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?

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 Aug 17 '16 at 16:11