I'm looking for a reference for the following fact.


Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic zero. Let $S^n(A)$ be the $n$-th symmetric power of $A$. That is, $S^n(A)$ is the subalgebra of $A^{\otimes n}$ fixed by the natural action of $S\_n$ on this tensor product by permuting the factors. I would like to know a reference for the fact that $S^n(A)$ is generated by elements of the form


$(a \otimes 1 \otimes \cdots \otimes 1) + (1 \otimes a \otimes 1 \otimes \cdots \otimes 1) + \cdots + (1 \otimes \cdots \otimes 1 \otimes a)$


for $a$ in $A$.


Note that if $A = \mathbb{C}[t]$, then $S^n(A)$ is the ring of symmetric polynomials in $n$ variables and then the result follows from the fact that the power sum polynomials generate this ring.


I call the above a "fact" since I've seen it used in papers (without proof -- and I've contacted the authors and they don't know of a reference) and a colleague of mine says he could write out a proof. However, I can't help but think that there must be a reference for this in some text. But I'm unable to find one...


Alternatively, if someone knows of a very short proof, that would be the next best thing to a reference. I want to use this fact in a paper I'm writing and I'd like to avoid having to digress from the main thread of the exposition by including a long proof of this fact.



A nice generating set for the symmetric power of an algebra
Since your algebra is finitely generated, you only really need this result in the case when $A=\mathbb{C}[t\_1,\ldots,t\_n]$. 


It seems that this result first appeared in 


F. Junker, *Über symmetrische Funktionen von mehreren Reihen von Veränderlichen*, Math. Ann. 43 (1893), 225-270 


(see [here](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0043&DMDID=DMDLOG_0024) for a scanned version of it). 


A much more modern reference (from which I know about Junker's paper) is 


J. Dalbec, *Multisymmetric functions*, Beiträge zur Algebra und Geometrie, Vol. 40 (1999), No. 1, pp. 27-51. 


(see [here](http://www.emis.de/journals/BAG/vol.40/no.1/3.html) for an electronic version).


**Edit:** as Abdelmalek Abdesselam suggests, this result goes back to 


L. Schläfli, *Über die Resultante eines Systemes mehrerer algebraischer Gleichungen Ein Beitrag zur Theorie der Elimination.* (A. d. IV Bd. der Denkschriften der math.-naturw. Classe der kais. Acad. d. Wiss. bes. abgedruckt), Wien 1852


(see [here](http://www.bsb-muenchen-digital.de/~web/web1094/bsb10942369/images/index.html?digID=bsb10942369&pimage=00001&v=pdf&nav=0&l=de) for a scanned version).


