What is a reference in the literature for the following fact?
Let $A$ be a commutative $\mathbb{Q}$-algebra. Then every monomial in $A$ of degree $n$ may be written as a linear combination of $n$th powers.
For example, $a \cdot b = \frac{1}{2} ((a+b)^2-a^2-b^2)$ for $n=2$. For $n=3$, we have $ab^2=\frac{1}{6} ((a+b)^3+(a-b)^3-2a^3)$, hence also $abc=\frac{1}{2} (a(b+c)^2-a b^2-a c^2)$ may be generated.
Of course it suffices to consider $A=\mathbb{Q}[T_1,\dotsc,T_n]$ and to write $T_1 \cdot \dotsc \cdot T_n$ as a linear combination of $n$th powers.
PS: I'm not looking for a proof, I'm looking for a reference.
PPS: It seems to be known as the Waring decomposition.
The following argument, if correct, is not original.
Let $X_1,\dots,X_n$ be indeterminates, let $V_m$ be the $\mathbb Q$-vector space formed by the degree $m$ homogeneous polynomials in $\mathbb Q[X_1,\dots,X_n]$, and let $W\subset V$ be the subspace generated by the $\ell^m$ with $\ell$ in $V_1$.
It suffices to show $W=V_m$.
Put $$ D_i:=\frac\partial{\partial X_i}\ . $$ The bilinear form $(f,g)\mapsto f(D)g(X)$ on $V_m$ being non-degenerate, it suffices to show that if $f$ in $V_m$ be such that $f(D)\ell(X)^m=0$ for all $\ell$ in $V_1$, then $f=0$. This is straightforward.
EDIT 1. Here are more details about the last sentence. If we set, using multi-index notation, $$ f(D)=\sum_\alpha a_\alpha\,D^\alpha,\quad \ell(X)=\sum_i b_i\,X_i, $$ then we get, by the Binomial Theorem, $$ 0=f(D)\ell(X)^m=m!\ f(b). $$ for all $b$ in $\mathbb Q^n$, which implies indeed $f=0$.
EDIT 2. Let's check the equality $f(D)\ell(X)^m=m!\ f(b)$ above. We have $$ f(D)\ell(X)^m=\sum_{\alpha,\beta}a_\alpha D^\alpha\binom m\beta b^\beta X^\beta=\sum_{\alpha,\beta}a_\alpha\binom m\beta b^\beta D^\alpha X^\beta $$ $$ =\sum_\alpha a_\alpha\binom m\alpha b^\alpha\alpha!=m!\sum_\alpha a_\alpha b^\alpha=m!f(b). $$ Throughout this post, $\alpha$ and $\beta$ are $n$-indices of length $m$.