## Symmetric Polynomials

Hi there (again),

I tried going about a proof to the following proposition,

Any symmetric integral polynomial in $d\beta_1, \ldots, d\beta_n$ is a symmetric polynomial in $d\beta_1, \ldots, d\beta_n, 0, 0 \dots, 0$,

Let us denote our symmetric integral polynomial by $f(d\beta_1, \ldots, d\beta_n).$ Then, we know that there exists some polynomial g that satisfies the relation $f(d\beta_1, \ldots, d\beta_n) = g(d\beta_1, \ldots, d\beta_n 0,0 , \dots, 0).$ We conclude that g is also a symmetric integral polynomial, since the zeros are not labelled, and so are unaffected by any permutation $\sigma$ on the set $\{1,2,\ldots,n\}$
EDIT: Of course, all of $d, \beta_1, \ldots, \beta_n$ are integers