TITLE: A nice generating set for the symmetric power of an algebra
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: 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 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 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 for a scanned version).