First of all, a comment, before this gets marked as a duplicate:

I have searched this website for the question I’m asking and I’m aware that this exact question has been asked before. However, Eric Temple Bell (1915) was not the first time the Unique Prime Factorization Theorem was called the “Fundamental Theorem of Arithmetic”.

Bell’s 1915 book called it the FTA for the first time in the English language.

Another mathematician, probably a German one, called it “Fundamental Theorem of Arithmetic”, and I know it was a few years before Eric Temple Bell.

I have spent quite a few hours researching this for something, but can’t seem to find the original “FTA” reference.

Thanks in advance for any answer, help or suggestion.

  • 1
    $\begingroup$ "Fundamentalsatz der Arithmetik" is probably what the German speaking write about.. $\endgroup$
    – sand1
    Sep 9, 2019 at 12:17
  • $\begingroup$ How do you know it was a few years before Bell? Did you see it in some old book a while ago? And how do you know that Bell was the first one in English? $\endgroup$
    – Conifold
    Sep 9, 2019 at 17:30
  • 2
    $\begingroup$ In his "Gedächtnisrede auf Ernst Kummer" written in 1910, Hensel mentions that the Gaussian integers satisfy the same "Fundamentalsatz der Arithmetik" as the real integers, namely to decompose uniquely into primes; he points to Kummer's investigations into whether such "Grundsätze der Arithmetik" hold true for more general rings of integers in number fields. link.springer.com/chapter/10.1007%2F978-3-7091-9534-5_3 $\endgroup$ Sep 10, 2019 at 4:00
  • $\begingroup$ Added: Hensel also stresses the "Fundamentalsatz" in his 1913 book "Zahlentheorie" (gutenberg.org/ebooks/38986, p. 41/56 of the file). If there is anything before Hensel, my bet would be Kroneckers lectures on Zahlentheorie/Arithmetik, which I don't have access to right now. $\endgroup$ Sep 11, 2019 at 18:37
  • 2
    $\begingroup$ @TorstenSchoeneberg I think you should convert these into an answer, it is better than one can usually hope for with such questions. Kronecker has a paper Ein Fundamentalsatz der allgemeinen Arithmetik which refers to something closer to fundamental theorem of algebra. $\endgroup$
    – Conifold
    Sep 13, 2019 at 0:28

1 Answer 1


Kurt Hensel gave a eulogy commemorating the 100th birthday of E. Kummer in 1910 ("Gedächtnisrede auf Ernst Eduard Kummer"; free transcribed version). On page 20 he lists the irreducible elements among the Gaussian integers $\mathbb Z[i]$ and then writes

Dies sind nun aber auch alle Primzahlen im Bereiche dieser komplexen Zahlen, und hier, wie in der Theorie der reellen Zahlen besteht der Fundamentalsatz [emphasis added]: Jede komplexe Zahl kann stets und nur auf eine einzige Weise in ein Produkt von komplexen Primzahlen zerlegt werden.

My translation:

But these, now, are all the primes among these complex numbers, and here, as in the theory of real numbers [sc. integers], we have the Fundamental Theorem: Each complex number [sc. Gaussian integer] can always be decomposed in one and only one way into a product of complex prime numbers [sc. irreducible Gaussian integers].

He goes on to explain, following Kummer, that (in modern language) if such number rings are UFD's, one can prove cases of Fermat's Last Theorem with that, but notes how in general such integer rings of numbers fields are not. Page 21:

Will man also auf dieses allgemeine FERMATsche Problem die Methoden und Ergebnisse der Zahlenlehre anwenden, so muß man zunächst fragen, ob auch für diese Zahlen, in $o$, wie ich sie nennen will, die Grundsätze der Arithmetik [emphasis added] gelten, zunächst also, ob sich jede solche Zahl stets als Produkt von nicht weiter zerlegbaren Zahlen in $o$ darstellen läßt.

My translation:

So if one wants to apply the methods and results of number theory to this general FERMAT Problem, one has to ask first whether for these numbers, in $o$, as I want to call them, the foundational theorems of arithmetic remain true, first and foremost, whether each such number can always be written as a product of numbers without further decomposition in $o$.

Further, in his book "Zahentheorie" published in 1913, Hensel indents the theorem as "Fundamentalsatz" again (http://www.gutenberg.org/files/38986/38986-pdf.pdf, bottom of p. 41 (56 of the pdf file)) and calls it (with his own emphasis) "foundational for the entire multiplicative theory of numbers".

So if we are very precise, he does not actually call it the Fundamental Theorem of Arithmetic, but "Fundamentalsatz" (without "der Arithmetik"), and a few lines later he describes it as the first among the "Grundsätze der Arithmetik". But otherwise this might be what your source had in mind, as Hensel was German, and this was a few years before Bell's 1915 book.


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