)
with
Bn
. The unique
A
-algebra homomorphism
εx
is called the
substitution homomorphism or the evaluation homomorphism defined by x.
The image of
εx
is the smallest
A
-subalgebra of
B
containing
{xi|i∈I}
and is denoted by
A[xi|i∈I]
.
We call it the
A
-subalgebra generated by the family
xi
,
i∈I
. We say that
B
is an
A
-algebra generated
by the family
xi
,
i∈I
, if
B=A[xi|i∈I]
. Further, we say that
B
is a finitely generated
A
-algebra or an
A
-algebra of finite type or an affine algebra over
A
if there exists a finite family
x1,...,xn
of elements of
B
such that
B=A[x1,...,xn]
. A ring homomorphism
ϕ:A→B
is called a homomorphism of finite type if
B
is
an A-algebra of finite type with respect to ϕ.
(3) Prime, maximal and radical Ideals
Let
A
be a ring. The set
SpecA
(resp.
Spm A
) of prime (resp.
maximal) ideals in
A
is called the prime (resp. maximal)spectrum of
A
. Then
Spm A⊆SpecA
and a
well-known theorem asserts that if
A6=0
then
Spm A6=/0
. For example,
Spm Z
is precisely the set
P
of
positive prime numbers and
SpecZ={0}∪P
. The ring
R
is a field if and only if
Spm A={0}
. The ring
A
is an integral domain if and only if
{0} ∈ SpecA
. For an ideal
a
in
R
, the ideal
√a:={f∈R|fr∈
afor some integer r≥1}
is called the radical of
a
. Clearly
a⊆√a
. If
√a=a
, then
a
is called a radical
ideal. Obviously,
p√a=√a
. Therefore the radical of an ideal is a radical ideal. Prime ideals are radical
ideals. An ideal ain Zis a radical ideal if and only if a=0 or ais generated by a square-free integer.
The radical
nA:=√0
of the zero ideal is the ideal of nilpotent elements and is called the nilradical of
A
.The
nilradical
nA=∩p∈SpecAp
is the intersection of all prime ideals in
A
.More generally, (formal Nullstellensatz)
√a=∩p∈SpecA{p|a⊆p}for every ideal ain A.
The intersection
mA:=∩m∈Spm Am
of maximal ideals in
A
is called the Jacobson radical of
A
. Clearly,
nA⊆mA. The Jacobson radical of Z(resp. the polynomial algebra K[X1,...,Xn]over a field K) is 0.
(4) Integral Extensions. Let A⊆Bbe an extension of rings. We say that an element b∈Bis integral over
A
if
b
is a zero of a monic polynomial
a0+···+an−1Xn−1+Xn∈A[X]
, i. e. if
a0+···+an−1bn−1+bn=0
with
a0,...,an−1∈A
. We say that
B
is integral over
A
if every element of
B
is integral over
A
. The concept of
an integral extension is a generalization of that of an algebraic extension. For example, an algebraic field
extension
E|K
is an integral extension. Moreover, if a ring extension
A⊆B
is an integral extension, then the
polynomial extension
A[X1,...,Xn]⊆B[X1,...,Xn]
is also integral. It is easy to see that : If
B
is a finite type
algebra over a ring A, then B is integral over A if and only if B is a finite A-module. Later we shall use the
following simple proposition in the proof of the classical form of HNS :
Proposition
Let
A⊆B
be an integral extension of rings and
a(A
be a non-unit ideal in
A
. Then the
extended ideal aB (in B) is also a non-unit ideal.
Proof
Note that
aB=B
if and only if
1∈aB
. Moreover, if
1∈aB
then since
B
is integral over
A
, already
1∈aB0
for some finite
A
-subalgebra
B0
of
B
. Therefore, we may assume that
B
is a finite
A
-module. But,
gpv2018-ToJournal.tex September 11, 2018 ; 12:38 a.m. 3/35
Page 4 §2 Nullstellensätze Goel-Patil-Verma
then by the Lemma
1
, there exists an element
a∈a
such that
(1−a)B=0
, in particular,
(1−a)·1=0
, i. e.
1=a∈awhich contradicts the assumption. •
(5) The K-Spectrum of a K-algebra
(see [
34
]) Let
K
be a field. Then using the universal property of
the polynomial algebra
K[X1,...,Xn]
, the affine space
Kn
can be identified with the set of
K
-algebra homo-
morphisms
HomK-alg(K[X1,...,Xn],K)
by identifying
a= (a1,...,an)∈Kn
with the substitution homomor-
phism
ξa:K[X1,...,Xn]→K
,
Xi7→ ai
. The kernel of
ξa
is the maximal ideal
ma=hX1−a1,...,Xn−ani
in
K[X1,...,Xn]
. Moreover, every maximal ideal
m
in
K[X1,...,Xn]
with
K[X1,...,Xn]/m=K
is of the type
ma
for a unique a= (a1,...,an)∈Kn; the component aiis determined by the congruence Xi≡aimod m.
The subset
K-Spec K[X1,...,Xn]:={ma|a∈Kn}
of
Spm K[X1,...,Xn]
is called the
K
-spectrum of
K[X1,...,Xn]. We have the identifications:
Kn←−−−−−−−−→ HomK-alg(K[X1,...,Xn],K)←−−−−−−−−→ K-Spec K[X1,...,Xn],
a←−−−−−−−−−−−−−−−−−−→ ξa←−−−−−−−−−−−−−−−−−−→ ma=Kerξa.
More generally, for any
K
-algebra
A
, the map
HomK-alg(A,K)−→ {m∈Spm A|A/m=K}
,
ξ7→ Kerξ
, is
bijective. Therefore we make the following definition :
For any
K
-algebra
A
, the subset
K-Spec A:={m∈Spm A|A/m=K}
is called the
K
-spectrum of
A
and is
denoted by
K-Spec A
. Under the above bijective map, we have the identification
K-Spec A=HomK-alg(A,K)
.
For example, since
C
is an algebraically closed field,
Spm C[X] = C-SpecC[X]
, but
R-SpecR[X](
Spm R[X]
. In fact, the maximal ideal
m:=hX2+1i
does not belong to
R-SpecR[X]
. More generally,
a field Kis algebraically closed 2if and only if Spm K[X] = K-Spec K[X].
(6) Polynomial maps
Let
A
be an algebra over a field
K
. For a polynomial
f∈K[X1,...,Xn]
, the function
ϕ∗
f:An→A
,
a7→ f(a)
, is called the polynomial function (over
K
)defined by
f
. If
A
is an infinite integral
domain, then the polynomial function
ϕ∗
f
defined by
f
determines the polynomial
f
uniquely. This follows
from the following more general observation :
Identity Theorem for Polynomials
Let
A
be an integral domain and
f∈A[X1,...,Xn]
,
f6=0
. If
Λ1,...,Λn⊆
A
with
|Λi|>degXif
for all
i=1,...,n
, then
Λ:=Λ1×···×Λn6⊆ VA(f):={a∈An|f(a) = 0}
, that is,
there exists
(a1,...,an)∈Λ
such that
f(a1,...,an)6=0
. In particular, if
A
is infinite, then
f:An→A
,
a7→ f(a)
, is not a zero function. If
A
is infinite, then the evaluation map
ε:A[X1,...,Xn]−→ Maps (An,A)
,
f7→ ε(f):a7→ f(a)is injective.
Proof
We prove the assertion by induction on
n
. If
n=0
, it is trivial. For a proof of the inductive step
from
n−1
to
n
, write
f=∑d
k=0fk(X1,...,Xn−1)Xk
n
with
fd(X1,...,Xn−1)6=0
in
A[X1,...,Xn−1]
. Since
degXifd≤degXif<|Λi|
for all
i=1,...,n−1
, by induction hypothesis, there exists
(a1,...,an−1)∈An−1
with
fd(a1,...,an−1)6=0
. Therefore
f(a1,...,an−1,Xn)
is a non-zero polynomial in
A[Xn]
of degree
d<|Λn|
and hence there exists an∈Λnwith f(a1,...,an−1,an)6=0. •
If
A=K
, then the identifications in (5) above allow us to write
f(a) = ξa(f)≡fmod ma
for any
a∈Kn
;
f(a)is called the value of f at a, or at ξa, or at ma.
Let
ϕ:K[Y1,...,Ym]→K[X1,...,Xn]
be a
K
-algebra homomorphism and let
fi:=ϕ(Yi),1≤i≤m
. Then the
map
ϕ∗:Kn→Km
defined by
ϕ∗(a1,...,an) = ( f1(a),..., fm(a))
is called the polynomial map associated
to
ϕ
. Under the identifications in (5), the polynomial map
ϕ∗
is described as follows :
ξa7→ ϕ∗ξa=ξa◦ϕ
or by ma7→ ϕ∗ma=ϕ−1(ma) = mf(a),a∈Kn. For every G∈K[Y1,...,Ym], we have ϕ∗
G◦ϕ∗=ϕ∗
ϕ(G).
More generally, for any
K
-algebra homomorphism
ϕ:A→B
, we define the map
ϕ∗:K-Spec B→K-Spec A
by
ϕ∗ξ:=ξ◦ϕ
or by
ϕ∗m=ϕ−1(m)
,
m=Kerξ∈K-Spec B=HomK-alg (B,K)
. Further, if
ψ:B→C
is an
another K-algebra homomorphism then (ψ◦ϕ)∗=ϕ∗◦ψ∗.
1Lemma of Dedekind-Krull-Nakayama
Let
a
be an ideal in a commutative ring
A
and
V
be a finite
A
-module. If
aV=V
, then there exists an element
a∈a
such that
(1−a)V=0
,i. e.
(1−a)∈AnnAV
.For a proof one uses the
well-known “Cayley-Hamilton trick”.
2
A field
K
is called algebraically closed if every non-constant polynomial in
K[X]
has a zero in
K
or equivalently,
every irreducible polynomial in
K[X]
is linear. The Fundamental Theorem of Algebra asserts that : the field of complex
numbers
C
is algebraically closed. This was first stated in 1746 by J. d’Alembert (1717-1783), who gave an incomplete
proof — with gaps at that time. The first complete proof was given in 1799 by Carl Friedrich Gauss (1777–1855).
gpv2018-ToJournal.tex September 11, 2018 ; 12:38 a.m. 4/35
Goel-Patil-Verma §2 Nullstellensätze Page 5
2.2
In general, we are interested in studying the solution set of a finite system of polynomials
f1,..., fm∈K[X1,...,Xn]
over a given field
K
(for example,
K=Q
,
R
,
C
, or any finite field, more
generally, even over a commutative ring, e.g. the ring of integers
Z
) in the affine
n
-space
Kn
over
K
or even in bigger affine n-space Lnover a field extension Lof K. Typical cases are :
(a) K=R,L=C. (Classical Algebraic Geometry).
(b) K=Q,L=Cor Q:=the algebraic closure of Qin C. (Arithmetic Geometry)
(c) Kis a finite field, L=K:=the algebraic closure of K.
2.3 Affine K-algebraic sets Let L|Kbe a field extension of a field K. The solution space
VL(fj,j∈J) = {a∈Ln|fj(a) = 0 for all j∈J} ⊆ Ln
of a family
fj
,
j∈J
, of polynomials in
K[X1,...,Xn]
is called an affine
K
-algebraic set in
Ln
, the
family
fj
,
j∈J
is called a system of defining equations, the field
K
is called the field of definition
and the field
L
is called the coordinate field of
VL(fj,j∈J)
. The points of
VL(fj,j∈J)∩Kn
are
called the K-rational points of V.
Note that
VL(fj,j∈J)=∩j∈JVL(fj)
and
VL(fj,j∈J)
depends only on the radical
√a
of the ideal
a:=hfj|j∈Ji
generated by the family
fj,j∈J
in
K[X1,...,Xn]
. By Hilbert’s Basis Theorem every
ideal in the polynomial ring
K[X1,...,Xn]
is finitely generated and so there exists a finite subset
J0⊆J
such that
a:=hfj|j∈J0i
. This shows that
VL(fj,j∈J)=VL(fj,j∈J0)=∩j∈J0VL(fj)
. In
other words, every affine
K
-algebraic set in
Ln
is a set of common zeros of finitely
many polynomials.
2.4 Examples Let L|Kbe a field extension of a field K.
(1) Linear K-algebraic sets
For linear polynomials
fi=∑n
j=1ai jXj−bi
,
ai j ,bi∈K
,
i=1,...,m
,
j=1,... n
,
the affine
K
-algebraic set
VL(f1,..., fm)
is called a linear
K
-algebraic set defined by the
m
linear equations
f1,..., fm
over
K
. This is precisely the solution space of the system of
m
linear equations in
X1,...,Xn
written
in the matrix notation :
A·X=b,where A= (ai j)1≤i≤m
1≤j≤n∈Mm,n(K),X=
X1
.
.
.
Xn
and b=
b1
.
.
.
bm
∈Mm,1(K).
Their investigation is part of Linear algebra. For example, if
L=K
,
r
is the rank of the matrix
A
, then
VK(f1,..., fm)has d=n−rlinearly independent solutions. In fact, there is a parametric representation :
VK(f1,..., fm) = {x0+
d
∑
i=1
ti·xi|t1,...,td∈K},
where
x0∈Kn
is a special solution and
xi∈Kn
,
i=1,...,d
are
d
linearly independent solutions of the given
system AX=b.
(2) K-Hypersurfaces
For
f∈K[X1,...,Xn]
, the affine
K
-algebraic set
VL(f) = {a∈Ln|f(a) = 0}
is called
the
K
-hypersurface defined by
f
. For
n=1
, since
K[X]
is a PID, every affine
K
-algebraic set is defined
by one polynomial
f∈K[X]
. Moreover, if
L
is algebraically closed and if
deg f
is positive, then
VL(f)
is a non-empty finite subset of
L
of cardinality
≤deg f
. Furthermore, if every
a∈VL(f)
is counted with
its multiplicity
νa(f):=Min{r∈N|f(r)(a)6=0}
, where for
r∈N
,
f(r)∈K[X]
denote the (formal)
r
-th
derivative of
f
, then we have a nice formula :
deg f=∑
a∈VL(f)
νa(f).
Therefore
VL(f) = {a1,...,ar}
if
f=a(X−a1)ν1···(X−ar)νrwith a∈Kand a1,...,ar∈Ldistinct and νi:=νai(f),i=1,...,r.
For n=2, 3, 4, K-hypersurfaces are called plane curves,surfaces, 3-folds, defined over K, respectively.
(3)
If
L
is infinite and
n≥1
, then the complement
LnrVL(f)
of the
K
-hypersurface
VL(f)
,
f∈K[X1,...,Xn]
,
is infinite. In particular, if
V=VL(a)(Ln
is a proper
K
-algebraic set, then its complement
LnrVL(a)
is
infinite.
Proof
By induction on
n
. If
n=1
, then clearly
VL(f)
is finite and hence the assertion is trivial, since
L
is infinite. Assume that
n≥2
. We may assume that the indeterminate
Xn
appears in
f
; then we have
a representation :
f=f0+f1Xn+··· +fdXd
n
with
f0,..., fd∈K[X1,...,Xn−1]
,
d>0
and
fd6=0
. By the
gpv2018-ToJournal.tex September 11, 2018 ; 12:38 a.m. 5/35
Page 6 §2 Nullstellensätze Goel-Patil-Verma
induction hypothesis, we may assume that there is
(a1,...,an−1)∈LnrVL(fd)
. Then the polynomial
f(a1,...,an−1,Xn)6=0
and hence it has only finitely many zeroes in
L
. In other words, there are infinitely
many an∈Lsuch that f(a1,...,an−1,an)6=0. •
(4)
If
L
is algebraically closed and
n≥2
, then every
K
-hypersurface
VL(f)
,
f∈K[X1,...,Xn]
contains
infinitely many points.
Proof
Since
L
is algebraically closed, it is infinite. Further, since
n≥2
, we may assume that
f
has
representation as in the above proof of (3) and hence by (3)
fd(a1,...,an−1)6=0
for infinitely many
(a1,...,an−1)∈Ln−1
. Now, since
L
is algebraically closed, for each of these
(a1,...,an−1)
, there exists
an
such that f(a1,...,an−1,an) = 0. •
(5) Conic Sections
The
K
-hypersurfaces
VK(f)⊆K2
defined by polynomials
f(X,Y)∈K[X,Y]
of degree
2
are called conic sections.
3
There are two possibilities. First
f
is not prime, then the (degenerated) conic
f(x,y) = 0
is a double line, or a union of two distinct straight lines. Second,
f
is prime, in this case, we
assume that
K
is an infinite field of
charK6=2
. Then by an affine
K
-automorphism
4
of
K[X,Y]
,
f(X,Y)
can
be brought into one of the forms
Y2−X
,
aX2+bY 2−1
,
a,b∈K×
, see Lemma 2.9. These are called parabola,
ellipse or hyperbola according as
aX2+bY 2
is prime or not prime. Further, the defining polynomial of a
hyperbola can be transformed into
XY −1
. Note that a polynomial
aX2+bY 2−1
,
a,b∈K×
, is always prime
and, if it has at least one zero
5
, then it has infinitely many zeros and hence is a defining polynomial of a
hyperbola or an ellipse.
2.5 The K-Ideals,K-coordinate rings and (Classical) Algebra-Geometry correspondences
We use the following notation : For a set
S
, let
P(S)
denote the power set of
S
, for any ring
A
, let
I(A)
(resp.
Rad-I(A)
) denote the set of ideals (resp. radical ideals) in
A
and for a field extension
L|K, let Aff-AlgK(Ln)⊆P(Ln)denote the set of all affine K-algebraic sets in Ln.
The definition in 2.3 defines a map
VL:Rad-I(K[X1,...,Xn]) −→ Aff-AlgK(Ln)⊆P(Ln),a7−→ VL(a),
Note that if
L
is infinite, then not every subset
V∈P(Ln)
is an affine algebraic
K
-set. For instance, if
n=1
and if
V
is infinite and
6=L
. On the other hand if
K=L
is finite, then every subset
V∈P(Ln)
is an affine algebraic K-set.
To understand the map
VL
better, for a subset
W∈P(Ln)
, we define the vanishing
K
-ideal of
W
:
IK(W):={f∈K[X1,...,Xn]|f(a) = 0 for all a∈W}.
Clearly,
IK(W)
is a radical ideal in
K[X1,...,Xn]
. The affine
K
-algebra
K[V]:=K[X1,...,Xn]/IK(V)
is called the K-coordinate ring of V. We therefore have defined the map :
IK:P(Ln)−→ Rad-I(K[X1,...,Xn]),W7−→ IK(W).
With these definitions, we are looking for answers to the following questions :
3
The discovery of conic sections is attributed to Menaechmus (350 B. C.). They were intensively investigated by
Apollonius of Perga (225 B. C.).
4
An affine
A
-automorphism of a polynomial algebra
A[X1,...,Xn]
is an
A
-algebra automorphism
ϕ
defined by
ϕ(Xj)=∑n
i=1ai jXi+bj
,
1≤j≤n
, where
(ai j)∈GLn(A)
and
(bj)∈An
. If
(ai j)
is the identity matrix then
ϕ
is called a
translation, if (bj)=0, then ϕis called a linear K-automorphism.
5Depending on the ground field K, it can be very difficult to decide whether such a polynomial has a zero or not.
gpv2018-ToJournal.tex September 11, 2018 ; 12:38 a.m. 6/35
Goel-Patil-Verma §2 Nullstellensätze Page 7
(a)
For which subsets
W∈P(Ln)
the equality
VL◦IK(W) = W
holds? The answer to this question
is provided by the introduction of the Zariski
K
-topology on
Ln
( see 2.6 below). This topology
is weaker than the usual topology (for instance, if
L=K=R
or
C
). The Zariski topology on
Ln
reflects the algebraic structure of
K
-regular
L
-valued functions on
Ln
. More generally, a function
ϕ:V→L
on a
K
-algebraic set
V⊆Ln
is called
K
-regular function if there exists a polynomial
f∈K[X1,...,Xn]
such that
ϕ(a) = f(a)
for all
a∈V
. The set
ΓK(V,L)
of all
K
-regular functions on
V
with values in
L
is, obviously, an affine
K
-algebra. Further, for every affine
K
-algebraic set
V⊆Ln
, the canonical map
K[X1,...,Xn/IK(V)∼
−→ ΓK(V,L)
is an isomorphism of
K
-algebras and the map
V∼
−→ HomK-alg(K[V],L)
,
a7→ ξa:f7→ f(a)is well-defined and is bijective.
(b)
When is the composite map
IK◦VL=idRad-I(K[X1,...,Xn])
? The answer to this question is
provided by the Hilbert’s Nullstellensatz (HNS 2 Geometric version), see Theorem 2.10(2).
(c)
When does the system
f1=0,..., fm=0
have a common solution, i. e.
VL(f1,..., fm)6=/0
. The
answer to this question is provided by the Hilbert’s Nullstellensatz (HNS 1), see Theorem2.10 (1).
(d)
When exactly is
VL(f1,..., fm)
a finite set? The answer to this question is provided by the
Finiteness Theorem, see Theorem 2.12.
Therefore if
L
is algebraically closed, then we have the algebra-geometry bijective correspondences
VLand IKwhich are inclusion reversing inverses of each other :
Rad-I(K[X1,...,Xn]) VL
←−−−−−−−−−−−−−→
IK
Aff-AlgK(Ln)
a=IK(VL(a)) ←−−−−−−−−−−−−−→ VL(a) = VL(IK(a)),
One can therefore study algebraic objects — ideals in the polynomial ring
K[X1,...,Xn]
and geo-
metric objects — affine
K
-algebraic sets together by using these algebra-geometry bijective corre-
spondences. These correspondences are the starting point of classical algebraic geometry. HNS 1
0
extends the Fundamental Theorem of Algebra to certain polynomials in many variables over alge-
braically closed fields. To establish the above bijective correspondences, the fundamental step is
provided by the Hilbert’s Nullstellensatz (HNS 2).
2.6 Zariski K-topology It is easy to check that the set Aff-AlgK(Ln)of affine K-algebraic sets in
Ln
satisfy the axioms for closed sets in a topological space. Therefore the affine
K
-algebraic sets in
Ln
form the closed sets of a topology on
Ln
. This topology is called the Zariski
K
-topology on
Ln
.
The open sets are the complements
LnrVL(Fj,j∈J) =: DL(Fj,j∈J) = [j∈JDL(Fj).
In particular,
DL(F) = {a∈Ln|F(a)6=0}=LnrVL(F)
for every polynomial
F∈K[X1,...,Xn]
.
These open subsets are called the distinguished open subsets in
Ln
.They form a basis for the Zariski
topology on Ln.
2.7 Abstract algebraic geometry
In a more general set-up one can replace affine
K
-algebraic set
by the prime spectrum SpecAof a commutative ring A.
The subsets of the form
V(a):={p∈SpecA|a⊆p}
,
a∈Rad-I(A)
of
SpecA
are called affine
algebraic sets in
SpecA
. The subset
FZ(SpecA) = {V(a)|a∈Rad-I(A)} ⊆ P(Spec A)
form the
closed sets of a topology on
SpecA
which is called the Zariski topology on
SpecA
. This topology is
not Hausdorff in general, but it is compact. The open subsets
D(f):=SpecArV(A f )
,
f∈A
, are
basic open sets for the Zariski topology on SpecA.
Similarly, for every subset
W⊆SpecA
, we define the ideal
I(W):=∩p∈Wp
of
W
.This is clearly a
radical ideal in
A
. Further, the equalities :
V(I(W)) = W=
the closure of
W
in the Zariski topology
of SpecAand (Formal Hilbert’s Nullstellensatz)a=I(V(a)) are rather easy to prove, see 2.1(3).
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Page 8 §2 Nullstellensätze Goel-Patil-Verma
With these general definitions, we have the abstract algebra-geometry bijective correspondences
|rmV and I which are inclusion reversing inverses of each other:
Rad-I(A)V
←−−−−−−−−−−−−→
IFZ(SpecA)
a=I(V(a)) ←−−−−−−−−−−−−→ V(a) = V(I(a)),
One can therefore study algebra and geometry together by using this abstract algebra-geometry
bijective correspondence which is the starting point of abstract algebraic geometry.
We now prove the classical version of Hilbert’s Nullstellensatz (HNS 1
0
), see Theorem 2.10 (1
0
). We
shall present a proof based on the ideas of the proof given by E. Arrondo in [
3
]. It uses the classical
notion of resultant of two polynomials over a commutative ring and the so-called “tilting of axes
lemma” (see Lemma 2.9 below) which is of independent interest.
2.8 Hilbert’s Nullstellensatz
( HNS 1
0
: C l a s s i c a l Ve r s i o n ) Let
K
be an algebraically closed
field and let a(K[X1,...,Xn]be a non-unit ideal. Then VK(a)6=/0 .
Proof
Let
a(K[X1,...,Xn]
be a non-unit ideal. We shall prove that
VK(a)6=/0
by induction on
n
. For
n=0
, there is nothing to prove. If
n=1
, then
a=hfi
for some polynomial
f∈K[X1]
which is not a unit in
K[X1]
, i. e.
f6∈ K×:=Kr{0}
. Since
K
is algebraically closed and
f∈
K[X1]rK×
, obviously
VK(f)6=/0
. Now, assume that
n≥2
. Since
a(K[X1,...,Xn]
, the contraction
a0:=a∩K[X1,...,Xn−1](K[X1,...,Xn−1]
too. Therefore by induction hypothesis
VK(a0)6=/0
.
Choose
a0= (a0
1,...,a0
n−1)∈VK(a0)
. We consider the surjective
K
-algebra homomorphism
ϕ:
K[X1,...,Xn]−→ K[Xn],f7→ f(a0
1,...,a0
n−1,Xn).
(∗)We claim that the image ϕ(a) = {f(a0
1,...,a0
n−1,Xn)|f∈a}is a non-unit ideal in K[Xn].
Since
ϕ
is surjective,
b=ϕ(a)
is an ideal in
K[Xn]
. We now prove that
b6=K[Xn]
. Suppose, on the
contrary that,
b=K[Xn]
.
Then f(a0
1,...,a0
n−1,Xn) = 1 for some f=f0+f1Xn+···+fdXd
n∈a,
where
f0,..., fd∈K[X1,...,Xn−1]
,
d∈N
with
f0(a0
1,...,a0
n−1) = 1
and
fi(a0
1,...,a0
n−1) = 0
for
every i=1,...,d.
Remember that we are now looking for a contradiction. For this, since
K
is algebraically closed,
it is infinite and hence by Lemma 2.9 below, we may assume that the ideal
a
contains a monic
polynomial g=g0+···+gr−1Xr−1
n+Xr
n∈awith g0,...,gr−1∈K[X1,...,Xn−1]and r≥1. Now
consider the Xn-resultant of the polynomials fand g
ResXn(f,g) = Det
f0f1··· fd0 0 ··· 0
0f0··· fd−1fd0··· 0
.
.
..
.
.....
.
..
.
..
.
.....
.
.
0 0 ··· f0f1f2··· fd
g0g1··· gr−11 0 ··· 0
0g0··· gr−2gr−11··· ·
.
.
..
.
.....
.
..
.
..
.
.....
.
.
0 0 ··· g0g1g2... 1
| {z }
r+d-columns
r-rows
d-rows
∈K[X1,...,Xn−1].
Since
f0(a0
1,...,a0
n−1) = 1
and
fi(a0
1,...,a0
n−1) = 0
for every
i=1,...,d
,
ResXn(f,g)(a0
1,...,a0
n−1)
is the determinant of the lower triangular matrix with all diagonal entries equal to
1
and so
ResXn(f,g)(a0
1,...,a0
n−1) = 1
. On the other hand, note that expanding the determinant of the
above
(d+r)×(d+r)
matrix ( the Sylvester’s matrix of polynomials
f
and
g
) by using the
first column after replacing the first column by adding
Xi
n
-times the
(i+1)
-th column for all
i=1,...d+r−1
, we get
ResXn(f,g) = Φf+ψg
for some polynomials
Φ,Ψ∈K[X1,...,Xn−1]
.
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Goel-Patil-Verma §2 Nullstellensätze Page 9
In particular,
ResXn(f,g)∈a∩K[X1,...,Xn−1]=a0
and hence
ResXn(f,g)(a0
1,...,a0
n−1)=0
, since
(a0
1,...,a0
n−1)∈VK(a0)
. This is a contradiction. Therefore
b(K[Xn]
and so
b=hh(Xn)i
for some
h(Xn)∈K[Xn]rK×
. Once again, since
K
is algebraically closed,
h(Xn)
has a zero
an∈K
. This
proves that f(a0
1,...,a0
n−1,an)=0 for all f∈a, i. e. (a0
1,...,a0
n−1,an)∈VK(a).•
2.9 Lemma
( T i t l t i n g o f A x e s L e m m a ) Let
K
be a field and
f∈K[X1,...,Xn]
be a
non-constant polynomial. Then there exists a
K
-automorphism
ϕ:K[X1,...,Xn]→K[X1,...,Xn]
such that
ϕ(Xn) = Xn
and
f=aXd
n+gd−1Xd−1
n+···+g0
, where
a∈K×
and
gj∈K[Y1,...,Yn−1]
,
0≤j≤d−1
,
Yi:=ϕ(Xi)
,
1≤i≤n−1
. Moreover, if
K
is infinite, then one can also choose a
linear K-automorphism ϕsatisfying the above conclusion. See Footnote 4.
Proof
First assume that
K
is infinite. Let
f=f0+f1+···+fd
, where
fm∈K[X1,...,Xn]
is the
homogeneous component of degree
m
of
f
,
0≤m≤d:=deg f
. For any
a1,...,an−1∈K
, put
Yi:=Xi−aiXn, 1 ≤i≤n−1. Then
f=
d
∑
m=0
fm(Y1+a1Xn,...,Yn−1+an−1Xn,Xn) =
d
∑
m=0 fm(a1,...,am−1,1)Xm
n+
m−1
∑
j=0
fm j Xj
n!
where
fm j ∈K[Y1,...,Yn−1]
are homogeneous polynomials of degree
m−j
. Since
fd
is homoge-
neous,
fd(X1,...,Xn−1,1)6=0
. Therefore, since
K
is infinite, we can choose (see Example 2.4 (3))
a1,...,an−1∈Ksuch that a:=fd(a1,...,an−1,1)6=0.
In the general case, let
f=∑α∈ΛaαXα
where
Λ
is a finite subset of
Nn
and
aα∈K×
for every
α= (α1,...,αn)∈Λ
. For any positive integers
γ1,...,γn−1
, put
Yi:=Xi−Xγi
n
,
1≤i≤n−1
and
γ:= (γ1,...,γn−1,1)∈Nn. Then
f=∑
α∈Λ
aαXα1
1···Xαn
n=∑
α∈Λ
aα(Y1+Xγ1
n)α1···(Yn−1+Xγn−1
n)αn−1Xαn
n.
For a natural number
r∈N
bigger than all the components of all
α= (α1,...,αn)∈Λ
, we have
degγXα=αn+α1r+···+αn−1rn−1
is the
r
-adic expansion of
degγXα
with digits
αn,α1,...,αn−1
.
Therefore by the uniqueness of the
r
-adic expansion,
degγXα6=degγXβ
for all
α,β∈Λ
,
α6=β
and hence there exists a unique
ν∈Λ
such that
d:=degγF=degγXν(>0)
. Therefore
f=
aνXd
n+fd−1Xd−1
n+···+f0with fj∈K[Y1,...,Yn−1].•
In the proof of the Tilting Axes Lemma for an infinite field
K
, we have used a simple linear
transformation of K[X1,...,Xn].
We now formulate several versions of Hilbert’s Nullstellensatz and prove their equivalence:
We say that a field extension E|Kis of finite type if the K-algebra Eis of finite type.
2.10 Theorem (Versions of HNS) The following statements are equivalent :
(1) HNS 1 :
Let
L|K
be an algebraically closed field extension of a field
K
and let
a(
K[X1,...,Xn]be a non-unit ideal. Then VL(a)6=/0 .
(10) HNS 10:
( C l a s s i c a l Ve r s i o n ) Let
K
be an algebraically closed field and let
a(
K[X1,...,Xn]be a non-unit ideal. Then VK(a)6=/0 .
(100) HNS 100:
Let
L|K
be an algebraically closed field extension of a field
K
and let
A
be a nonzero
K-algebra of finite type. Then HomK-alg(A,L)6=/0.
(2) HNS 2 :
(Geometric Version) Let
L|K
be an algebraically closed field extension of a
field K and let a⊆K[X1,...,Xn]be an ideal. Then IK(VL(a)) = √a.
(3) HNS 3 :
( F i e l d T h e o r e t i c F o r m
—
Z a r i s k i ’ s L e m m a ) Let
K
be a field and
E|K
be a finite type field extension of K. Then E |K is algebraic. In particular, E |K is finite.
gpv2018-ToJournal.tex September 11, 2018 ; 12:38 a.m. 9/35
Page 10 §2 Nullstellensätze Goel-Patil-Verma
(30) HNS 30:If K is an algebraically closed field, then the map
ξ:Kn−→ Spm(K[X1,...,Xn]) ,a= (a1,...,an)7−→ ma:=hX1−a1,...,Xn−ani,
is bijective.
Proof To prove the equivalence of these statements, we shall prove the implications :
(30)⇐= (3)
⇓ ⇑
(10) =⇒(1)⇐⇒ (100)
m
(2)
(10)⇒(1) :
Since
L
is algebraically closed, there is an algebraic closure
K
of
K
with
K⊆L
. Further,
since
K[X1,...,Xn]⊆K[X1,...,Xn]
is an integral extension and
a(K[X1,...,Xn]
is a non-unit
ideal, by the Proposition in (2.1) (4) the extended ideal
aK[X1,...,Xn]
is also a non-unit ideal in
K[X1,...,Xn]. Therefore by (10) we have /0 6=VK(aK[X1,...,Xn]) = VK(a)⊆VL(a).
(1) ⇒(3) :
By the given condition,
E=K[X1,...,Xn]/m
with
m∈Spm K[X1,...,Xn]
and hence by
(1) (applied to
L=K
the algebraic closure of
K
) there exists
a= (a1,...,an)∈Ln
such that
a∈
VK(m)
. Now, clearly the (substitution)
K
-algebra homomorphism
εa:K[X1,...,Xn]−→ K
,
Xi7→ai
,
i=1,...,n
, has kernel
=ma=m
and hence
εa
induces an injective
K
-algebra homomorphism
E=K[X1,...,Xn]/m−→ K. Therefore Eis algebraic over K.
(3) ⇒(30) :
Clearly, the map
ξ
is always injective. To prove that
ξ
is surjective, if
K
is an
algebraically closed field, let
m∈Spm(K[X1,...,Xn])
. Then
E:=K[X1,...,Xn]/m
is a finite type
field extension of
K
and hence
E|K
is algebraic by (3). Since
K
is algebraically closed,
E=K
and
hence there exists
(a1,...,an)∈Kn
such that
Xi≡aimod m
, for every
i=1,...,n
. This proves
that ma⊆mand hence m=ma∈Im ξ, since mais maximal.
(30)⇒(10) :
Since
a
is a non-unit ideal in
K[X1,...,Xn]
, by Krull’s Theorem there exists a maximal
ideal
m∈Spm K[X1,...,Xn]
with
a⊆m
. Now, by (3
0
)
a⊆m=ma
for some
a∈Kn
, or equivalently
a∈VK(a).
(1) ⇐⇒ (100) :
Let
L|K
be an algebraically closed field extension of
K
. Let
A=K[x1,...,xn]
be a
K
-algebra of finite type and
εx:K[X1,...,Xn]→A
be the surjective
K
-algebra homomor-
phism with
ϕ(Xi) = xi
for all
i=1,...,n
, and
a=Kerεx
. Note that for
a= (a1,...,an)∈Ln
,
the substitution homomorphism
εa:K[X1,...,Xn]−→ L
induces a
K
-algebra homomorphism
ϕ:A=K[x1,...,xn]−→ Lsuch that the diagram
K[X1,...,Xn]
εx
εa//L
A=K[x1,...,xn]
ϕ
88
is commutative if and only if a=Kerεx⊆Kerεa, or equivalently εa(a) = 0,i. e. a∈VL(a).
(1) ⇒(2) :
Clearly,
a⊆IK(VL(a)).
Conversely, suppose that
f∈IK(VL(a))
. Put
g=1−Xn+1f∈
K[X1,...,Xn+1]
and consider the
K
-algebraic set
W=VL(ha,gi)⊆Ln
. If
(a,an+1)∈W
with
a∈Ln
and
an+1∈L
, then
a∈VL(a)
. Therefore
f(a) = 0
, since
f∈IK(VL(a))
and so
0=g(a,an+1) =
1−an+1f(a) = 1
, a contradiction. This proves that
W=/0
and hence
ha,gi
is the unit ideal in
K[X1,...,Xn+1]
by (1). Therefore there exist
f1,..., fr∈a
and
h1,...,hr,h∈K[X1,...,Xn+1]
such
that
1=
r
∑
i=1
hi(X1,...,Xn+1)fi+h(X1,X2,...,Xn+1)g.
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Goel-Patil-Verma §2 Nullstellensätze Page 11
Since g(X1,...,Xn,1/f) = 0, substituting Xn+1=1/fin the above equation we get :
1=
r
∑
i=1
hi(X1,...,Xn,1/f)fi
Now, clearing the denominator in all
hi(X1,...,Xn,1/f)
,
i=1,...,r
, we get
fs∈ hf1,..., fri ⊆ a
for some s∈N,s≥1. This proves that f∈√a.
Remark :
The idea to use an additional indeterminate was introduced by J. L. Rabinowitsch [
37
] and is
known as Rabinowitsch’s trick.
(2) ⇒(1) :
Let
a
be a non-unit ideal in
K[X1,...,Xn]
. If
VL(a) = /0
, then by (2)
√a=IK(VL(a)) =
K[X1,...,Xn]and hence 1 ∈awhich contradicts the assumption. •
2.11 Remarks
(1) In 1947 Oscar Zariski [
46
] proved the following elegant result : ( Z a r i s k i ’ s L e m m a )
Let
A
be an algebra of finite type over a field
K
and
m∈Spm A
. Then
A/m
is a finite field extension of
K
.In
spite of its innocuous appearance, it is a useful result in affine algebras over any field.
(2) Note that since we have already proved the classical version of Hilbert’s Nullstellensatz (HNS 1
0
) in 2.8,
Theorem 2.10 proves all the versions of Hilbert’s Nullstellensatz.
(3) One can also prove that the following general form of Hilbert’s Nullstellensatz is equivalent to any one of
the forms of HNS mentioned in Theorem 2.10 :
HNS 4 :
( G e n e r a l F o r m ) Let
A
be a Jacobson ring
6
and let
B
be an
A
-algebra of finite type. If
n∈Spm B
is a maximal ideal in
B
, then its contraction
m=A∩n∈Spm A
is also a maximal ideal in
A
and
the residue field B/nof B is a finite field extension of the residue field A/mof A.
We give two applications of HNS 2 (see [
43
]) for solving systems of polynomial equations with
finitely many solutions. First, we prove a criterion for a system of polynomial equations to have
finitely many solutions.
2.12 Theorem
(Finiteness Theorem) Let
K
be an algebraically closed field and
a
be a
non-unit ideal in K[X1,...,Xn]. Then the following statements are equivalent:
(i) VK(a)is a finite set.
(ii) K[X1,...,Xn]/ais a finite dimensional K-vector space.
(iii) There exist polynomials f1(X1),..., fn(Xn)∈a.
Proof
Since
(K[X1,...,Xn]/a)red =K[X1,...,Xn]/√a
,
K[X1,...,Xn]/a
is Artinian if and only if
K[X1,...,Xn]/√ais Artinian. Also VK(a)= VK(√a). So we may assume that ais a radical ideal.
(i)
⇒
(iii) : Suppose that
VK(a) = {a1= (a1i),...,ar= (ari)} ⊆ Kn
is finite. Consider the poly-
nomials
fi(Xi) = (Xi−a1i)···(Xi−ari)
,
i=1,...,n
. Clearly,
fi(Xi)
vanishes on
VK(a)
and hence
fi(Xi)∈IK(VK(a)) = aby HNS 2.
(iii) ⇒(i) : By (iii) VK(a)⊆ ∩n
i=1VK(fi(Xi)) which is finite of cardinality ≤∏n
i=1deg fi.
(ii)
⇒
(iii) : Since
K[X1,...,Xn]/a=K[x1,...,xn]
is a finite dimensional
K
-vector space by (ii),
x1,...,xn
are algebraic over
K
. Let
µxi∈K[X]
be the minimal polynomial of
xi
over
K
and
fi(Xi):=
µxi(Xi)
,
i=1,...,n
. Then
fi(Xi)∈a
, since
fi(xi) = µxi(xi) = 0
in
K[x1,...,xn] = K[X1,...,Xn]/a
for all i=1,...,n.
(iii)
⇒
(ii) : Let
di=deg fi
for all
i=1,...,n
, and let
d=max{d1,...,dn}
. Then for all
i
,
xd
i
can
be expressed as a polynomial in
xm
i
for
m=0,1,...,di−1
. Since the monomials in
x1,...,xn
form
a generating set of the
K
-vector space
K[x1,...,xn]
, it follows that it is a finite dimensional
K
-vector
space. •
6
Recall that a ring
A
is called a Jacobson ring if every prime ideal in
A
is the intersection of maximal ideals in
A
.
Clearly, fields and the ring of integers,
Z
are Jacobson rings. Further, by Zariski’s Lemma2.11 (1) every finite type
algebra over a field
K
is a Jacobson ring. — The name Jacobson ring is used by Wolfgang Krull (1899-1971) to honour
Nathan Jacobson (1910-1999). The name Hilbert ring also appears in the literature.
gpv2018-ToJournal.tex September 11, 2018 ; 12:38 a.m. 11/35
Page 12 §2 Nullstellensätze Goel-Patil-Verma
As an application of HNS 2 we describe finite
K
-algebraic sets in
Kn
over an algebraically closed
field
K
using the eigenvalues of some commuting linear operators. For
n=1
, this is taught in the
undergraduate course on linear algebra, namely : Let
f(X) = Xn+an−1Xn−1+···+a1X+a0∈K[X]
be a monic polynomial of degree
n
over a field
K
and
a∈K
. Then
a∈VK(f)
if and only if
a
is an eigenvalue of the
K
-linear operator
λx:K[x]→K[x]
,
y7→ xy
on the
K
-vector space
K[x]:=K[X]/hf(X)i
of dimension
n=deg f
.For a proof use division with remainder in
K[X]
to
note that
a∈VK(f)
if and only if
f(X) = (X−a)g(X)
with
g(X)∈K[X]rhf(X)i
, equivalently,
0=f(x)=(x−a)g(x) = λx(g(x)) −ag(x)
with
g(x)6=0
in
K[x]
, i. e.
λx(g(x)) = ag(x)
with
g(x)6=0 which means ais an eigenvalue of λxwith eigenvector g(x).
Ludwig Stickelberger generalized the above observation to the case of polynomials in
n
indetermi-
nates over an algebraically closed field. More precisely, we prove the following :
2.13 Theorem
( S t i c k e l b e r g e r) Let
K
be an algebraically closed field and let
a⊆K[X1,...,Xn]
be a radical ideal in
K[X1,...,Xn]
with
VK(a)
finite. Then there exists
06=g(x):=g(x1,...,xn)∈
K[x1,...,xn] = K[X1,...,Xn]/a
such that
VK(a) = {a1= (a1i)...,ar= (ari)}
, where for each
j=1,...,r
, the
i
-th coordinate
aj i
of
aj
is an eigenvalue of
λxi
with eigenvector
g(x)
,i. e.
xig(x) =
λxi(g(x)) = aj i g(x)for each j =1,...,r and for all i =1,...,n.
Proof
Suppose that
VK(a) = {a1= (a1i),...,ar= (ari)} ⊆ Kn
is finite. For every
j=2,...,r
,
there exists
kj
such that
a1kj6=aj k j
and hence
g1(X1,...,Xn):=∏r
j=2(Xkj−aj kj)/(a1kj−aj k j)6∈ a
,
since
g1(a1)6=0
. Further,
(Xig1−aj i g1)(aj) = aj i g1(aj)−aj i g1(aj) = 0
for all
i=1,...,n
.
Therefore
g1(x) = g1(x1,...,xn)6=0
,
Xig1−aj ig1∈IK(VK(a)) = √a=a
by HNS 2 and hence
xig1(x) = λxi(g1(x)) = aj i g1(x) = 0
for all
j=1,...r
. Conversely, let
b= (b1,...,bn)∈Kn
be
such that
xig(x) = λxi(g(x)) = big(x)
for some
06=g(x)∈K[X1,...,Xn]/a
for all
i=1,...,n
.
Then
bν1
1···bνn
ng(x) = (λν1
x1◦···◦λνn
xn)(g(x)) = xν1
1···xνn
ng(x)
for every
ν= (ν1,...,νn)∈Nn
, and
hence
f(b1,...,bn)g(x) = f(x)g(x) = 0
for every
f∈a
. Therefore, since
f(b1,...,bn)∈K
and
g(x)6=0
in the
K
-vector space
K[x1,...,xn]
, it follows that
f(b1,...,bn) = 0
for every
f∈a
, i. e.
(b1,...,bn)∈VK(a).•
2.14 Examples of PIDs which are not EDs
In most textbooks it is stated that there are examples
of principal ideal domains which are not Euclidean domains. However, concrete examples are
almost never presented with full details. In this subsection we use HNS 3 to give a family of
such examples with full proofs which are accessible even to undergraduate students. The main
ingredients are computations of the unit group
A×
and the
K-Spec A
for an affine algebras over a
field K. First we recall some standard definitions and preliminary results :
(1) Unit Groups
For a ring
A
, the group
A×
of the invertible elements in the multiplicative monoid
(A,·)
of the ring
A
is called the unit group ; its elements are called the units in
A
. The determination of the unit
group of a ring is an interesting problem which is not always easy. Some simple examples are :
Z×={−1,1}
;
if
n≥2
, then
Z×
n={m∈N|0≤m<nand gcd(m,n) = 1}
; if
K
is a field then
K×=Kr{0}
; if
A
is an
integral domain, then
(A[X1,...,Xn])×=A×
; if
K
is a field, then
(K[T,T−1])×={λTn|λ∈K×and n∈Z}∼
=
the product group K××Z;(A[[X1,...,Xn]])×={f∈A[[X1,...,Xn]] |f(0)∈A×}.
(2) Norm
The notion of the norm is very useful for the determination of the unit groups of some domains.
Let
R
be a (commutative) ring and let
A
be a finite free
R
-algebra. For
x∈A
, let
λx:A→A
denote the
(left) multiplication by
x
. The norm map
NA
R:A→R
,
x7→ Det λx
, contains important information about the
multiplicative structure of Aover R. The following properties of the norm map are easy to verify :
The norm map
NA
R:A→R
is multiplicative, i. e.
NA
R(xy)=NA
R(x)·NA
R(y)
for all
x,y∈A
,
NA
R(a)=an
for every
a