Let $s_1, s_2, \ldots, s_n$ be the $n$ elementary symmetric polynomials in the $n$ variables $X, X_2, X_3, \ldots, X_n$ in the polynomial ring $\mathbb{Q}\left[X, X_2, X_3, \ldots, X_n\right]$.
I want to find that $X^n-1$ is a $\mathbb{Q}[X]$-linear combination of $s_1,...,s_{n-1}$ and $s_n + (-1)^n$. I know that $$X^n=\sum \limits_{i=1}^n(-1)^ks_kX^{n-k}=(-1)^ns_n+\sum \limits_{i=1}^{n-1}(-1)^ks_kX^{n-k} $$. So $X^n-1=((-1)^ns_n-1)+\sum \limits_{i=1}^{n-1}(-1)^ks_kX^{n-k} $.
How do rewrite this with $s_n+(-1)^{n-1}$ ?
Cordialy, doeup