I wanted some help understanding the proof of Kronecker's Theorem (Bombieri) more specifically, the part involving symmetric elementary functions. I don't know how to reach the inequality (I'm not very familiar with symmetric elementary functions). I appreciate the help.
Let $d$ be the degree of and let $\zeta=(\zeta_{1},\zeta_{2},\dotsc,\zeta_{d})$ be a full set of conjugates of $\zeta$ . Now consider, for every positive integer $m$, the elementary symmetric functions $s_{i}(\zeta^{m})$ of $\zeta_{1}^{m},\zeta_{2}^{m},\dotsc,\zeta_{d}^{m}$, where $i=0,\dotsc,d$. Since $\zeta$ is an algebraic integer, we have $s_{i}(\zeta^{m})\in Z$ for every $m$. Since $|\zeta_{j}|_{v} =1$ for every $j$ and $v$ , and since $s_{i}(\zeta^{m})$is the sum of ${d \choose i}$ terms each of which is a product of factors not exceeding 1 in absolute value, it is now clear that $$\sum\limits_{i=0}^{d}{|s_{i}(\zeta^{m})|}\leq\sum\limits_{i=0}^{d}{d \choose i}.$$