# First use of term “Hilbert's Nullstellensatz”

The post below first appeared on hsm.stackexchange over a week ago and has received no answers there yet, so by now I think it is okay to ask it here.

This year (2021) marks the 100th anniversary of Emmy Noether's 1921 paper in which she introduced Noetherian rings and proved the primary ideal decomposition for them. The original version of her paper is here and an English translation is on the arXiv here. She of course does not call rings "Noetherian" but instead refers to rings satisfying "the finiteness condition" (die Endlichkeitsbedingung). When referring to results of Hilbert that we'd call Hilbert's basis theorem and Hilbert's Nullstellensatz at the start of Section 10 of the paper, she writes "Hilbert’s Module Basis Theorem" (Hilbertschen Theorem von der Modulbasis) and "a famous theorem of Hilbert’s" (... eines bekannten Hilbertschen Satzes). In two footnotes on this page she also refers to "Hilbert's theorem" instead of the Nullstellensatz.

The Nullstellensatz was published by Hilbert in 1893 here (see the theorem starting on the bottom of p. 320), and that was nearly 30 years earlier. I am surprised that Noether refers to that result simply as "Hilbert's Theorem". Thus my question:

What is the earliest known reference to Hilbert's theorem as the Nullstellensatz or Hilbert's Nullstellensatz (in German, presumably)?

The answer is no later than 1927, when a paper by van der Waerden here refers to Hilbert's Nullstellensatz on the top of p. 202. This is the earliest of two papers mentioned in Zentralblatt that has "Nullstellensatz" in a review. The other earliest paper in Zentralblatt is by Grete Hermann from 1926 here, which refers to Hentzelt's Nullstellensatz in the 2nd and 3rd paragraphs of the third page (Hentzelt's Nullstellensatz also appears in van der Waerden's paper on p. 203). I didn't find her mentioning Hilbert's Nullstellensatz. Hermann's paper is in English translation here -- see the first column of the second page. The paper by Hentzelt is from 1923 here (it was edited by Noether since Hentzelt went missing in Belgium in 1914 during WW1), and I did not find the term "Hilbert's Nullstellensatz" used in it.

For what it's worth, the earliest article in Zentralblatt with Nullstellensatz in a title is the paper by Rabinowitsch here (1930) in which he used his trick to prove the Nullstellensatz. That is also the earliest paper in MathSciNet's records with Nullstellensatz appearing anywhere. It's not from a real MathSciNet review since it predates the birth of MathSciNet (well, the birth of Mathematical Reviews) in 1940. The earliest real review in MathSciNet mentioning the Nullstellensatz is Zariski's paper from 1947 with his new proof of the Nullstellensatz by "Zariski's lemma".

• Do you know the statement of Hentzelt's Nullstellensatz? – auniket May 3 at 20:24
• @auniket : see Hentzelt's Nullstellensatz – Carlo Beenakker May 4 at 6:00
• @Carlo Google does not show that page. – Emil Jeřábek May 4 at 6:59
• apologies; it does show the page for me, but it's a book and sometimes Google limits the pages that can be displayed for copyright reasons; it's page 166 of Algebra, volume II by Van der Waerden – Carlo Beenakker May 4 at 9:41
• google.com/books/edition/Nieuw_Archief_Voor_Wiskunde/… is claiming a 1925 example (Google books are not reliable for dates) in a Dutch publication Nieuw Archief Voor Wiskunde, though the surrounding words look German to me – Henry May 4 at 12:16

I think it is indeed van der Waerden, but in the earlier paper [1926], where he sounds just like one does when introducing terminology: translated from p. 143,

the proposition in question is an almost immediate consequence of two fundamental propositions of HILBERT, which can be referred to as HILBERT’s Nullstellensatz and HILBERT’s Basissatz.

An extra twist is that this all presumably first appeared in Dutch, in a first version [1925] of the paper that would be interesting to find and check; it doesn’t seem to be online.

• Perhaps Carlo Beenakker will be able to procure the 1925 paper and check it when he sees this page. – KConrad May 2 at 18:59
• I found it in the HathiTrust Library, since the chronology of the 1925 and 1927 Van der Waerden papers requires some further explanation, I will post it separately. – Carlo Beenakker May 2 at 20:02

Below is the Dutch paper mentioned by Francois Ziegler; This paper did indeed appear before the 1927 paper mentioned in the OP, but Van der Waerden does refer to that forthcoming publication in a footnote; I translate:

This theorem is a special case of the "Nulpuntenstelling" of HILBERT$$^{5})$$

$$^{5})$$ Hilbert assumes that $$\Omega$$ is the field of complex numbers; his proof actually holds more generally. For a different proof see my forthcoming paper in the Mathematische Annalen entitled "Zur Nullstellentheorie der Polynomideale".

The Mathematische Annalen paper was submitted 14 August 1925, while the paper below was read at the Netherlands Academy on 28 November 1925, which explains why Van der Waerden could refer to former as a an earlier work.

B.L. Van der Waerden, Een algebraies kriterium voor de oplosbaarheid van een stelsel homogene vergelijkingen, Verslag van de Gewone Vergadering van de Afdeling Natuurkunde van de Koninklijke Akademie van Wetenschappen 34, 1123-1130 (1925)

[source] -- only accessible online from a US IP number.

In response to a request in a comment: Van der Waerden wrote again on this topic in 1927, Neue Begründung der Eliminations– und Resultantentheorie, Nieuw Archief Wiskunde 15, 302–320 (1927) [source]

• The image link at the end of your post is broken. – KConrad May 2 at 20:06
• In Dutch today, are the terms "Nullstellensatz" and "Nulpuntenstelling" interchangeable, or does the second term sound archaic? – KConrad May 2 at 20:10
• they are not interchangeable, the first is German and the second is Dutch; professionally the first term is used more often than the second term (see for example, the Dutch Wikipedia entry) , but if one would speak for audience of non-experts one would certainly use the second term. – Carlo Beenakker May 2 at 20:16
• If I recall correctly (my Dutch was never very good), the official Dutch for "algebraies" is now "algebraïsch", but I find a lot of Google results for "algebraïes" coming from South Africa, so presumably the old spelling is kept in Afrikaans. – Robert Furber May 3 at 6:08
• indeed, "algebraies" and "mathematies" was an uncommon spelling even for that era; the replacement of the ending -isch by -ies was advocated in a proposed phonetic spelling reform, adopted by South Africa in 1905, but never officially adopted in Dutch. – Carlo Beenakker May 3 at 8:54