# Early proofs of Hilbert’s Nullstellensatz

Jan Stevens  Department of Mathematical Sciences, Chalmers University of
Technology and University of Gothenburg. SE 412 96 Gothenburg, Sweden
<EMAIL_ADDRESS>
###### Abstract.

By Rabinowitsch’ trick Hilbert’s Nullstellensatz follows from the weak
Nullstellensatz (Rabinowitsch 1929). The weak version can be shown with
elimination theory. Hilbert’s original proof is also based on successive
elimination. Lasker obtained a new proof using primary decomposition. We
describe these early proofs and place them in the development of commutative
algebra up to the appearance of van der Waerden’s Moderne Algebra. We also
explain Hentzelt’s Nullstellensatz.

###### Key words and phrases:

Polynomial ideals, primary decomposition, Nullstellensatz, elimination theory,
resultant, Hilbert, Lasker

###### 2020 Mathematics Subject Classification:

14A05, 14-03, 13-03, 01A60, 01A55

## Introduction

Hilbert’s theorem of zeros, or Nullstellensatz as it is usually called, states
that if a polynomial $f\in P=k[X_{1},\ldots,X_{n}]$, where $k$ is an
algebraically closed field, vanishes in all common zeros of an ideal $I\subset
P$, then $f^{r}\in I$ for some natural number $r$. Usually the proof is
reduced to a special case, the weak Nullstellensatz, that an ideal without
zeros is the whole ring, by an argument due to Rabinowitsch [Rab29]. The weak
Nullstellensatz follows by elimination. Hilbert’s original proof [Hil93] is
also based on elimination. A different proof based on primary decomposition is
due to Lasker [Las05a]. We place these proofs in the early development of
commutative algebra.

Rabinowitsch’s proof [Rab29] appeared just in time to be included in the
second volume of van der Waerden’s Moderne Algebra [vdW31b]. This book can be
seen as marking the end of the early period. It made the subject widely known,
and in fact it is still a good introduction to the results we discuss in this
paper. Afterwards new proofs of the weak Nullstellensatz appeared with a
totally different flavour, like Zariski’s proof, based on his lemma that if a
finite integral domain over a field $K$ is a field then it is an algebraic
extension of $K$ [Zar47]. The most common modern proofs are variations of this
proof.

Rabinowitsch’s half page paper claims in fact to give a (complete) proof of
the Nullstellensatz and does not use the term weak Nullstellensatz. It refers
to Emmy Noether’s paper [HN22] for the statement that an ideal without zeros
is the whole ring, with a footnote that it also follows from Kronecker’s
elimination theory. Both the Hentzelt-Noether and the Kronecker theory are
based on successive elimination of variables. This is also the technique
Hilbert uses in his proof; he adapts Kronecker’s device to the homogeneous
case. In line with his intended application in invariant theory Hilbert
formulates and proves in [Hil93] the Nullstellensatz for homogeneous
polynomials.

The Nullstellensatz was instrumental to the creation of the concept of primary
ideal [Las05a]. Lasker’s definition is different from the modern one, which is
due to Emmy Noether [Noe21]. Macaulay paraphrases Lasker as follows: if the
product of two ideals is contained in a primary ideal, and if one does not
contain its zero set the other is contained in the ideal [Mac16]. To be able
to work with this definition it is essential to know that a prime ideal
consists of all polynomials vanishing on its zero set. This is a special case
of the Nullstellensatz. It is also possible to show the result directly and
use it in turn to prove the Nullstellensatz. Both Lasker [Las05b, Las05a] and
Macaulay [Mac16] do this. Their methods are different. Macaulay uses more
specific computations and he uses Kronecker’s theory of the resolvent to
describe the zero set of ideals. Lasker reasons more abstractly.

The Moderne Algebra [vdW31b] contains a second proof, in Chapter 13 on the
theory of polynomial ideals, based on van der Waerden’s earlier paper [vdW26].
The proof uses specialisation in fields of algebraic functions and avoids
elimination theory. In the paperback edition Algebra II [vdW67] of the Moderne
Algebra the chapter on elimination theory is eliminated; only the resultant of
two polynomials in one variable has been retained and moved to the first
volume [vdW71a].

We witness here the rise of commutative algebra as separate discipline, in a
period which starts with Noether and ends with Noether, to borrow from the
title of the essay [Gra97]. An important motivation for Lasker and Macaulay
was the generalisation of Max Noether’s “fundamental theorem on algebraic
functions”. The first proved special case of the Nullstellensatz (Netto’s
theorem [Net85]) follows from Bertini’s refinement of Noether’s theorem
[Ber90]. The most most far-reaching generalisation is Hentzelt’s
Nullstellensatz, proved by Emmy Noether’s first official PhD student Grete
Hermann [Her26]. Only after this theorem Hilbert’s theorem is referred to as
Nullstellensatz. The influence of Emmy Noether on van der Waerden is well
known. She also influenced Rabinowitsch: Noether spent the winter 1928/29 in
Moscow and led a seminar on algebraic geometry at the Communist Academy
[Ale36].

From the early proofs we first give Rabinowitsch’s proof, which does not prove
the weak Nullstellensatz. We describe Kronecker’s elimination theory, from
which the result does follow. It also gives ingredients for Hilbert’s original
proof of the theorem. These proofs use induction, but otherwise not more than
the basic properties of the resultant of two binary forms. We recall those in
the Appendix. Less elementary are the proofs using primary decomposition. We
describe the background needed. The last proof is van der Waerden’s proof by
specialisation. Finally we formulate but do not prove Hentzelt’z
Nullstellensatz. Before discussing the proofs we place them in a historic
context and describe the main characters and main developments, from Noether’s
fundamental theorem until Noether’s modern algebra.

##  1\. Notation

We use modern terminology, in particular the word ideal for the older term
module, or modular system. Lasker [Las05a] uses the term module for ideals in
$\mathbb{C}[x_{1},\dots,x_{n}]$, whereas his ideals are ideals in
$\mathbb{Z}[x_{1},\dots,x_{n}]$. The use of ideal was propagated by Emmy
Noether, see [HN22, Footnote 11]. The name modular system is explained by the
notation introduced by Kronecker [Kro82, §20]. He writes

$G\equiv 0\bmod(F_{1},F_{2},\dots,F_{k})$

to express that a polynomial $G$ can be written in the form

$P_{1}F_{1}+P_{2}F_{2}+\dots+P_{k}F_{k}\;,$

that is, $G\in(F_{1},F_{2},\dots,F_{k})$. This is an extension of the notation
$a\equiv b\bmod c$ introduced by Gauss for integers to express that $a-b$ is
divisible by $c$ without remainder.

We do not use the arithmetic terminology that an ideal $a$ divides $b$ if
$b\subset a$. Macaulay [Mac16] says in this case that $b$ contains $a$.

The term polynomial is used for the the older term whole rational function.
Homogeneous polynomials will also be called forms. The coefficients will be
taken in $K$, an algebraically closed field. In the older literature the base
field is tacitly assumed to be that of the complex numbers. Most proofs work
in the general case.

We have tried to use a uniform notation. In particular, following Emmy Noether
we use a Fraktur font to denote ideals. Hermann Weyl addressed Noether at her
funeral service: “Mit vielen kleinen deutschen Buchstaben hast Du Deinen Namen
in die Geschichte der Mathematik geschrieben”111With many German lower-case
letters you wrote your name in the history of mathematics. (this is the
version Grete Hermann gives in a letter to van der Waerden in 1982 [Roq07, p.
20]; the originaltext of the speech is also in loc. cit.).

Throughout this paper we use the name Nullstellensatz, although it is of
relatively recent date, see the MathOverflow question [KCo]. Before [vdW26]
the theorem was referred to as a known theorem of Hilbert, or an explicit
reference was given to [Hil93, §3]. Macaulay [Mac16] calls it the Hilbert-
Netto Theorem. Interestingly van der Waerden refers to it as a well known
theorem of Hilbert in the paper [vdW31a] (an extract of a letter to J. F.
Ritt), which has the footnote “See Macaulay, Modular Systems, p. 46. (J. F.
R.)”. The first mention of a Nullstellensatz concerns that of Hentzelt, in
Noether’s report on the thesis of Grete Hermann (Noether, 1.2.1925,
Promotionsakte Hermann) [Kor15, p. 320].

The use of the German name originates in the US. Emmy Noether advised her
students “don’t bother to translate, just read the German”, according to Ruth
McKee [QMLT83]. Hodge and Pedoe [HP47] use the term Hilbert’s zero-theorem,
but Zariski states in his review in Math Reviews [MR0028055] that they prove
Hilbert’s Nullstellensatz. Miles Reid explains the name as theorem of zeros,
adding: “but stick to the German if you don’t want to be considered as
ignorant peasant” [Rei88].

##  2\. From Noether to Noether

###  2.1. Noether’s fundamental theorem

Lasker spends nine pages of his paper [Las05a] to give an overview of the
development of ideal theory. It was Max Noether who with his fundamental
theorem “clearly and sharply apprehended and demonstrated the central
position, which the method of modular systems has in all questions of algebra”
[Las05a, p. 44].

Max Noether’s fundamental theorem on algebraic functions [Nöt73] concerns
plane curves. The equation of an algebraic curve $f=0$ passing through the
intersection of two curves $\varphi=0$, $\psi=0$ can be written in the form
$f=A\varphi+B\psi$, if the intersections are simple, but as Noether noticed,
the statement ceases to be valid otherwise. An easy example is provided by two
conics, tangent in two points. Then the line passing through these two points
is not contained in the pencil spanned by the two conics, but twice the line
(given by the square of a linear function) is. The correct condition for $f$
to lie in the ideal $(\varphi,\psi)$ is that in each of the intersection
points $f$ can be written as $f=a\varphi+b\psi$ with $a$ and $b$ power series;
it suffices to check this identity up to a certain order $\rho$ only depending
on the ideal $(\varphi,\psi)$, as was first observed by Bertini [Ber89].

In a paper “Zur Theorie der Elimination” [Net85] Netto gives a different
answer for the case of non-simple intersection, without making a connection to
Noether’s theorem; this is done by F. Meyer in his Jahrbuch review [JFM
17.0096.01]. Netto expresses his result geometricallyas follows: if an
algebraic curve $f(x,y)=0$ passes through all intersection points of two other
algebraic curves $\varphi(x,y)=0$, $\psi(x,y)=0$, then some power of the
polynomial $f(x,y)$ can be expressed as linear homogeneous function of
$\varphi(x,y)$ and $\psi(x,y)$, i.e.

$f(x,y)^{\rho}=A(x,y)\varphi(x,y)+B(x,y)\psi(x,y)\;,$

where $A(x,y)$ and $B(x,y)$ are also polynomials. As Hilbert [Hil93] remarks,
this is the special case of his Nullstellensatz for two inhomogeneous
variables. Netto’s proof gives that the required power is bounded by the
highest intersection multiplicity.

To generalise Noether’s fundamental theorem to $n$ dimensions was one of the
problems van der Waerden worried about when he came to Göttingen in 1924. In
his paper [vdW75] on the sources of his Moderne Algebra he says that a new
world opened for him. Already in Amsterdam van der Waerden “discovered that
the real difficulties of algebraic geometry cannot be overcome by calculating
invariants and covariants” [vdW75, p. 32]. In Göttingen he learned from Emmy
Noether that suitable tools had been developed by Dedekind and Weber, by
Hilbert, Lasker and Macaulay, by Steinitz and by Emmy Noether herself.

Noether’s theorem was generalised by König, Lasker and Macaulay. The most
general form is Hentzelt’s Nullstellensatz, which provides a criterion
indicating how much a polynomial has to vanish in the zeros of an ideal in
order to belong to it. It was proved by Emmy Noether’s first PhD student Grete
Hermann. Van der Waerden gives a non-constructive proof in [vdW26].

###  2.2. Kronecker’s elimination theory

The word ideal has its origin in Kummer’s work on the factorisation of
algebraic numbers. His theory has been developed by Kronecker, and in a
different way by Dedekind. The story is told in detail by Edwards [Edw80]. A
more elementary account is in Klein [Kle26, Ch. 7]. Kronecker’s role in
development of the algebraic tools in the period from Noether to Noether is
discussed in [Gra97].

Kronecker’s theory of algebraic quantities applies not only to number fields
but also to polynomial rings, and in particular to Elimination Theory. It
gives a method to find all solutions of a set of algebraic equations by
successive eliminations. We describe it in Section 4.

Kronecker lectured on his theory and disseminated it in private conversations,
but published his results only in 1882 in a Festschrift for Kummer’s doctor
jubilee [Kro82]. A part of Kronecker’s theory was treated in detail in a long
paper in Acta Mathematica by his student Jules Molk [Mol85]. In this paper he
mentions the result of Netto, which prompted Netto to publish his theorem also
in Acta Mathematica [Net85]. Netto gives in the second volume of his book
[Net00] an extensive account of elimination theory, also of the older work of
Bézout, Liouville, Poisson and Cayley. He generalises his theorem from [Net85]
to the case of a complete intersection of $n$ variables, adding that for $n=2$
Noether had shown this in a bit different, for his purposes less suitable
form. Indeed, by Bertini’s bound on Noether’s theorem it follows that a
sufficiently high power of the function satisfies the condition [Ber90]. With
a flawed argument Netto goes on to show that the result still holds if there
are more equations than variables, and derives the Nullstellensatz with
Hilbert’s argument for zero sets of higher dimensions (see Section 5).

Kronecker’s theory (and much more) was presented in the 564 page book [Kön03b]
by Julius (Gyula) König, published simultaneously in Hungarian [Kön03a] and
German. His goal (as stated in the Introduction of [Kön03b]) was to popularise
Kronecker’s ideas (“if this expression for this difficult area of mathematics
is allowed”). He indeed succeeded, as shown by the many comtemporary
references. A footnote to the published paper [Mac05] of Macaulay’s talk at
the ICM in Heidelberg (1904) states that Professor Noether and Professor Brill
have kindly drawn his attention to the recently published book by König. This
work is “remarkable for its precision and comprehensiveness and the large
additions it makes to the subject”. König’s book is still mentioned in a
footnote in [vdW67, p. 167]. Nowadays it is almost completely forgotten. As
[Gra97] puts it, a useful book on a new field will, if succesful, draw others
to discover better results, simpler and more general methods and if it does
not become a classic the work will gradually be covered up and forgotten. For
a modern reader König’s book is hard to read.

König gives applications to geometry, notably a generalisation of Noether’s
theorem to higher dimensions. He also treats several fundamental results of
Hilbert. In particular, he gives a new, simpler proof of the Nullstellensatz;
but his proof is flawed, as he “makes an absurdly false assumption concerning
divisibility” [Mac16, p. 35]. Actually, König gives a faulty proof of an
absurd divisibility statement [Kön03b, p. 399].

###  2.3. Hilbert

The next important development comes from Hilbert’s work on invariant theory,
in his two famous papers in Mathematische Annalen [Hil90, Hil93]. Klein
[Kle26, p. 329] writes that Hilbert takes up ideas from Kronecker with
Dedekind’s way of thinking, and applies them brilliantly on the problems of
invariant theory. Indeed, Hilbert states explicitly in the Introduction of
[Hil93] that he uses methods of the general theory of polynomial ideals, so
that the theory of invariants becomes a striking example of that theory, just
as cyclotomic fields constitute a striking example in number theory, where the
most important theorems about general number fields have first been found and
proven. In the first paper [Hil90], where he proves the basis theorem (stating
that every ideal is finitely generated) and his syzygy theorem and introduces
the Hilbert polynomial, many geometric examples are given, and only in the
last section the results are applied to prove the finiteness of the system of
invariants. One of the examples solves a problem of Salmon. Lasker points to
the influence of the work of Salmon and Cayley and comments:

> Man hat Salmons Werk unterschätzt, weil seinen Methoden die Strenge der
> Beweisführung abging. Wie groß dieser Fehler auch sein mag, so darf man
> niemals die Bedeutung Salmons als des großen Problemstellers und Wegweisers
> vergessen.222Salmon’s work has been underestimated because his methods
> lacked rigor of proof. However great this error may be, one must never
> forget the importance of Salmon as the great problem poser and guide.
> ([Las05a, p. 44])

At several places Hilbert [Hil90] stresses the importance of generalising
Noether’s fundamental theorem to higher dimensions.

Hilbert formulates the basis theorem in a different way from what is usual
nowadays.

######  Theorem 2.1.

Given a non-terminating sequence of forms in the $n$ variables
$x_{1},\dots,x_{n}$, say $F_{1},F_{2},F_{3},\dots$, there always exists a
number $m$ such that every form of the sequence can be written as

$F=A_{1}F_{1}+A_{2}F_{2}+\dots+A_{m}F_{m}\;$

where $A_{1},A_{2},\dots,A_{m}$ are suitable forms in the same $n$ variables.

Hilbert also gives a second version of the basis theorem, for forms with
integral coeffcients. Hilbert’s formulation seems natural if one thinks about
the explicit computation of invariants in special cases, which leads to lists.
Moreover, Hilbert treats only homogeneous polynomials, or forms, whereas the
modern formulation works with inhomogeneous polynomials. The theorem can be
extended to the inhomogeneous case by making all polynomials homogeneous with
a new variable of homogeneity [Mac16, p. 38].

Hilbert explicitly states that the basis theorem applies in particular to
homogeneous ideals in polynomial rings; he uses Dedekind’s term module.
Hilbert makes the connection with Kronecker’s concept of modular systems, but
stresses that his syzygy theory and the characteristic function (in modern
terms the Hilbert polynomial) use homogeneity in an essential way.

Conversely the basis theorem for ideals, that every ideal in the polynomial
ring $K[x_{1},\dots,x_{n}]$ is finitely generated, implies the theorem in
Hilbert’s formulation [vdW31b, § 80], [vdW67, § 115]: the ideal generated by
the $F_{i}$ has a finite set of generators, each of which is a linear
combination of only finitely many $F_{j}$.

The basis theorem is the first step in proving that the ring of invariants is
finitely generated. The invariants in question concern in modern terms the
action of the group $G=SL_{n}(\mathbb{C})$ on a vector space $V$ which is a
direct sum of the type $S^{d_{1}}{\mathbb{C}^{n}}\oplus\dots\oplus
S^{d_{k}}{\mathbb{C}^{n}}$. The result is that the ring $\mathbb{C}[V]^{G}$ of
$G$-invariant polynomials on $V$ is finitely generated. By the basis theorem
every invariant $i$ can be expressed as $i=A_{1}i_{i}+\dots A_{m}i_{m}$. By an
averaging procedure the $A_{i}$ can themselves be taken as invariants, of
lower degree than $i$. By applying the same reasoning to these invariants one
finally obtains that $i$ is a sum of products of the $i_{j}$. Nowadays one
uses the Reynolds operator, which is a $G$-invariant projection
$\mathbb{C}[V]\to\mathbb{C}[V]^{G}$, but Hilbert had to construct it using
Cayley’s $\Omega$-process; for details we refer to [DK15].

Hilbert’s proof was criticised for its nonconstructive character. The goal of
[Hil93] is to give a method to find the generators (in principle). By the
homogeneous version of Noether normalisation (proved by Hilbert for this
purpose) the ring of invariants is an integral extension of a polynomial ring
$k[J_{1},\dots,J_{\kappa}]$ with the $J_{i}$ invariants (of the same degree).
The quotient field of the ring of invariants is the field of rational
invariants and by the theorem of the primitive element it is an extension of
the field $L=K(J_{1},\dots,J_{\kappa})$ of the form $L(J)$ with $J$ a
polynomial invariant; Hilbert shows how to construct such a $J$. To find the
ring of invariants Hilbert gives three steps, of which the first is the most
difficult, namely to find the system $\\{J_{1},\dots,J_{\kappa}\\}$ of
invariants, such that every other invariant is integral over the
$J_{1},\dots,J_{\kappa}$, that is, satisfies a monic equation with
coefficients which are polynomials in the $J_{1},\dots,J_{\kappa}$. The second
step is to find $J$, such that all invariants are rational functions of
$J,J_{1},\dots,J_{\kappa}$. The third step is to find the integral elements of
the field $L(J)$, which can be done according to a general theory of
Kronecker: “If the invariants $J,J_{1},\dots,J_{\kappa}$ are known, finding
the full invariant system only requires the solution of an elementary problem
from the arithmetic theory of algebraic functions” [Hil93, p. 320].

The system $\\{J_{1},\dots,J_{\kappa}\\}$ has the property that all invariants
vanish if the $J_{1},\dots,J_{\kappa}$ vanish. Of fundamental importance for
the whole theory is that the converse can be proved: if invariants
$I_{1},\dots,I_{\mu}$ have the property that their vanishing implies the
vanishing of all other invariants, then every invariant is integral over the
$I_{1},\dots,I_{\mu}$. To prove this Hilbert first shows the Nullstellensatz
(see Theorem 5.1 for Hilbert’s version). This gives that powers of the
generators of the ring of invariants lie in the ideal $(I_{1},\dots,I_{\mu})$
and therefore every invariant of degree at least some fixed $\rho$. The
coefficients can again be taken as invariants. A finite number invariants of
lower degree form therefore a basis of the ring of invariants as
$K[I_{1},\dots,I_{\mu}]$-module, so this ring is integral over
$K[I_{1},\dots,I_{\mu}]$. Hilbert shows this with the now standard determinant
trick.

A form for which all invariants vanishes is called a null-form. In the space
of all forms they form an algebraic subset, and knowing it helps determining
the invariants $(I_{1},\dots,I_{\mu})$. For binary forms Hilbert determines
the null-forms with elementary means: a form $f(x,y)$ of degree $d$ is a null-
form if and only if $f$ has a zero of multiplicity bigger than $\frac{d}{2}$.
This can easily shown with the Hilbert-Mumford criterion, see [DK15, Example
2.5.4]; Hilbert proved the criterion later in his paper to handle forms of
more variables. In fact, this part of the theory was only taken up 70 years
later by Mumford in his Geometric Invariant Theory [Mum65]. For these
developments and their relation to Hilbert’s text we refer to the comments by
V.L. Popov to his Russian translation of Hilbert’s paper [Pop98].

###  2.4. Lasker

Little is known about the origins of the highly original paper “Zur Theorie
der Moduln and Ideale” [Las05a], by the world chess champion Emanuel Lasker.
Van der Waerden [vdW75] states that Lasker took his Ph.D. degree under
Hilbert’s guidance in 1905, but that is not correct.

Lasker (1868–1941) studied mathematics from 1888 in Berlin and later in
Göttingen (when Hilbert still was in Königsberg), but 1891 he interrupted his
studies and concentrated on chess, becoming world champion in 1894. He took up
his studies again in 1897, first in Heidelberg (taking courses with Landsberg)
and later in Berlin (courses with Hensel) [Ros09].

Lasker submitted a manuscript for the Grand prix des sciences mathématiques in
1898, where the question was “Chercher à étendre le rôle que peuvent jouer en
analyse les séries divergentes”, but it was considered to be a bit beside the
question [Prix98]. He used the first 23 pages of this manuscript to get a
doctoral degree. Max Noether in Erlangen was prepared to help him. Staying at
the Hotel Erlanger Hof Lasker wrote to the dean on Monday January 29, 1900,
who convened the examining committee for the next Wednesday. On the same
Monday Noether already delivered his report. Lasker passed magma cum laude
[Ull17]. Lasker submitted the paper to the Philosophical Transactions of the
Royal Society of London, where it was published in 1901 in German (Über Reihen
auf der Convergenzgrenze) [Las01]. So Lasker was neither a student of Hilbert
nor of Noether.

Lasker wrote a small paper [Las04] on the theory of canonical forms, dated New
York May 1903. His main mathematical work [Las05a] might have been an attempt
to start an academic career; he never had a permanent academic position in
mathematics. The paper is dated Charlottenburg March 1904. Right after Lasker
travelled to the US to play the chess tournament at Cambridge Springs.

Albert Einstein came to know Lasker in later life. He wrote on occasion of
Lasker’s sixtieth birtday:

> “Emanuel Lasker ist einer der stärksten Geister, denen ich auf meinem
> Lebenswege begegnet bin. Renaissance-Mensch, mit einem unbändigen
> Freiheitsdrang begabt, jeder sozialen Bindung abhold. So wurde er
> Schachmeister, wohl weniger aus besonderer hingebender Liebe zum Spiel.
> Letztere galt vielmehr der Philosophie, dem Verstehen überhaupt. Er liebt
> als ächter Eigenbrödler und Eigenwilliger die Deduktion und steht der
> induktiven Forschung fremder gegenüber. Kein Wunder es liegt ihm nicht, im
> Objekt den Richter über die Kinder seines Geistes zu sehen, sondern die
> Schönheit des Gedankens geht ihm über jene Wahrheit, die ihren Anspruch aus
> der Beobachtung des Objektes ableitet. Der Amor dei intellektualis ist sein
> einziger Gott, verkörpert in Mathematik und spekulativer Philosophie. Ich
> liebe seine Schriften unabhängig von ihrem Wahrheitsgehalt als die Früchte
> eines grossen originalen und freien Geistes.”333 Emanuel Lasker is one of
> the strongest minds I have encountered in the course of my life. A
> Renaissance man, gifted with a boundless desire for freedom, averse to any
> social obligation. Thus he became a chess master probably not so much
> because of any particular devoted love for the game. What he loves, rather,
> is philosophy, understanding in general. As a true maverick with a mind of
> his own, he loves deduction, and inductive research is foreign to him. That
> is not surprising he does not see the object as the judge of his mind’s
> offspring instead, for him the beauty of the idea is more important than the
> truth, which derives its claim from the observation of the object. The amor
> dei intellectualis is his sole god, embodied in mathematics and speculative
> philosophy. I love his writings independently of their truth content, as the
> product of a great original and free mind. ([Ein21])

Lasker’s paper [Las05a] is famous for the introduction of primary ideals, but
contains much more. It does not have an Introduction, but from the “final
remarks about some applications of the Theorems” we can conclude that the main
objective of the theory is the extension of Noether’s fundamental theorem to
the case of several variables. The paper contains a new approach to the
Hilbert polynomial, based on multiplication with non-zero divisors; the
Hilbert polynomial is used for a new proof of the Nullstellensatz. There is
also an extension of the theory to the ring of convergent power series in
several variables, which is used to prove Lasker’s generalisation of Noether’s
theorem. The last application sketched in the paper concerns Plücker formulas
for curves with arbitrary singularities.

###  2.5. Macaulay

Lasker proved that an ideal is the intersection of primary ideals, but gave no
methods to compute these. This was the goal of Macaulay’s paper [Mac13]. F. S.
Macaulay (1862–1937) was a school teacher until his retirement in 1911. For
his mathematical work see [EG23].

While in [Mac13] Macaulay uses the theories of Kronecker, Hilbert and Lasker,
the goal of first three chapters of his 1916 Tract [Mac16] is to present them.
In the preface Macaulay writes:

> The present state of our knowledge of the properties of Modular Systems is
> chiefly due to the fundamental theorems and processes of L. Kronecker, M.
> Noether, D. Hilbert, and E. Lasker, and above all to J. König’s profound
> exposition and numerous extensions of Kronecker’s theory. ([Mac16, Preface])

In this slim volume Macaulay only treats the case of polynomial ideals in
$\mathbb{C}[x_{1},\dots,x_{n}]$, what he calls the algebraic theory of modular
systems; the “absolute theory” concerns the case of integer coefficients. This
is the same distinction as Lasker makes between modules and ideals. The last
chapter of the Tract introduces the Inverse system, according to Paul Roberts
(in his Introduction to the 1994 reprint) one of the most original ideas in
the book. A simplified treatment is given in Macaulay’s last paper [Mac34]
(not mentioned by Roberts). In her Zentralblatt review [Zbl. 0008.29105] Emmy
Noether writes, among other things, that again the many examples and counter-
examples are important.

Macaulay’s Tract was one of the works Emmy Noether advised B. L. van der
Waerden to study when he came to Göttingen in 1924. It is the direct source
for several sections in Moderne Algebra, according to [vdW75]. Elsewhere van
der Waerden recollects:

> Most important work on the theory of Polynomial Ideals was done by Lasker,
> the famous chess champion, who had got his problem from Hilbert, and by
> Macaulay, a schoolmaster who lived near Cambridge, England, but who was
> nearly unknown to the Cambridge mathematicians when I visited Cambridge in
> 1933. I guess the importance of Macaulay’s work was known only in Göttingen.
> ([vdW71b])

###  2.6. Noether

Meanwhile a different treatment of explicit elimination theory was given by
Kurt Hentzelt in his 1914 Ph.D. thesis “Zur Theorie der Polynomideale und
Resultanten” under E. Fischer. Kurt Hentzelt (1889–1914, he went missing in
action near Diksmuide) studied first in Berlin and then 1919-1913 in Erlangen
[Kor15, p. 320]. Presumably he was also a student of Emmy Noether [Kor15, p.
15]. She acknowledges him in her paper on “fields and systems of rational
functions” (dated May 1914, before the start of the First World War) [Noe15].
In [Noe21] she gives in a footnote the simplest example of a non-unique
decomposition in primary ideals, adding that she got it from K. Hentzelt.
Noether published a conceptual version [HN22] of Hentzelt’s thesis, which she
characterises:

> Diese ganz auf Grund eigener Ideen verfaßte Dissertation ist lückenlos
> aufgebaut; aber Hilfssatz reiht sich an Hilfssatz, alle Begriffe sind durch
> Formeln mit vier und fünf Indizes umschrieben, der Text fehlt fast
> vollständig, so daß dem Verständnis die größten Schwierigkeiten bereitet
> werden.’444This dissertation, entirely based on own ideas, is structured
> without gaps; but Lemma follows lemma, all concepts are represented by
> formulas with four and five subscripts, the text is almost completely
> absent, so that the greatest difficulties are caused for understanding.
> ([Noe21, p. 53])

The part concerning computation in a finite number of steps was reserved for a
later publication. Noether gave this problem to her first PhD student Grete
Hermann.

In [vdW26] van der Waerden gives a new foundation for the theory of zeros of
polynomial ideals, independent of elimination theory. Even though he later
pleaded for the use of elimination theory in algebraic geometry [vdW54], van
der Waerden contributed to the elimination of elimination theory. In [vdW26]
he also gives a new proof of Hilbert’s Nullstellensatz. Van der
Waerden[vdW71b] recalls his use of generic points:

> I wrote a paper [vdW26] based upon this simple idea and showed it Emmy
> Noether. She at once accepted it for the Mathematische Annalen, without
> telling me that she had presented the same idea in a course of lectures just
> before I came to Göttingen. I heard it later from Grell, who had attended
> her course. ([vdW71b])

The above quote shows a greater participation of Emmy Noether in the
development of algebraic geometry than visible from her published papers. Emmy
Noether spent the winter 1928/29 in Moscow. She gave a course on abstract
algebra at Moscow University and led a seminar on algebraic geometry at the
Communist Academy (in 1936 merged with the Academy of Sciences of the Soviet
Union) [Ale36]. It is probable that J. L. Rabinowitsch was one of the
participants. In her report on the thesis of Hans Fitting [Kor15, p. 316]
Noether mentions unpublished work of “Rabinowitsch-Moskau” on the subject of
[Noe29], which probably her Moscow lectures took up. Not much is known about
him, but it can be Juli Lasarewitsch Rabinowitsch (in German transliteration),
who was born in 1904 and graduated from Moscow State University in 1924, where
he worked since 1932 and was awarded a Ph.D. and a title of Docent in 1935
[Tes22]. He later mainly worked on Differental Equations, but one can imagine
Noether attracting a rather wide audience. Curiously van der Waerden always
writes A. Rabinowitsch, in his Moderne Algebra but also in [vdW75]. Zariski
repeats this mistake in [Zar47], which shows that his source is van der
Waerden’s book.

It is not unlikely that Noether took Rabinowitsch’ paper back to Germany and
arranged for its publication in Mathematische Annalen, and that she provided
the references.

##  3\. Rabinowitsch’ paper

The text of [Rab29] consists of only 13 lines. Figure 1 shows it in
translation.

On Hilbert’s Nullstellensatz. By J. L. Rabinowitsch in Moscow.
Theorem: If the polynomial $f(x_{1},x_{2},\dots,x_{n})$ vanishes in all zeros
— in an algebraically closed field — of a polynomial ideal $\mathfrak{a}$,
then there is a power $f^{\rho}$ of $f$ belonging to $\mathfrak{a}$. Proof:
Let $\mathfrak{a}=(f_{1},\dots,f_{m})$, where $f_{i}$ contain the variables
$x_{1},\dots,x_{n}$. Let $x_{0}$ be an auxiliary variable. We form the ideal
$\bar{\mathfrak{a}}=(f_{1},\dots,f_{m},x_{0}f-1)$. As by assumption $f=0$
whenever all $f_{i}$ vanish, the ideal $\bar{\mathfrak{a}}$ has no zeros.
Therefore $\bar{\mathfrak{a}}$ has to coincide with the unit ideal. (Cf. for
example K. Hentzelt, “Eigentliche Eliminationstheorie”, § 6, Math. Annalen
88555Follows also already from Kronecker’s elimination theory..) If then
$1=\sum_{i=1}^{m}F_{i}(x_{0},x_{1},\dots,x_{n})f_{i}+F_{0}(x_{0}f-1)$ and we
put $x_{0}=\frac{1}{f}$ in this identity, so results:
$1=\sum_{i=1}^{m}F_{i}\left(\frac{1}{f},x_{1},\dots,x_{n}\right)f_{i}=\frac{\sum_{i=1}^{m}\bar{F}_{i}f_{i}}{f^{\rho}}\;.$
Therefore $f^{\rho}\equiv 0\pmod{\mathfrak{a}}$, q.e.d.

(Received on 8. 5. 1929)

Figure 1. Rabinowitsch’s paper [Rab29]

The statement that a nontrivial ideal has zeros is nowadays called weak
Nullstellensatz, but in the text and earlier it is not considered to be a
special case. The proof of the result can be found in many places (e.g.
[Mac16, p. 21]), but the connection with Hilbert’s theorem is not made. Only
in [Mac13] Macaulay gives a footnote to the statement that the only ideal
without zeros is the unit ideal, where he writes that this follows as a
particular case of the theorem in [Hil93, §3] but was known earlier from the
theory of the resultant. In the accounts of Kronecker’s elimination theory by
Kronecker himself [Kro82], Molk [Mol85], Netto [Net00] and König [Kön03b] the
conclusion is not drawn. In [HN22] we find in §6 on p. 76 the statement that
Theorem XII of that paper in particular shows that every ideal without zeros
becomes the unit ideal, again without mentioning the Nullstellensatz.

##  4\. Kronecker’s elimination theory

In this section we explain Kronecker’s theory, following the account given by
Macaulay [Mac16]. We show in particular how it implies the weak
Nullstellensatz.

Let $F_{1},\dots,F_{k}\in K[x_{1},\dots,x_{n}]$ with $K$ an algebraically
closed field and consider the equations $F_{1}=\dots=F_{k}=0$. The problem is
to find all solutions. Coincidences due to special values of the coefficients,
like equations not being regular (a polynomial of degree $l$ is regular in
$x_{1}$ if the monomial $x_{1}^{l}$ occurs with non-zero coefficient), can be
avoided by a (general) linear change of coordinates. As Macaulay remarks this
transformation is seldom needed in specific examples, but always assumed in
theoretical reasoning.

If the polynomials $F_{i}$ have a non-constant greatest common divisor $D$,
then the hypersurface $D=0$ gives solutions of the equations. This greatest
common divisor can be found with the Euclidean algorithm, by considering the
polynomials as elements of $K(x_{2},\dots,x_{n})[x_{1}]$. If $D$ is a constant
we take it equal to $1$. We divide the $F_{i}$ by $D$ and write
$F_{i}=D\phi_{i}$. We eliminate $x_{1}$ from the equations
$\phi_{1}=\dots=\phi_{k}=0$, using the following device. We form the
expressions

$\displaystyle\Phi_{1}$ $\displaystyle=u_{1}\phi_{1}+\dots+u_{m}\phi_{m}$
$\displaystyle\Phi_{2}$ $\displaystyle=v_{1}\phi_{1}+\dots+v_{m}\phi_{m}$

with the $u_{i}$ and $v_{i}$ indeterminates, and write the resultant
$R(\Phi_{1},\Phi_{2})$ (see Appendix A) as

$w_{1}F_{1}^{(1)}+w_{2}F_{2}^{(1)}+\dots+w_{k_{1}}F_{k_{1}}^{(1)}\;,$

where the $w_{i}$ are monomials in the $u_{j}$ and $v_{j}$ and $F_{i}^{(1)}\in
K[x_{2},\dots,x_{n}]$. Any solution of $F_{1}=\dots=F_{k}=0$ is a solution of
$D=0$ or of $\phi_{1}=\dots=\phi_{k}=0$, and any solution of
$\phi_{1}=\dots=\phi_{k}=0$ is a solution of
$F_{1}^{(1)}=\dots=F_{k_{1}}^{(1)}=0$, since by the property A.4 of the
resultant $\sum w_{i}F_{i}^{(1)}=A_{1}\Phi_{1}+A_{2}\Phi_{2}$; by equating
coefficients of the $w_{i}$ we conclude that
$F_{i}^{(1)}\in(\phi_{1},\dots,\phi_{k})$. Therefore
$D\,F_{i}^{(1)}\in(F_{1},\dots,F_{k})$. Conversely, if
$(\xi_{2},\dots,\xi_{n})$ is a solution of
$F_{1}^{(1)}=\dots=F_{k_{1}}^{(1)}=0$, then the resultant
$R(\Phi_{1},\Phi_{2})$ vanishes for
$(x_{2},\dots,x_{n})=(\xi_{2},\dots,\xi_{n})$ and the equations
$\Phi_{1}=\Phi_{2}=0$ have a solution $(\xi_{1},\xi_{2},\dots,\xi_{n})$ (and
we find all solutions). As $(x_{1}-\xi_{1})$ is a factor of
$\Phi_{1}(x_{1},\xi_{2},\dots,\xi_{n})$, it does not depend on the $v_{i}$;
nor does it depend on the $u_{i}$, being a factor of $\Phi_{2}$. Therefore
$(\xi_{1},\xi_{2},\dots,\xi_{n})$ is a solution of
$\phi_{1}=\dots=\phi_{k}=0$, so also of $F_{1}=\dots=F_{k}=0$.

We may assume that the $F_{i}^{(1)}$ are regular in $x_{2}$; the needed linear
transformation could have been performed at the start. We apply the same
procedure and find the greatest common divisor $D^{(1)}$ of the
$F_{i}^{(1)}=D^{(1)}\phi_{i}^{(1)}$ considered as polynomials in $x_{2}$, and
eliminate $x_{2}$ to get polynomials $F_{1}^{(2)},\dots,F_{k_{2}}^{(2)}$ in
$x_{3},\dots,x_{n}$. Any solution of $F_{1}=\dots=F_{k}=0$ is a solution of
$D\,D^{(1)}=0$ or of $F_{1}^{(2)}=\dots=F_{k_{2}}^{(2)}=0$ and
$D\,D^{(1)}F_{i}^{(2)}\in(F_{1},\dots,F_{k})$

We continue and successively find $D^{(j)}$ and eliminate $x_{j+1}$. After
eliminating $x_{n-1}$ we have polynomials $F_{i}^{(n-1)}$ in one variable
$x_{n}$, with greatest common divisor $D^{(n-1)}$ and after dividing with this
common factor the polynomials $\phi_{i}^{(n-1)}$ have no common root. We find
that any solution of $F_{1}=\dots=F_{k}=0$ is a solution of the single
equation $DD^{(1)}\dots D^{(n-1)}=0$. Conversely, from the solutions of
$DD^{(1)}\dots D^{(n-1)}=0$ we can find all solutions of
$F_{1}=\dots=F_{k}=0$. As the $DD^{(1)}\cdots
D^{(n-2)}F_{i}^{(n-1)}=DD^{(1)}\cdots D^{(n-1)}\phi_{i}^{(n-1)}$ lie in the
ideal $(F_{1},\dots,F_{k})$ and
$1\in(\phi_{1}^{(n-1)},\dots,\phi_{k}^{(n-1)})$, we conclude that

$DD^{(1)}\cdots D^{(n-1)}\in(F_{1},\dots,F_{k})\;.$

######  Definition 4.1.

$DD^{(1)}\cdots D^{(n-1)}$ is the complete (total) resolvent of the equations
$F_{1}=\dots=F_{k}=0$, and $D^{(i-1)}$ is the complete partial resolvent of
rank $i$. Any factor of $D^{(i-1)}$ is a partial resolvent of rank $i$.

The weak Nullstellensatz follows.

######  Proposition 4.2 ([Mac16, p. 21]).

If the equations $F_{1}=\dots=F_{k}=0$ have no solution then the complete
resolvent is equal to $1$ and consequently $1\in(F_{1},\dots,F_{k})$.

Macaulay continues to show by examples that the resolvent does not always
detect embedded components or may indicate such when they do not exist. This
problem does not occur with Hentzelt’s elimination theory. Noether [HN22]
describes how to form a resultant form with better properties, depending only
on the ideal. We refer for details to Krull’s report in the 1939 edition of
the Enzyklopädie [Kru39]. Whereas Kronecker’s method seeks to solve the
equations $F_{1}=\dots=F_{k}=0$, Hentzelt looks for zeros of the ideal
$\mathfrak{a}=(F_{1},\dots,F_{k})$. We may suppose that $\mathfrak{a}$
contains a polynomial $F$, which is regular in $x_{1}$ of order $r$. Let
$R_{1},\dots,R_{t}$ denote the remainders obtained by dividing the polynomials
$x_{1}^{j}F_{i}$, $j=0,\dots,r-1$, $i=1,\dots,k$, by $F$, as polynomials in
$x_{1}$. Let $M$ be the set of all polynomials in $\mathfrak{a}$ with degree
less than $r$ in $x_{i}$. It is a submodule of the free
$K[x_{2},\dots,x_{n}]$-module with basis $1,\dots,x_{1}^{r-1}$, with
$R_{1},\dots,R_{t}$ generators of $M$. The rank of $M$ is less than $r$ if and
only if the polynomials in $I$ have a common factor of positive degree in
$x_{1}$; this holds for undeterminate $x_{2},\dots,x_{n}$ but also for
specialised values. Let $\mathfrak{a}_{1}$ be the ideal of the minors of size
$r$ of the coefficient matrix of $M$. Then
$\mathfrak{a}_{1}\subset\mathfrak{a}$ and $\mathfrak{a}_{1}$ depends only on
$\mathfrak{a}$. If $\mathfrak{a}_{1}$ is not the zero ideal, then we can
proceed in the same way. This process stops if some ideal
$\mathfrak{a}_{r}=0$, or with $\mathfrak{a}_{n}$. If $\mathfrak{a}_{r}=0$,
then $x_{r+1},\dots,x_{n}$ can be chosen arbitrarily, and the value of the
other variables can be found by successively solving equations. As
$\mathfrak{a}_{n}$ does not depend on the variables, it can only be the zero
or unit ideal. In particular, if $\mathfrak{a}$ has no zeroes, then it is the
unit ideal [HN22, p. 76].

######  Example 4.3.

To illustrate the difference in the elimination procedures we consider
Macaulay’s example iii in section 17 [Mac16, p. 23]. He considers the ideal
$\mathfrak{a}=(x_{1}^{3},x_{2}^{3},x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3})$. A
less symmetric, but more convenient basis of $\mathfrak{a}$ is

$(x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3},x_{1}x_{2}^{2}(1-x_{3}^{2}),x_{2}^{3})\;.$

The ideal $\mathfrak{a}$ has one isolated component,
$\mathfrak{a}^{\prime}=(x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3},x_{1}x_{2}^{2},x_{2}^{3})$
and two embedded components
$\mathfrak{a}^{\prime\prime}=(x_{3}-1,x_{1}^{2}+x_{2}^{2}+x_{1}x_{2},x_{2}^{3})$
and
$\mathfrak{a}^{\prime\prime\prime}=(x_{3}+1,x_{1}^{2}+x_{2}^{2}-x_{1}x_{2},x_{2}^{3})$.
As the polynomial $f_{1}=x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3}$ is regular in
$x_{1}$, we may in Kronecker’s method take the resultant of $f_{1}$ and
$v_{2}f_{2}+v_{3}f_{3}$, where $f_{2}$ and $f_{3}$ are the other two
generators of $\mathfrak{a}$ [vdW31b, §73, Remark 1]. We get the determinant

$\begin{vmatrix}1&x_{2}x_{3}&x_{2}^{2}\\\
v_{2}x_{2}^{2}(1-x_{3}^{2})&v_{3}x_{2}^{3}&0\\\
0&v_{2}x_{2}^{2}(1-x_{3}^{2})&v_{3}x_{2}^{3}\end{vmatrix}$

which equals

$x_{2}^{6}\left(v_{2}^{2}(1-x_{3}^{2})^{2}-v_{2}v_{3}x_{3}(1-x_{3}^{2})+v_{3}^{2}\right)\;.$

It follows that the complete resolvent is $x_{2}^{6}$. For
$\mathfrak{a}^{\prime}$ the computation is almost the same, except that the
factors $(1-x_{3}^{2})$ are to be removed from the determinant. Although
$\mathfrak{a}\varsubsetneq\mathfrak{a}^{\prime}$ both ideals have the same
complete resolvent.

To eliminate according to Hentzelt-Noether we divide
$f_{2},x_{1}f_{2},f_{3},xf_{3}$ by $f_{1}=x_{1}^{2}+x_{2}^{2}+x_{1}x_{2}x_{3}$
and find that $f_{2},x_{1}f_{2},f_{3}$ form a basis of the module of
polynomials in $\mathfrak{a}$ of degree at most 1 in $x_{1}$. The coefficient
matrix is

$\begin{bmatrix}x_{2}^{3}&0&0\\\
0&x_{2}^{3}&x_{2}^{2}(1-x_{3}^{2})\end{bmatrix}$

with minors $x_{2}^{6}$ and $x_{2}^{5}(1-x_{3}^{2})$. For
$\mathfrak{a}^{\prime}$ we find the ideal generated by $x_{2}^{5}$.

Successive elimination requires that the variables are general. This is
achieved by Noether [Noe23] by adjoining the coefficients $u_{ij}$ of an
indeterminate $n\times n$ matrix $U$ to the field $K$ and the change of
variables $y_{i}=\sum u_{ij}x_{j}$. The same device is used by Hermann
[Her26].

######  Remark 4.4.

Proposition 4.2 is here a consequence of a more general theory. A direct proof
of the weak Nullstellensatz is shorter [vdW31b, §74]. We use induction on the
number of variables. For one variable the result is true by the extended
Euclidean algorithm. Suppose the ideal $\mathfrak{a}=(F_{1},\dots,F_{k})$ has
no common zeros, and that $\mathfrak{a}$ contains a polynomial $F$ regular in
$x_{1}$. We eliminate $x_{1}$ as above (with Kronecker’s or Hentzelt’s method)
and find polynomials
$F_{1}^{(1)},\dots,F_{k_{1}}^{(1)}\in(F_{1},\dots,F_{k})$, again without
common zeros. By the induction hypothesis
$1\in(F_{1}^{(1)},\dots,F_{k_{1}}^{(1)})\subset(F_{1},\dots,F_{k})$.

##  5\. Hilbert’s proof

Hilbert’s original proof is also based on elimination. The theorem is
formulated with the application to invariants in mind. It looks different from
the theorem stated by Rabinowitsch.

######  Theorem 5.1.

Given $m$ homogeneous polynomials $f_{1},\dots,f_{m}$ in $n$ variables
$x_{1},\dots,x_{n}$, let $F,F^{\prime},F^{\prime\prime},\dots$ be homogeneous
polynomials in the same variables with the property that they vanish for all
those values of these variables for which the given $m$ polynomials
$f_{1},\dots,f_{m}$ all are equal to zero: then it is always possible to
determine an integer $r$ such that any product $\Pi^{(r)}$ of $r$ arbitrary
polynomials of the sequence $F,F^{\prime},F^{\prime\prime},\dots$ can be
expressed in the form

$\textstyle{\Pi^{(r)}}=a_{1}f_{1}+a_{2}f_{2}+\dots+a_{m}f_{m}\;,$

where $a_{1},a_{2},\dots,a_{m}$ are suitably chosen polynomials in the
variables $x_{1},\dots,x_{n}$.

######  Remark 5.2.

  1. 1.

Hilbert formulates the projective Nullstellensatz. As the polynomials are
homogeneous, their zeros are taken in projective space. The inhomogeneous
version follows by making all polynomials in the sequence homogeneous with a
new variable $x_{0}$, and applying the homogeneous theorem to a set of
generators of the homogenisation of the ideal $(f_{1},\dots,f_{m})$. Putting
then $x_{0}=1$ leads to the sought relation.

  2. 2.

The Theorem in the above form implies in particular that for any polynomial
among the $F,F^{\prime},F^{\prime\prime},\dots$ the $r$-th power lies in the
ideal $(f_{1},\dots,f_{m})$. Hilbert remarks that this fact was stated and
proved for inhomogeneous polynomials of two variables by Netto [Net85]. The
special case that the $r$-th power lies in the ideal implies the general case.
Firstly, by Hilbert’s basis theorem (Theorem I of [Hil90]), the polynomials in
the sequence are expressable in finitely many of them, say
$F^{(1)},\dots,F^{(k)}$. Any product of $r$ polynomials
$F,F^{\prime},F^{\prime\prime},\dots$ becomes a sum of products of $r$
polynomials $F^{(1)},\dots,F^{(k)}$ with polynomials coefficients. If
$(F^{(i)})^{r_{i}}\in(f_{1},\dots,f_{m})$, then put
$r=(r_{1}-1)+(r_{2}-1)+\dots+(r_{k}-1)+1$. Every product
$(F^{(1)})^{l_{1}}\cdots(F^{(k)})^{l_{k}}$ with $\sum l_{i}=r$ contains at
least one factor $(F^{(i)})^{r_{i}}$; otherwise $\sum
l_{i}\leq\sum(r_{i}-1)=r-1$. [vdW31b, §75], [vdW67, §130].

  3. 3.

The statement that $r$ is independent of $f$, but only depends on the $f_{i}$,
is nowadays normally not included. It follows from the fact that the ideal of
polynomials vanishing at the zero set of the $f_{i}$ is finitely generated.

For the proof of the theorem we first reduce to the case that there are only
finitely many $F^{(i)}$. By the basis theorem every polynomial in the sequence
is a linear combination of say $F^{(1)},\dots,F^{(k)}$, and a product of $r$
polynomials is a sum of products of $r$ polynomials $F^{(1)},\dots,F^{(k)}$.
Hilbert writes this reduction at the end of his proof, and starts by assuming
that there are only finitely many polynomials in the sequence.

The proof then splits in two parts. In the first it is assumed that the
polynomials $f_{1},\dots,f_{m}$ have only finitely many common zeros. The main
tool is elimination using the resultant of two binary forms (see Appendix A).
Substituting in the $f_{i}$ the expressions
$x_{1}\xi_{1},\dots,x_{n-1}\xi_{1},\xi_{2}$ for the variables
$x_{1},\dots,x_{n}$ makes them binary forms in the variables $\xi_{1}$,
$\xi_{2}$, of degrees $\nu_{1},\dots,\nu_{m}$. Let
$\nu=\max_{j}\\{\nu_{j}\\}$.

To eliminate $\xi_{1},\xi_{2}$ Hilbert uses Kronecker’s device [Kro82] and
forms the expressions

$\displaystyle F_{1}$ $\displaystyle=u_{1}f_{1}+\dots+u_{m}f_{m}$
$\displaystyle F_{2}$ $\displaystyle=v_{1}f_{1}+\dots+v_{m}f_{m}$

where the $u_{i}$ and $v_{i}$ are binary forms in $\xi_{1}$, $\xi_{2}$ of
degree $\nu-\nu_{i}$ with indetermined coefficients, making $F_{1}$ and
$F_{2}$ homogeneous of degree $\nu$. The resultant $R(F_{1},F_{2})$ is a
polynomial in the indeterminates occurring in the $u_{i}$ and $v_{i}$, whose
coefficients are polynomials $f_{1}^{\prime},\dots,f_{m^{\prime}}^{\prime}$
depending only on the variables $x_{1},\dots,x_{n-1}$, and by putting
$\xi_{1}=1$, $\xi_{2}=x_{n}$ one sees (from Proposition A.3) that the
$f_{i}^{\prime}$ lie in the ideal $(f_{1},\dots,f_{m})$.

Hilbert does not consider the possibility that all $f_{i}^{\prime}$ are
identically zero. This happens if one of the common zeros is the point
$(0:\dots:0:1)$, for then $(\xi_{1}:\xi_{2})=(0:1)$ is a common zero of
$F_{1}$ and $F_{2}$. The standard trick is to apply a general linear
transformation. We may therefore assume that no common zero of the transformed
system lies in a coordinate hyperplane, simplifying somewhat Hilbert’s
argument.

If the polynomials $f_{i}$ have the common zeros
$(\alpha_{1}:\ldots:\alpha_{n-1}:\alpha_{n})$,
$(\beta_{1},:\ldots:\beta_{n-1}:\beta_{n})$, …,
$(\kappa_{1}:\ldots:\kappa_{n-1}:\kappa_{n})$, then the $f_{i}^{\prime}$ have
only the common zeros $(\alpha_{1}:\ldots:\alpha_{n-1})$, …,
$(\kappa_{1}:\ldots:\kappa_{n-1})$.

In the same way the variable $x_{n-1}$ can be eliminated from the
$f_{i}^{\prime}$, leading to polynomials
$f_{1}^{\prime\prime},\dots,f_{m^{\prime\prime}}^{\prime\prime}$, and so on
until a system of binary forms $f_{1}^{(n-2)},\dots,f_{m^{(n-2)}}^{(n-2)}$ in
the variables $x_{1}$, $x_{2}$ is reached.

Hilbert uses this procedure to prove the result by induction on the number of
common zeros. The base of the induction is the case that the $f_{i}$ have no
common zeros at all. Then the forms
$f_{1}^{(n-2)},\dots,f_{m^{(n-2)}}^{(n-2)}$ have no common zeros. This implies
that every binary form of sufficiently high degree lies in the ideal
$(f_{1}^{(n-2)},\dots,f_{m^{(n-2)}}^{(n-2)})$ and therefore in the ideal
$(f_{1},\dots,f_{m})$; in particular $x_{1}^{r_{1}}$ and $x_{2}^{r_{2}}$ lie
in the ideal for some $r_{1}$ and $r_{2}$. In the same way it follows that
$x_{3}^{r_{3}}$, …, $x_{n}^{r_{n}}$ lie in the ideal for sufficiently large
$r_{3}$, …, $r_{n}$. Therefore every homogeneous polynomial in
$x_{1},\dots,x_{n}$ of degree at least $\sum(r_{i}-1)+1$ lies in the ideal,
proving the base case.

This result can be called the weak projective Nullstellensatz, and we
formulate it separately.

######  Proposition 5.3.

A homogeneous ideal $\mathfrak{a}$ has no zeros if and only if there exists an
integer $r$ such that every form of degree at least $r$ lies in
$\mathfrak{a}$.

We continue with the proof of the theorem. The induction step is that the
statement holds if the polynomials have a given number of common zeros, say
$(\beta_{1}:\ldots:\beta_{n-1}:\beta_{n})$, …,
$(\kappa_{1}:\ldots:\kappa_{n-1}:\kappa_{n})$. Suppose that there is an
additional common zero $(\alpha_{1}:\ldots:\alpha_{n-1}:\alpha_{n})$. Then
every $F^{(i)}$ can be written in the form

$(\alpha_{2}x_{1}-\alpha_{1}x_{2})F^{(i)}_{12}+(\alpha_{3}x_{1}-\alpha_{1}x_{3})F^{(i)}_{13}+\dots+(\alpha_{n}x_{n-1}-\alpha_{n-1}x_{n})F^{(i)}_{n-1,n}.$

By our assumption $\alpha_{1}$ and $\alpha_{2}$ are both non-zero, so
elimination as above leads to forms
$f_{1}^{(n-2)},\dots,f_{m^{(n-2)}}^{(n-2)}$ in the variables $x_{1}$, $x_{2}$
lying in the ideal $(f_{1},\dots,f_{m})$, which only have the common zeros
$(\alpha_{1}:\alpha_{2})$, …, $(\kappa_{1}:\kappa_{2})$. Choose one of these
forms and write it in the form
$(\alpha_{2}x_{1}-\alpha_{1}x_{2})^{r_{12}}\varphi_{12}$ with $\varphi_{12}$ a
binary form not vanishing for $x_{1}=\alpha_{1}$, $x_{2}=\alpha_{2}$. In the
same way one finds $r_{ij}$ and $\varphi_{ij}$ for the other $1\leq i<j\leq
n$.

Put $r^{\prime}=r_{12}+r_{13}+\dots+r_{n-1,n}$ and
$\Phi=\varphi_{12}\,\varphi_{13}\cdots\varphi_{n-1,n}$. Then

$\Phi\,\textstyle{\Pi^{(r^{\prime})}}\in(f_{1},\dots,f_{m})\;,$

where $\Phi$ is a polynomial that does not vanish in the point
$(\alpha_{1}:\ldots:\alpha_{n})$ and $\Pi^{(r^{\prime})}$ is an
$r^{\prime}$-fold product of $F^{(i)}$. The polynomials
$\Phi,f_{1},\dots,f_{m}$ have only the common zeros
$(\beta_{1}:\ldots:\beta_{n})$, …, $(\kappa_{1}:\ldots:\kappa_{n})$. Therefore
there exists a number $r^{\prime\prime}$ such that
$\Pi^{(r^{\prime\prime})}\in(\Phi,f_{1},\dots,f_{m})$. Then
$\Pi^{(r)}\in(f_{1},\dots,f_{m})$ for $r=r^{\prime}+r^{\prime\prime}$, which
proves the induction step.

In the second step the theorem is proved in general by induction on the number
of variables. The induction hypothesis is that the result holds for $n-1$
variables and that the number $r$ can be chosen below a bound only depending
on the degrees and the (finite) number of the forms
$f_{1},\dots,f_{m},F,F^{\prime},\dots$, but not on their coefficients. The
base of the induction is $n=2$, where the result is true by the first part of
the proof, as binary forms can only have a finite number of zeros.

For the induction step put $x_{1}=tx_{2}$. The polynomials
$f_{1},\dots,f_{m},$ $F,F^{\prime},\dots$ become polynomials
$g_{1},\dots,g_{m},G,G^{\prime},\dots$ in the $n-1$ variables
$x_{2},\dots,x_{n}$ with coefficients polynomials in the parameter $t$. If $t$
takes a specific value then every $G^{(i)}$ vanishes whenever all $g_{i}$
vanish (as polynomials in $x_{2},\dots,x_{n}$). By the induction hypothesis
there is a number $r_{12}$ such that every product $\Pi^{(r_{12})}$ of
polynomials $G^{(i)}$ for every special value of $t$ has a representation

$\Pi^{(r_{12})}=b_{1}g_{1}+\dots+b_{m}g_{m}$

with the $b_{i}$ polynomials in $x_{2},\dots,x_{n}$. Considering the
coefficients of the $b_{i}$ as indeterminates $u_{j}$ and taking in this
equation coefficients of the monomials in $x_{2},\dots,x_{n}$ yields a system
of linear inhomogeneous equations for the $u_{j}$. The coefficients of these
linear equations are polynomials in $t$, and for every value of $t$ solutions
exist.

At this point Hilbert uses an “easily proved” lemma: If a given a system of
linear equations

$\displaystyle c_{11}u_{1}+\dots+c_{1p}u_{p}$ $\displaystyle=c_{1},$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$ $\displaystyle
c_{q1}u_{1}+\dots+c_{qp}u_{p}$ $\displaystyle=c_{q}$

with $c_{ij}$, $c_{k}\in K[t]$, has solutions for every value of $t$ ($K$
being an infinite field) then there exists a solution with $u_{i}\in K(t)$.

Indeed, as Popov remarks in his comments [Pop98], a solution exists if and
only if the rank of the coefficient matrix is equal to the rank of the
augmented matrix. As $k$ is infinite, one can find a $t_{0}\in k$ such the
rank of these matrices over $k(t)$ is the same as the rank over $k$ with
$t_{0}$ substituted for $t$. Applying this lemma to the equations for the
coefficients of the $b_{i}$ and substituting $t=\frac{x_{1}}{x_{2}}$ gives
after clearing denominators that

$\psi_{12}\Pi^{(r_{12})}\in(f_{1},\dots,f_{m})$

with $\psi_{12}$ a binary form in $(x_{1},x_{2})$, and $\Pi^{(r_{12})}$ the
product of $r_{12}$ polynomials $F^{(i)}$ corresponding to the chosen
$G^{(i)}$. In the same way one finds $r_{ij}$ and a binary form $\psi_{ij}$ in
$(x_{i},x_{j})$ with $\psi_{ij}\Pi^{(r_{ij})}\in(f_{1},\dots,f_{m})$. Now put
$r^{\prime}=\max\\{r_{ij}\\}$ and choose $r^{\prime}$ polynomials $F^{(i)}$.
The corresponding binary forms $\psi_{12},\dots,\psi_{n-1,n}$ have only
finitely many common zeros, so also the polynomials
$\psi_{12}\dots\psi_{n-1,n},f_{1},\dots,f_{m}$. By the first part of the proof
there exists a number $r^{\prime\prime}$ such that

$\Pi^{(r^{\prime\prime})}\in(\psi_{12}\dots\psi_{n-1,n},f_{1},\dots,f_{m})\;.$

As $\psi_{ij}\Pi^{(r_{ij})}\in(f_{1},\dots,f_{m})$ one has that for
$r=r^{\prime}+r^{\prime\prime}$ that

$\Pi^{(r)}\in(f_{1},\dots,f_{m})\;.$

This concludes the induction step, and with that the proof of the
Nullstellensatz.

##  6\. Proofs using primary decomposition

Primary ideals were introduced by Lasker [Las05a] in the setting of polynomial
rings. He used primary decomposition to give a new proof of the
Nullstellensatz. Macaulay [Mac16] follows this strategy, but with different
proofs.

The modern definition of primary ideals is due to Emmy Noether [Noe21], and
applies to all Noetherian rings.

######  Definition 6.1.

An ideal $\mathfrak{q}$ in a Noetherian ring $R$ is primary if whenever
$ab\in\mathfrak{q}$ but $a\notin\mathfrak{q}$ it follows that
$b^{k}\in\mathfrak{q}$ for some $k>0$.

The radical $\sqrt{\mathfrak{q}}$ of $\mathfrak{q}$, that is $\\{a\in R\mid
a^{k}\in\mathfrak{q}\text{ for some }k>0\\}$, is a prime ideal $\mathfrak{p}$
and $\mathfrak{q}$ is said to be $\mathfrak{p}$-primary.

By the Lasker–Noether Theorem (which Lasker proposed to call the
Noether–Dedekind Theorem) every ideal $\mathfrak{a}$ has an irredundant
primary decomposition into primary ideals
$\mathfrak{a}=\mathfrak{q}_{1}\cap\cdots\cap\mathfrak{q}_{n}$ (for a proof see
van der Waerden’s Algebra II [vdW67, Ch. 15]). The ideals
$\mathfrak{p}_{i}=\sqrt{\mathfrak{q}_{i}}$ are the associated primes of
$\mathfrak{a}$.

The decomposition is not unique. The simplest example is the ideal
$(x^{2},xy)\subset K[x,y]$, which can be written as $(x)\cap(x^{2},xy,y^{2})$
but also as $(x)\cap(x^{2},y)$, and even $(x)\cap(x^{2},y+\lambda x)$. The
number of components is always the same, and also the associated prime ideals;
in the example the ideals $(x)$ and $(x,y)$. According to [Noe21, footnote 10]
Noether learned this example from K. Hentzelt. This means that Emmy Noether
occupied herself with primary ideals already in 1913/14. Macaulay [Mac13,
Mac16] has more complicated examples. Maybe Noether knew about the manuscript
for [Mac13] through her father; the paper has footnotes on p. 71 and p. 86,
mentioning Max Noether, saying among others: “I am also indebted to Professor
Noether for kindly suggesting other alterations which I have carried out”.
This sounds as a reaction to a (non-anonymous) referee report.

By the definition of a primary ideal and the fact that the associated prime
ideal is finitely generated one immediately obtains

######  Proposition 6.2.

If $\mathfrak{q}$ is a primary ideal and $\mathfrak{p}$ the associated prime
ideal, then some finite power of $\mathfrak{p}$ is contained in
$\mathfrak{q}$.

Noether [Noe21] applies her results to polynomial ideals; in this paper she
still only considers complex coefficients. The connection with elimination
theory and ideal theory, say as described by Macaulay in his Tract [Mac16], is
given by the following special case of the Nullstellensatz.

######  Proposition 6.3.

A prime ideal $\mathfrak{p}$ consists of all polynomials vanishing on its zero
set.

Conversely, the Nullstellensatz follows from this Proposition. Let
$\mathfrak{q}_{i}$ be a primary ideal in the decomposition of $\mathfrak{a}$
with associated prime ideal $\mathfrak{p}_{i}$. If $\mathfrak{b}$ is an ideal
vanishing on the zero set of $\mathfrak{a}$, then it vanishes on the zero set
of $\mathfrak{p}_{i}$ and therefore $\mathfrak{b}\subset\mathfrak{p}_{i}$ and
$\mathfrak{b}^{k_{i}}\subset\mathfrak{p}_{i}^{k_{i}}\subset\mathfrak{q}_{i}$.
Let $k$ be the maximum of the $k_{i}$, then
$\mathfrak{b}^{k}\subset\bigcap\mathfrak{q}_{i}=\mathfrak{a}$.

Lasker stated Proposition 6.3 in [Las05a], adding that it follows, say, from
the Nullstellensatz, but in the addendum [Las05b] he explained how to prove it
directly. It seems that Macaulay [Mac16] did not notice this, as he criticises
Lasker’s proof of Proposition 6.2, saying that Lasker first assumes the result
and then proves it.

Macaulay and Lasker have a different definition of primary ideals, which makes
the proof of Proposition 6.2 non-trivial. Macaulay [Mac16] defines a primary
ideal by the property that no product of two ideals is contained in it without
one of them contained in it or both containing its zero set. Hence if one does
not contain the zero set the other is contained in the ideal. By the
Nullstellensatz Macaulay’s definition is equivalent to Noether’s.

Lasker’s original definition was stated for homogeneous ideals in
$S=K[x_{1},\dots,x_{n}]$, making the statements about primary decomposition
more complicated. A primary ideal $\mathfrak{q}$ and the associated prime
ideal $\mathfrak{p}$ occur both in his definition: whenever
$ab\in\mathfrak{q}$ and $a\notin\mathfrak{p}$ it follows that
$b\in\mathfrak{q}$. To make the zero set $C$ of $\mathfrak{p}$ an irreducible
component of the zero set of $\mathfrak{q}$ it is required that the dimension
of $\mathfrak{q}$ is at most that of $\mathfrak{p}$. According to Lasker an
algebraic set $C$ has dimension $m$ if it has a finite number of points in
common with $m$ general linear forms, and if the forms in an ideal
$\mathfrak{a}$ vanish on sets of dimension $m$, but not on sets of higher
dimension, then $\mathfrak{a}$ has dimension $m$. Actually Lasker uses the
quantity $m+1$, which he calls “Mannigfaltigkeit” and Macaulay [Mac13]
translates with manifoldness or dimensionality. The value 0 is allowed for
dimensionality, and it occurs in [Las05a] in an essential way, although it is
not defined what it means.

Lasker’s approach to primary decomposition is as follows. Let $C_{1}$, …,
$C_{j}$ be the irreducible components of the zero set of $\mathfrak{a}$ of
highest dimension and let $\mathfrak{p}_{i}$ be the prime ideal corresponding
to $C_{i}$. Define $\mathfrak{a}_{i}$ as the set of all $f\in S$ such that
$C_{i}$ is not a component of the zero set of the ideal quotient
$\mathfrak{a}:(f)=\\{g\in S\mid gf\in\mathfrak{a}\\}$, so there exists a
$\phi$ not vanishing on $C_{i}$ with $f\phi\in\mathfrak{a}$. Then
$\mathfrak{a}_{i}$ is an ideal, whose zero set only consists of $C_{i}$.
Furthermore $\mathfrak{a}_{i}$ is primary, for if $ab\in\mathfrak{a}_{i}$ with
$a\notin\mathfrak{p}_{i}$, then $ab\phi\in\mathfrak{a}$ for a $\phi$ not
vanishing on $C_{i}$, and as $a\phi\notin\mathfrak{p}_{i}$, we have
$b\in\mathfrak{a}_{i}$ by the definition of $\mathfrak{a}_{i}$.

The set $C_{i}$ is not a component of the zero set of the ideal quotient
$\mathfrak{a}_{j}^{\prime}=\mathfrak{a}:\mathfrak{a}_{j}$. Let
$\psi\in\mathfrak{a}_{1}^{\prime}+\dots+\mathfrak{a}_{j}^{\prime}$ be a form
which does not vanish on any of the $C_{i}$. Then we claim that
$\mathfrak{a}=\mathfrak{a}_{1}\cap\dots\cap\mathfrak{a}_{j}\cap(\mathfrak{a},\psi)$.
If $f$ is an element of the right hand side, then $f\in(\mathfrak{a},\psi)$ so
$f-g\psi\in\mathfrak{a}\subset\mathfrak{a}_{i}$, and as
$f\in\mathfrak{a}_{i}$, we get $g\psi\in\mathfrak{a}_{i}$; because
$\mathfrak{a}_{i}$ is primary, and $\psi$ does not vanish on $C_{i}$, we get
$g\in\mathfrak{a}_{i}$ for all $i$. As
$\psi\in\mathfrak{a}_{1}^{\prime}+\dots+\mathfrak{a}_{j}^{\prime}$ we find
that $g\psi\in\mathfrak{a}$ and therefore $f\in\mathfrak{a}$. The dimension of
$(\mathfrak{a},\psi)$ is lower, and we can repeat the process.

In this proof essential use is made of Proposition 6.3. Lasker proves it in
[Las05b], Macaulay in [Mac16, Section 31] and formulates it as follows: there
is only one prime ideal with a given (irreducible) zero set, viz. the ideal
consisting of all polynomials vanishing on the zero set. In [Mac16, Section
32] he shows Proposition 6.2. With primary decomposition the Hilbert-Netto
Theorem (Macaulay’s name for the Nullstellensatz) then follows.

Macaulay proves Propositions 6.3 and 6.2 with classical methods of elimination
theory, which are unfamiliar to the modern reader. The main ingredient is the
so-called $u$-resolvent. The goal is to describe the irreducible components of
the zero set of an ideal. Consider $\mathfrak{a}=(F_{1},\dots,F_{k})$, which
is as always supposed to be prepared by a general linear coordinate change.
Macaulay shortly writes “The solutions of $F_{1}=F_{2}=\dots=F_{k}=0$ are
obtained in the most useful way by introducing a general unknown $x$ standing
for $u_{1}x_{1}+u_{2}x_{2}+\dots+u_{n}x_{n}$, where $u_{1},u_{2},\dots,u_{n}$
are undetermined coefficients”. This is known as the Liouville substitution
[Lio47], and its use is explained in detail in Netto’s book [Net00]. One
substitutes

$x_{1}=\frac{x-u_{2}x_{2}-\dots-u_{n}x_{n}}{u_{1}}$

in the equations and multiplies with suitable powers of $u_{1}$ to make the
new equations $f_{1}=f_{2}=\dots=f_{k}=0$ in $x,x_{2},\dots,x_{n}$ polynomial.
The solutions of the first system determine those of the second and vice
versa. The complete resolvent $D_{u}D_{u}^{(1)}\cdots D_{u}^{(n-1)}(=F_{u})$
of $(f_{1},\dots,f_{k})$ obtained by eliminating $x_{2},x_{3},\dots,x_{n}$ in
this order is called the complete $u$-resolvent of $(F_{1},\dots,F_{k})$.

The $u$-resolvent $F_{u}$ is a polynomial in
$x,x_{2},\dots,x_{n},u_{1},\dots,u_{n}$. As a polynomial in $x$, when the
$x_{i}$ and $u_{i}$ have specific values, it splits in linear factors. Such a
linear factor of $D_{u}^{(r-1)}$ has the form

$x-u_{1}\xi_{1}-u_{r}\xi_{r}-u_{r+1}x_{r+1}-\dots-u_{n}x_{n}$

where $\xi_{1},\dots,\xi_{r},x_{r+1},\dots,x_{n}$ is a solution of
$F_{1}=F_{2}=\dots=F_{k}=0$. Macaulay calls a linear factor a true linear
factor if $\xi_{1},\dots,\xi_{r}$ are independent of $u_{1},\dots,u_{n}$, that
is, if it is linear in $x,u_{1},\dots,u_{n}$.

######  Example 6.4.

Consider the ideal $(x_{1}^{2}+x_{2}^{2}-2,x_{1}^{2}-x_{2}^{2})$. Substituting
$x=u_{1}x_{1}+u_{2}x_{2}$ and eliminating $x_{2}$ using the Sylvester
determinant gives
$F_{u}=4u_{1}^{2}(x-u_{2}-u_{1})(x-u_{2}+u_{1})(x+u_{2}-u_{1})(x+u_{2}+u_{1})$.
From the factor $x-u_{2}-u_{1}$ one finds the solution $(x_{1},x_{2})=(1,1)$
by substituting the values $(1,0)$ and $(0,1)$ for $(u_{1},u_{2})$. Using the
same substitution for the principal ideal $(x_{1}^{2}+x_{2}^{2}-2)$ gives
$F_{u}=(x-u_{1}\sqrt{2-x_{2}^{2}}-u_{2}x_{2})(x+u_{1}\sqrt{2-x_{2}^{2}}-u_{2}x_{2})$,
which is indeed of the form
$(x-u_{1}\xi_{1}-u_{2}x_{2})(x-u_{1}\xi_{2}-u_{2}x_{2})$.

Macaulay goes on to prove that the solution supplied by a factor which is not
a true linear one is an embedded solution. In particular all the linear
factors of the first complete partial $u$-resolvent are true linear factors.
According to Macaulay Kronecker states without proof that all linear factors
of $F_{u}$ are true linear factors, while König’s proof contains an error, and
Macaulay doubts whether the statement is true. In fact, Kronecker’s claim is
true for the resultant form in the elimination theory of Hentzelt-Noether
[HN22, Satz XIII].

An irreducible factor $R_{u}$ of rank $r$ of $F_{u}$ having a true linear
factor leads to a parametrisation of the corresponding irreducible component
of the solution set. One can take $x_{r+1},\dots,x_{n}$ arbitrary and for each
set of values there are solutions $x_{1i},\dots,x_{ri}$, $i=1,\dots,d$, with
$d$ the degree of the component. Therefore we can (formally) write

$R_{u}=A\,\Pi_{i=1}^{d}(x-u_{1}x_{1i}-\dots-u_{r}x_{ri}-u_{r+1}x_{r+1}-\dots-
u_{n}x_{n})$

so

$(R_{u})_{x=u_{1}x_{1}+\dots+u_{n}x_{n}}=A\,\Pi_{i=1}^{d}(u_{1}(x_{1}-x_{1i})+\dots+u_{r}(x_{r}-x_{ri}))$

The last expression is independent of $u_{r+1},\dots,u_{n}$ and vanishes
identically at all points of the solutions set and at no other points, that is
irrespective of $u_{1},\dots,u_{r}$. The coefficients of the monomials in the
$u_{i}$ in $(R_{u})_{x=u_{1}x_{1}+\dots+u_{n}x_{n}}$ are polynomials in the
$x_{i}$ which all vanish at all points of the solution set and do not all
vanish at other points. This gives equations for the solution set. We single
out some of them. The coefficient of $u_{r}^{d}$ is
$\phi(x_{r},x_{r+1},\dots,x_{n})=A\,\Pi(x_{r}-x_{ri})$. The coefficient of
$u_{1}u_{r}^{d-1}$ is $\phi\sum\frac{x_{1}-x_{1i}}{x_{r}-x_{ri}}$, which we
write as $x_{1}\phi^{\prime}-\phi_{1}$, where $\phi^{\prime}$ is the
derivative of $\phi$ w.r.t. $x_{r}$ and
$\phi_{1}=\phi\sum\frac{x_{1i}}{x_{r}-x_{ri}}$. Similarly we have
$x_{2}\phi^{\prime}-\phi_{2}$, …, $x_{r-1}\phi^{\prime}-\phi x_{r-1}$. Outside
$\phi^{\prime}=0$ we have therefore the equations

$\phi=0,\quad x_{i}=\frac{\phi_{i}}{\phi^{\prime}},\;i=1,\dots,r-1\;.$

With these preparations we can give the proof of Proposition 6.3.

###### Macaulay’s proof of Proposition 6.3.

The zero set is irreducible, otherwise the complete $u$-resolvent would
contain at least two factors corresponding to different irreducible
components, contradicting that the ideal is prime.

Let $\mathfrak{p}=(F_{1},\dots,F_{k})$ be a prime ideal. It will be sufficient
to prove that $F\in\mathfrak{p}$ for every polynomial $F$ that vanishes on the
zero set of $\mathfrak{p}$. The first complete partial $u$-resolvent of
$\mathfrak{p}$ will be a power $R_{u}^{m}$ of an irreducible polynomial
$R_{u}$ in $x,x_{r+1},\dots,x_{n}$. The complete $u$-resolvent lies in the
prime ideal $(f_{1},\dots,f_{k})$, and for dimension reasons the other factors
do not vanish on the zero set of $(f_{1},\dots,f_{k})$. Hence $R_{u}^{m}$ and
therefore $R_{u}$ itself belongs to $(f_{1},\dots,f_{k})$. This gives that
$(R_{u})_{x=u_{1}x_{1}+\dots+u_{n}x_{n}}\in(F_{1},\dots,F_{k})=P$, and the
same holds for the polynomial coefficients of the monomials in the $u_{i}$. In
particular $\phi\in\mathfrak{p}$ and
$\psi_{i}:=x_{i}\phi^{\prime}-\phi_{i}\in\mathfrak{p}$, $i_{1},\dots,r-1$.

Let now $F$ vanish on the zero set of $\mathfrak{p}$ and substitute
$x_{i}=\phi_{i}/\phi^{\prime}$, $i=1,\dots,r-1$; then $F$ becomes a rational
function of $x_{r},x_{r+1},\dots,x_{n}$ with denominator
${\phi^{\prime}}^{l}$, where $l$ is the degree of $F$. This rational function
vanishes for all points of the zero set of $\mathfrak{p}$ where
$\phi^{\prime}$ does not vanish and its numerator is therefore divisible by
$\phi$. We conclude that

$\textstyle{\phi^{\prime}}^{l}F(\frac{\phi_{1}}{\phi^{\prime}},\dots,\frac{\phi_{r-1}}{\phi^{\prime}},x_{r},\dots,x_{n})=G\phi$

for some polynomial $G$ in $x_{r},\dots,x_{n}$. Therefore
${\phi^{\prime}}^{l}F(x_{1},\dots,x_{n})\in(\psi_{1},\dots,\psi_{r-1},\phi)\subset\mathfrak{p}$
and hence $F\in\mathfrak{p}$. ∎

Macaulay’s proof of Proposition 6.2 follows the same steps, but now we can
only conclude that $R_{u}^{m}\in(f_{1},\dots,f_{k})$. Taking suitable
coefficients of $(R_{u}^{m})_{x=u_{1}x_{1}+\dots+u_{n}x_{n}}$ we find that
$\phi^{m}\in\mathfrak{q}=(F_{1},\dots,F_{k})$ and
$\psi_{i}^{m}-G\phi\in\mathfrak{q}$ so $\psi_{i}^{m^{2}}\in Q$. If $F\in P$
with $P$ the associated prime ideal then we have just seen that $\phi^{\prime
l}F\in(\psi_{1},\dots,\psi_{r-1},\phi)$. Therefore $(\phi^{\prime
l}F)^{rm^{2}}\in(\psi_{1}^{m^{2}},\dots,\psi_{r-1}^{m^{2}},\phi^{m^{2}})\subset\mathfrak{q}$
and since $\mathfrak{q}$ is primary and no power of $\phi^{\prime}$ is
contained in $\mathfrak{q}$ we conclude that $F^{rm^{2}}\in\mathfrak{q}$.

Lasker gives in [Las05b] a totally different proof of Proposition 6.3, based
on properties of the Hilbert polynomial [Hil90]. Let $\mathfrak{a}$ be a
homogeneous ideal in $S=K[x_{1},\dots,x_{n}]$. Define the Hilbert function
$H_{\mathfrak{a}}(\nu)=\dim(S/\mathfrak{a})_{\nu}$, where
$(S/\mathfrak{a})_{\nu}$ is the degree $\nu$ part of $S/\mathfrak{a}$. Lasker
proves from scratch the now familar properties of this function [Las05a, Kap.
II]. For large $\nu$ the function $H_{\mathfrak{a}}(\nu)$ is a polynomial in
$\nu$. Lasker shows that $H_{\mathfrak{a}}(\nu)=0$ for sufficiently large
$\nu$ if the ideal $\mathfrak{a}$ has no zeros. This follows from the fact
that under this assumption all monomials of large degree belong to
$\mathfrak{a}$, that is, the weak projective Nullstellensatz 5.3. Lasker
proves it using his version of the resultant of $n-1$ forms in $n$ variables.
Furthermore, if $u$ is a form of degree $d$, which is not a zero divisor in
$S/\mathfrak{a}$, then
$H_{(\mathfrak{a},u)}(\nu)=\Delta_{d}H_{\mathfrak{a}}(\nu)$, where
$\Delta_{d}$ is defined by $\Delta_{d}f(\nu)=f(\nu)-f(\nu-d)$ for a function
$f$.

In Lasker’s version of Proposition 6.3 (the zero set $C$ of a prime ideal is
irreducible and every form vanishing on $C$ belongs to $\mathfrak{p}$)
irreducibility has to be proved. Suppose that the zero set of the prime ideal
$\mathfrak{p}$ consists only of points. If $u\notin\mathfrak{p}$ is a linear
form then $H_{\mathfrak{p}}(u)=\Delta_{1}H_{\mathfrak{p}}$. If $u$ does not
vanish in any of the points, then
$H_{(\mathfrak{p},u)}=0=\Delta_{1}H_{\mathfrak{p}}$ and $H_{\mathfrak{p}}$ is
constant. If there would exist a form $u\notin\mathfrak{p}$ vanishing in one
of the points, then $H_{(\mathfrak{p},u)}\neq 0$, so such a form does not
exist. We conclude that every linear form vanishing in one of the points
vanishes in all, which is only possible if there is only one point, showing
irreducibility. And every form vanishing in the point belongs to
$\mathfrak{p}$.

If $\mathfrak{p}$ has dimension 1 and $u$ does not contain any 1-dimensional
component of the zero set, then again
$H_{(\mathfrak{p},u)}=\Delta_{1}H_{\mathfrak{p}}$, so $H_{\mathfrak{p}}$ a
linear function of $\nu$. If there exists a form $u\notin\mathfrak{p}$,
vanishing on a 1-dimensional component $C$, then $H_{(\mathfrak{p},u)}$ is
independent of $\nu$. The forms in $(\mathfrak{p},u)$ all vanish on $C$ and
therefore are all contained in the prime ideal $\Pi$ of forms vanishing on
$C$. The Hilbert polynomial of $\Pi$ is linear, so $H_{(\mathfrak{p},u)}$
cannot be constant. This shows that every form vanishing on the zero set of
$\mathfrak{p}$ belongs to $\mathfrak{p}$. Irreducibility also follows: suppose
on the contrary that there exist $a,b$ with $ab$ vanishing while $a$ and $b$
do not vanish in all points. Let $a$ contain a 1-dimensional component, but
not all zeros. Then $a\notin\mathfrak{p}$, so such a form cannot exist, and
therefore $a$ vanishes in all points.

In this way the Proposition can be shown by induction.

The proof of Proposition 6.2 uses (or rather proves) general properties of
primary ideals, and is in this way closer to the modern approach than
Macaulay’s.

###### Lasker’s proof of Proposition 6.2.

Let $F_{1},\dots,F_{h}$ be a basis of $\mathfrak{p}$. Form
$F=p_{1}F_{1}+\dots+p_{h}F_{h}$ with indetermined coefficients, of suitable
degree. The ideal quotient $\mathfrak{q}^{\prime}=\mathfrak{q}:(F)=\\{a\in
S\mid aF\in\mathfrak{q}\\}$ can be found despite the fact that $F$ has
indetermined coefficients, as the condition $aF\in\mathfrak{q}$ leads in each
each degree to only linear equations, which can be solved with indetermined
coefficients. Put $\mathfrak{q}^{\prime\prime}=\mathfrak{q}^{\prime}:(F)$,
$\mathfrak{q}^{\prime\prime\prime}=\mathfrak{q}^{\prime\prime}:(F)$ and so on.
In this sequence every ideal is contained in the next, so there is a number
$k$ with $\mathfrak{q}^{(k)}=\mathfrak{q}^{(k+1)}$, by the Ascending Chain
Condition (proved from the Basis Theorem by Lasker [Las05a, p. 56]). Every
ideal $\mathfrak{q}^{(i)}$ is $\mathfrak{p}$-primary. Lasker proves this by
doing the example $\mathfrak{q}^{\prime}$. Let $a\notin\mathfrak{p}$ and
$ab\in\mathfrak{q}^{\prime}$, so $Fab\in\mathfrak{q}$ and because
$\mathfrak{q}$ is primary, $Fb\in\mathfrak{q}$, giving
$b\in\mathfrak{q}^{\prime}$. Also the dimension of $\mathfrak{q}^{\prime}$ is
at most that of $\mathfrak{p}$. Therefore $\mathfrak{q}^{\prime}$ is
$\mathfrak{p}$-primary (according to Lasker’s definition).

Moreover, according to Lasker’s definition, if an ideal $\mathfrak{q}$ is
$\mathfrak{p}$-primary then the zero set of $\mathfrak{q}$ contains the zero
set of $\mathfrak{p}$ or $\mathfrak{q}$ is the whole ring: if
$a\in\mathfrak{q}$, but $a\notin\mathfrak{p}$ and if $f$ is an arbitrary form,
then $af\in\mathfrak{q}$ so $f\in\mathfrak{q}$. Now the above constructed $F$
is not a zero divisor on $S/\mathfrak{q}^{(k)}$, so by the properties of the
Hilbert polynomial the dimension of $(\mathfrak{q}^{(k)},F)$ should be less
than that of $\mathfrak{q}^{(k)}$. The conclusion is that $\mathfrak{q}^{(k)}$
is the whole ring, so $1\in\mathfrak{q}^{(k)}$.

As $\mathfrak{q}^{(k)}=\mathfrak{q}^{(k-1)}:(F)$, we get
$F\in\mathfrak{q}^{(k-1)}$, and then $F^{2}\in\mathfrak{q}^{(k-2)}$, until
finally $F^{k}\in\mathfrak{q}$. As the coefficients of the $p_{i}$ are
indeterminates we conclude that $F_{1}^{k}$, $F_{1}^{k-1}F_{2}$, …,
$F_{h}^{k}$ lie in $\mathfrak{q}$. Therefore $f^{k}\in\mathfrak{q}$ for any
form $f=q_{1}F_{1}+\dots+q_{h}F_{h}$ in $\mathfrak{p}$. ∎

##  7\. Modern algebra

In Moderne Algebra II [vdW31b] van der Waerden gives two proofs of the
Nullstellensatz, the first one using Rabinowitsch’ trick and proving the weak
version by elimination theory. The second proof is based on [vdW26] and
belongs therefore to the proofs before Rabinowitsch. It proves Proposition
6.3, and using Noether’s definition of primary ideals the Nullstellensatz
follows as above. In later editions, in Algebra II [vdW67], the elimination
theory proof is removed, and the weak version is shown with same type of ideas
as in the second proof, but avoiding primary decomposition. This proof was
first described in [vdW54].

Whereas Noether in [Noe21] still considered complex coefficients in the
application to algebraic geometry, she takes in later papers always an
arbitrary field as base field. In [vdW26] this point is stressed by a footnote
(Footnote 13), stating that the new definition holds for unusual spaces, like
those where the fourth harmonic point always coincides with the third; this
happens in characteristic two: if the first three points on the line are
normalised to be $0,\infty$ and $1$, then the fourth harmonic has coordinate
$-1$.

Let $K$ be a field and $R=K[x_{1},\dots,x_{n}]$ the polynomial ring in $n$
variables over $K$. Consider points in the affine space $\mathbb{A}^{n}(L)$
with coordinates in an algebraic extension $L$ of $K$. Besides such points one
has also to consider ‘undetermined’ points, where the coordinates are
indeterminates or algebraic functions of parameters, that is elements in a
transcendental extension $\Omega$ of $K$.

Let therefore $\Omega=K(\xi_{1},\dots,\xi_{n})$ be a field extension. The
polynomials $f\in R$ for which $f(\xi_{1},\dots,\xi_{n})=0$ form a prime ideal
$\mathfrak{p}$ in $R$: if

$f(\xi_{1},\dots,\xi_{n})g(\xi_{1},\dots,\xi_{n})=0$

and $g(\xi_{1},\dots,\xi_{n})\neq 0$, then $f(\xi_{1},\dots,\xi_{n})=0$, as a
field does not contain zero divisors. Van der Waerden [vdW26] gives a simple
example: let $\xi_{1},\dots,\xi_{n}$ be linear functions of one indeterminate
$t$ with coefficients in $K$:

$\xi_{i}=\alpha_{i}+\beta_{i}t\;.$

Then $\mathfrak{p}$ consists of all polynomials vanishing on the line given by
the above parametrisation. This example is not contained in Moderne Algebra II
[vdW31b], but occurs again in Algebra II [vdW67].

The field $\Omega$ is isomorphic to the quotient field $\Pi$ of
$R/\mathfrak{p}$, in such a way that the $\xi_{1}$ correspond to the $x_{i}$.
Conversely, for every prime ideal $\mathfrak{p}\neq 0$ there exists a field
$\Omega=K(\xi_{1},\dots,\xi_{n})$ such that $\mathfrak{p}$ consists of all
polynomials $f\in R$ for which $f(\xi_{1},\dots,\xi_{n})=0$. The point
$(\xi_{1},\dots,\xi_{n})$ is the general zero of $\mathfrak{p}$.

The dimension of $\mathfrak{p}$ is the transcendence degree of $\Omega$ over
$K$. Let $t_{1},\dots,t_{r}$ be a transcendence basis of $\Omega$, so $\Omega$
is an algebraic extension of the field of rational functions
$(t_{1},\dots,t_{r})$. Let $f_{1},\dots,f_{s}$ be elements of the function
field $\Omega$. For given values $\tau_{1},\dots,\tau_{r}$ of the argument one
can solve and find values $\varphi_{1},\dots,\varphi_{s}$ in a suitable
extension of $k$, but only those systems of values are allowed for which all
relation $F(f_{1},\dots,f_{s},t_{1},\dots,t_{r})=0$ also hold for the specific
values, that is $F(\varphi_{1},\dots,\varphi_{s},\tau_{1},\dots,\tau_{r})=0$.
For example, if $f_{1}=\sqrt{t}$, $f_{2}=-f_{1}$, then we have the equations
$f_{1}^{2}=t$ and $f_{2}^{2}=t$, giving for $t=1$ the values $\varphi_{1}=\pm
1$ and $\varphi_{2}=\pm 1$, but it is not allowed to combine $\varphi_{1}=1$
with $\varphi_{2}=1$, as this violates the relation $f_{1}+f_{2}=0$ [vdW31b,
§88]. The existence of such systems is shown by adjoining the $f_{i}$
successively. The denominators in the resulting monic equations for the
$f_{i}$ can be taken to depend only on the $t_{j}$. Let $V(t_{1},\dots,t_{r})$
be the lowest common multiple of the denominators. Consider only parameter
values for which $V(\tau_{1},\dots,\tau_{r})\neq 0$. But also the converse is
valid: if a relation
$F(\varphi_{1},\dots,\varphi_{s},\tau_{1},\dots,\tau_{r})=0$ holds for all
regular systems ofvalues for the arguments and all admissable corresponding
function values, then also the relation
$F(f_{1},\dots,f_{s},t_{1},\dots,t_{r})=0$ holds in the function field
[vdW31b, §88].

Every system of algebraic functions $\xi_{1},\dots,\xi_{n}$ of
$t_{1},\dots,t_{r}$ can be specialised in the above way to
$\xi_{1}^{\prime},\dots,\xi_{n}^{\prime}$ and this determines a point
$\xi^{\prime}$ in affine space over a suitable algebraic extension of $k$. Let
$V$ be the Zariski closure of these points, that is the zero set of all
polynomials $F$ for which $F(\xi_{1}^{\prime},\dots,\xi_{n}^{\prime})=0$. This
means that $\xi_{1},\dots,\xi_{n}$ determines the algebraic variety $V$ in
parameter form and its prime ideal $\mathfrak{p}$ has the general zero
$(\xi_{1},\dots,\xi_{n})$. As every prime ideal $\mathfrak{p}$ has a general
zero $(\xi_{1},\dots,\xi_{n})$, where the $\xi_{i}$ are algebraic functions of
parameters $t_{1},\dots,t_{r}$, Proposition 6.3 follows: every prime ideal is
the ideal of its zero set. In particular, the only prime ideal without zeros
is the whole ring.

As application van der Waerden proves first the generalisation of Noether’s
fundamental theorem to zero dimensional ideals in arbitrary dimension. König
proved the theorem for the case of a complete intersection [Kön03b, p. 385],
and Macaulay observed that the general case easily follows using primary
decomposition [Mac16, p. 61].

######  Theorem 7.1.

Let $\mathfrak{a}$ be an ideal in $R=K[x_{1},\dots,x_{n}]$ with finitely many
zeros $P_{i}$ in $\mathbb{A}^{n}(K)$, $K$ algebraically closed. For a zero
$P=(\xi_{1},\dots,\xi_{n})$ let
$\mathfrak{m}_{P}=(x_{1}-\xi_{1},\dots,x_{n}-\xi_{n})$. There is an integer
$\rho$ depending only on $\mathfrak{a}$ such that
$f\in\mathfrak{a}+\mathfrak{m}_{P_{i}}^{\rho}$ for all $i$ implies
$f\in\mathfrak{a}$.

###### Proof.

Let $\mathfrak{a}=\bigcap_{i}\mathfrak{q}_{i}$ be the primary decomposition of
$\mathfrak{a}$. The associated prime ideal of $\mathfrak{q}_{i}$ is
$\mathfrak{m}_{P_{i}}$. For each $i$ there is an exponent $\rho_{i}$ such that
$\mathfrak{m}_{P_{i}}^{\rho_{i}}\subset\mathfrak{q}_{i}$ and then
$\mathfrak{q}_{i}=\mathfrak{a}+\mathfrak{m}_{P_{i}}^{\rho_{i}}$. With
$\rho=\max\rho_{i}$ the condition in the theorem implies that
$f\in\mathfrak{q}_{i}$ for all $i$ and therefore $f\in\mathfrak{a}$. ∎

Lasker generalised Noether’s theorem [Las05a, Satz XXVII] to what Macaulay
calls the Lasker-Noether Theorem [Mac16, p. 61]. He formulates it roughly as
follows.

######  Theorem 7.2.

If $\mathfrak{a}=(F_{1},F_{2},\dots,F_{k})$ and $F$ can be written as
$F=P_{1}F_{1}+P_{2}F_{2}+\dots+P_{k}F_{k}$ , where the $P_{i}$ are power
series, then there exists a polynomial $\phi$ not vanishing at the origin such
that $F\phi\in\mathfrak{a}$.

It follows that $F$ lies in every primary component containing the origin. For
a criterion that $F\in\mathfrak{a}$ it suffices to impose the power series
condition in a finite number of points.

According to van der Waerden both Lasker’s and Macaulay’s proofs are
insufficient; he adds a note in proof that the gaps in proof by Macaulay are
filled in correspondence between them [vdW26]. The proof still needs
convergence of the power series involved, a condition not necessary in the
proof van der Waerden says to have. The easiest proof seems to be due to Krull
[Kru28], and it is this proof which Macaulay gives in [Mac34], and refers to
as a hitherto unpublished result. This makes it probable that Macaulay learned
it from van der Waerden.

A different generalisation is due to Hentzelt, and elaborated by Hermann
[Her26]. We give it in the formulation of Krull [Kru39, Nr. 20].

######  Theorem 7.3.

For every ideal $\mathfrak{a}=(F_{1},\dots,F_{k})$ in $R=K[x_{1},\dots,x_{n}]$
exists an exponent $\rho$ depending only on $n$ and $k$ and the degrees of the
$F_{i}$ such that $F\in\mathfrak{a}$, if
$F\in\mathfrak{a}+\mathfrak{p}_{i}^{\rho}$ for all associated prime ideals
$\mathfrak{p}_{i}$ of $\mathfrak{a}$.

In this formulation the hard part is to establish that the bound only depends
on the stated quantities. To make clear that it is a Nullstellensatz, the
condition can be formulated as
$F\in\mathfrak{a}R_{i}+(x_{1}-\xi_{1},\dots,x_{n}-\xi_{n})^{\rho}$, where
$(\xi_{1},\dots,\xi_{n})$ is the general zero of the prime ideal
$\mathfrak{p}_{i}$ and $R_{i}=K(\xi_{1},\dots,\xi_{n})[x_{1},\dots,x_{n}]$.
Hentzelt originally formulated the condition for all (infinitely many)
geometric zeros $(\xi_{1},\dots,\xi_{n})$ of $\mathfrak{a}$, that
$F\in\mathfrak{a}+(x_{1}-\xi_{1},\dots,x_{n}-\xi_{n})^{\rho}$.

A non-constructive proof, not establishing the degree bound, was given by van
der Waerden [vdW26]. It uses reduction to the zero dimensional case. It is
explained in [vdW67, §133].

##  Appendix A The resultant

Let $A$ be a unique factorisation domain. We are interested in the question
when two binary forms $F(X,Y),G(X,Y)\in A[X,Y]$ have a common factor.

######  Proposition A.1.

The binary forms $F$ and $G$ in $A[X,Y]$ have a non-constant factor $H$ in
common, if and only if there exist forms $U$ and $V$ of degree less than $\deg
F$, resp. $\deg G$, not both vanishing, such that $VF+UG=0$.

###### Proof.

Suppose $VF=-UG$. All irreducible factors of $F$ have to occur in $UG$, and
not all can occur in $U$, because $\deg U<\deg F$; therefore $F$ and $G$ have
a factor in common. Conversely, given $H$ one finds a $U$ and a $V$ such that
$F=-UH$ and $G=VH$, so the equation $VF+UG=0$ is satisfied, with $\deg U<\deg
F$ and $\deg V<\deg G$. ∎

Suppose $\deg F=m$ and $\deg G=n$ and consider the free module
$A[X,Y]_{n+m-1}$ of forms of degree $m+n-1$,

The existence of a relation $VF+UG=0$ is equivalent to the fact that the forms
$X^{n-1}F$, $X^{n-2}YF$, …, $Y^{n-1}F$, $X^{m-1}G$, …, $Y^{m-1}G$ are linearly
dependent in vector space $Q(A)[X,Y]_{n+m-1}$ of dimension $m+n$, where $Q(A)$
is the quotient field of $A$. We represent a form
$c_{0}X^{n+m-1}+\dots+c_{n+m-1}Y^{n+m-1}$ by the row vector
$(c_{0},\dots,c_{n+m-1})$; multiplying with the column vector
$\mathcal{X}=(X^{n+m-1},\dots,Y^{n+m-1})^{t}$ gives back the form.

Put

$\displaystyle F$ $\displaystyle=a_{0}X^{m}+a_{1}X^{m-1}Y+\dots+a_{m}Y^{m},$
$\displaystyle G$ $\displaystyle=b_{0}X^{n}+b_{1}X^{n-1}Y+\dots+b_{n}Y^{n}.$

Writing out the forms $X^{n-1}f$, …, $Y^{n-1}F$, $X^{m-1}G$, …, $Y^{m-1}g$ in
the basis $X^{n+m-1},\dots,Y^{n+m-1}$ leads in this way to a matrix equation
$S_{F,G}\mathcal{X}=\mathcal{F}$, with
$\mathcal{F}=(X^{n-1}f,\dots,Y^{m-1}g)^{t}$ and $S_{F,G}$ the Sylvester matrix

$S_{F,G}=\begin{pmatrix}a_{0}&a_{1}&a_{2}&\dots&a_{m}&\\\
&a_{0}&a_{1}&a_{2}&\dots&a_{m}&\\\ &&\dotfill\hphantom{n}\\\
&&&a_{0}&a_{1}&a_{2}&\dots&a_{m}\\\ b_{0}&b_{1}&\dots&b_{n-1}&b_{n}&\\\
&b_{0}&b_{1}&\dots&b_{n-1}&b_{n}\\\ &&\dotfill\hphantom{nn}\\\
&&&b_{0}&b_{1}&\dots&b_{n-1}&b_{n}\end{pmatrix}\,.$

The determinant $R(F,G)$ of this matrix is called the _resultant_ of $F$ and
$G$.

From Proposition A.1 follows:

######  Proposition A.2.

The forms $G$ and $G$ have a non-constant factor $H$ in common, if and only if
$R(F,H)=0\in A$.

If the resultant does not vanish, the matrix equation
$S_{F,G}\mathcal{X}=\mathcal{F}$ can be “solved” by inverting the matrix
$S_{F,G}$. To avoid denominators it is better to multiply with the adjugate
matrix.

######  Proposition A.3.

For any two forms $F$ and $G$ of degrees $m$ and $n$ in $A[X,Y]$ and each
monomial $X^{k-1}Y^{n+m-k}$ there exist forms $U$ and $V$ such that:

$VF+UG=X^{k-1}Y^{n+m-k}R(F,G).$

The above results also apply to the inhomogeneous case, by putting $Y=1$. If
the (homogeneous) resultant vanishes, there is a common factor, but this
factor can be a power of $Y$. This means in particular that $a_{0}=b_{0}=0$,
so the degree of the inhomogeneous polynomials $f(x)=F(X,1)$ and $g(x)=G(X,1)$
are lower than $m$ and $n$. This situation can occur if the coefficients are
variables, which can be specialised to specific values. We call $m$ the formal
degree of the polynomial $f$ [vdW71a, §34]. The resultant $R(f,g)$ of two
polynomials $f(x)$ and $g(x)$ of formal degrees $m$ and $n$ vanishes if and
only if $f$ and $g$ have a non-constant factor in common or in both polynoials
the highest coefficient vanishes.

For inhomogeneous polynomials one obtains from Proposition A.3:

######  Proposition A.4.

For any two polynomials $f$ and $g$ there exist polynomials $u$ and $v$ with
$\deg u<\deg f$ and $\deg v<\deg g$, such that:

$vf+ug=R(f,g).$

The resultant is in fact a universal polynomial in the $a_{i}$ and $b_{j}$
with integral coefficients and the above relation also holds for indeterminate
$a_{i}$ and $b_{j}$.

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