"WITHOUT DOUBT THIS IS THE MOST IMPORTANT WORK ON GENERAL ALGEBRA"

HILBERT, DAVID.

Ueber die Endlichkeit des Invariantensystems für binäre grundformen (+) Ueber Büschel von binären Formen mit vorgeschriebener Functionaldeterminante.

Leipzig, B.G. Teubner, 1888. 8vo. Original printed wrappers, no backstrip and a small nick to front wrapper. In "Mathematische Annalen. Begründet 1888 durch Rudolf Friedrich Alfred Clebsch. XXXIII.[33] Band. 2. Heft." Entire issue offered. Internally very fine and clean. [Hilbert:] Pp. 223-6; Pp.227-36 [Entire issue: Pp. 161-316].

First printing of Hilbert's exceedingly important and groundbreaking paper in which he proved his famous Basis Theorem that is, if every ideal in a ring R has a finite basis, so does every ideal in the polynomial ring R[x]. Hilbert had thus connected the theory of invariants to the fields of algebraic functions and algebraic varieties. When Felix Klein read the paper he wrote "I do not doubt that this is the most important work on general algebra that the Mathematische Annalen has ever published."

Hilbert submitted a paper proving the finite basis theorem to Mathematische Annalen. However Gordan was the expert on invariant theory for the journal and he found Hilbert's revolutionary approach difficult to appreciate. He refereed the paper and sent his comments to Klein:
"The problem lies not with the form ... but rather much deeper. Hilbert has scorned to present his thoughts following formal rules, he thinks it suffices that no one contradict his proof ... he is content to think that the importance and correctness of his propositions suffice. ... for a comprehensive work for the Annalen this is insufficient."
Gordan rejected the article. His - now famous - comment was: Das ist nicht Mathematik. Das ist Theologie. (i.e. This is not Mathematics. This is Theology).

However, Hilbert had learnt through his friend Hurwitz about Gordan's letter to Klein and Hilbert wrote himself to Klein in forceful terms:
"... I am not prepared to alter or delete anything, and regarding this paper, I say with all modesty, that this is my last word so long as no definite and irrefutable objection against my reasoning is raised."

At the time Klein received these two letters from Hilbert and Gordan, Hilbert was an assistant lecturer while Gordan was the recognised leading world expert on invariant theory and also a close friend of Klein's. However Klein recognised the importance of Hilbert's work and assured him that it would appear in the Annalen without any changes whatsoever, as indeed it did. Hilbert expanded on his methods in a later paper, again submitted to the Mathematische Annalen [1893] and Klein,
after reading the manuscript, wrote to Hilbert saying:-I do not doubt that this is the most important work on general algebra that the Annalen has ever published.

Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself said: "I have convinced myself that even theology has its merits".(Klein. Development of mathematics in the 19th century. P. 311).

Sometimes Hilbert's first publication of the Basis Theorem is referred to as being published in the paper "Zur Theorie der algebraischen Gebilde" in Göottinger Nachrichten in 1888. This, however, was published in December 1888 and the present issue was published in March.

Order-nr.: 44430