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Do you know the history (esp. first use) of the name "Fundamental Theorem on Symmetric Polynomials" (or "FT of SP", or "FT on S Functions", etc.) for the statement that any polynomial (resp. rational function) symmetric in n variables is uniquely expressible as a polynomial (resp. rational function) in the elementary symmetric polynomials in those variables?

I got a comment from a reviewer (for an article about this theorem) asking for a reference for the name. Every 20th century source I consulted (back to van der Waerden) just presents the theorem with this name. The 18th and 19th century sources I've looked at (Waring, Galois, Gauss) do not give a name (nor do they uniformly articulate it as a theorem; Gauss does; Waring kind of does). I assume the theorem was christened with the name sometime between 1850 and 1950. Do you know where, when and by whom?

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  • $\begingroup$ Wikipedia's page for "Albert Girard" (of the Girard-Newton formulas) gives a reference to a Math Monthly article Funkhouser, H. Gray (1930). "A short account of the history of symmetric functions of roots of equations". Amer. Math. Monthly. 37 (7): 357–365. doi:10.2307/2299273. JFM 56.0005.02, which (I can't easily look at it from where I am) might conceivably be useful. $\endgroup$
    – paul garrett
    Commented Aug 15, 2016 at 22:09
  • $\begingroup$ @paulgarrett - I had a look. It traces some history of this theorem, esp. noting proofs by Waring and Meyer Hirsch, but it doesn't give any history of the name. (After mentioning Waring's proof, he refers to it as the "beginning of the fundamental theorem of modern symmetric functions"). $\endgroup$
    – benblumsmith
    Commented Aug 16, 2016 at 21:11
  • $\begingroup$ Tsk. And I've had a chance to look around, and found no attribution of the naming... nor clear attribution of the result. Lost in the mists of antiquity, apparently. $\endgroup$
    – paul garrett
    Commented Aug 16, 2016 at 21:23
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    $\begingroup$ Richard Stanley, in Enumerative Combinatorics, refers to the following reference for the history of symmetric functions, including the fundamental theorem: Karl Theodor Vahlen, IB3b, Rationale Funktionen der Wurzeln; symmetrische und Affektfunktionen, in Encyklopedia der Mathematischen Wissenschaften, Erster Band, Teubner, Leipzig, 1898-1904, pp.449-479. Unfortunately I don't have this reference handy nor have I found it online, so I can't say whether it answers your question. Note that Germans seem to use "Hauptsatz" rather than "Fundamentalsatz" for this particular theorem. $\endgroup$
    – Timothy Chow
    Commented Aug 17, 2016 at 16:11

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A good source is Muir's "The Theory of Determinants in the Historical Order of Development". The whole book has been digitized and put in the public domain (available through University of Michigan Historical Math Collection).

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  • $\begingroup$ Searching seems to indicate that "symmetric" and "fundamental" are never found on the same page. $\endgroup$
    – benblumsmith
    Commented Aug 16, 2016 at 21:12
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    $\begingroup$ I would try to search for the theorem itself. If Muir does not mention this name, that would constitute evidence that the name was adapted later than 1st edition of vol.2 was written. Incidentally, Bocher's "Introduction to Higher Algebra" (1907), which contains extensive treatment of symmetric functions, does not use the name "Fundamental Theorem", cf Theorem 1 in Sec 84, p 243. $\endgroup$
    – Victor Protsak
    Commented Aug 17, 2016 at 1:40
  • $\begingroup$ The theorem does not seem to appear. I searched for all occurrences of "symmetric function" (as "symmetric polynomial" does not appear) and also looked at most of the instances just of "symmetric". Symmetric functions seem mostly to come up in relation to alternating functions and don't seem to get any airtime on their own terms. $\endgroup$
    – benblumsmith
    Commented Aug 19, 2016 at 15:35

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