Generalizing an elementary problem, the author proves in this note the following two results. (i) Let \(K\subset {\mathbb{C}}\) be a field, \(P(X)=a_ 0+a_ 1X+...+a_ mX^ m\) a polynomial with \(a_ j\in K\) and let \(x_ 1,x_ 2,...,x_ m\in {\mathbb{C}}\) be the roots of \(P(X)=0\). Let \(q_ 1,q_ 2,...,q_ r\in K[X_ 1,X_ 2,...,X_ m]\) be given polynomials. Denote \(y_ j=q_ j(x_ 1,...,x_ m)\), \(j=1,2,...,r\). There exists a polynomial \(R\in K[X]\) such that \(R(y_ j)=0\) for \(j=1,2,...,r.\)
(ii) Let \(F\in {\mathbb{Q}}[X_ 1,X_ 2,...,X_ m]\). The polynomial F is symmetric if and only if \(F(x_ 1,...,x_ m)\in {\mathbb{Q}}\) for all \((x_ 1,x_ 2,...,x_ m)\in {\mathbb{C}}^ m\) for which there exists a polynomial \(P\in {\mathbb{Q}}[X]\) such that \(P(x_ j)=0\) for \(j=1\), 2,...,m.