I am trying to understand Dirichlet's Unit Theorem and I come across to a part that I do not have any idea. I will write down the arguments and then ask my question:
Let $K$ be a number field of degree $n=r_1 + 2r_2$ where $r_1$ denotes the number of real embedding of $K$ and $2r_2$ is the number of complex embeddings of $K$. Since complex emdeddings come in pairs, we can consider only the first $r_1$ one: $\sigma_{r_1+1},\dots,\sigma_{r_1+r_2}$. Let us define now the homomorphism $$\sigma:K \to \mathbb{R}^{r_1+r_2} ; \sigma(x)=(|\sigma_1(x)|,\dots,|\sigma_{r_1+r_2}(x)|).$$
My aim is to show that the image of units in $\mathcal{O}_K$ under $\sigma$ is a discrete subgroup of $ \mathbb{R}^{r_1+r_2}$.
Now, let us take a compact subset $B$ of $\mathbb{R}^{r_1+r_2}$. Set $B'$ to be the set $\{x \in \mathcal{O}_K: \sigma(x) \in B\}$. Since $B$ is bounded, so is $B'$. So I can find some $\epsilon$-ball in $\mathbb{R}^{r_1+r_2}$ so that $B$ fits in it. Which also means that the norm of elements of $B$ is bounded.
With this information, I can find some $\alpha >1$ such that for each $x \in B'$, $\alpha^{-1} \le |\alpha_i(x)| \le \alpha, (i=1,\dots,n)$-here, the book says $n$ but I guess it should be $r_1+r_2$.-
Finally, the part that I do not understand comes:
It follows that the elementary symmetric functions of $\sigma_i(x)$'s are bounded in absolute value. Since they belong to $\mathbb{Z}$, the set of possible values for the symmetric functions of the $\sigma_i(x)$'s is a finite set.
It seems like we can write for each $i$, $\sigma_i(x)=p_1(x) \times \dots \times p_k(x)$ where $k \le n$, $p_i(x)$'s are symmetric functions. So we can get the result, but how?
Any help is appreciated, thank you.
Consider the characteristic polynomial of $x$, which is
$$ \Phi_x(t) = \prod_{i = 1}^n (t - \sigma_i(x)).$$
(Here we are also considering the entire Galois group rather than just half of the complex embeddings.) If $\mu_x(t)$ is the minimal polynomial of $x$ then $\Phi_x(t) = \mu_x(t)^{[K:\mathbf{Q}(x)]}$. Both polynomials belong to $\mathbf{Z}[t]$ since $x \in \mathcal O_K$.
We can alternatively write
$$\Phi_x(t) = \sum_{k = 0}^n (-1)^{n - k} e_{n - k}(x) t^k $$
where
$$ e_k(x) = \sum_{1 \le i_1 < \cdots < i_k \le n} \sigma_{i_1}(x) \cdots \sigma_{i_k}(x). $$
We call $e_0(x), \dots, e_n(x)$ the elementary symmetric functions in $\sigma_1(x), \dots, \sigma_n(x)$.
Since $\Phi_x(t) \in \mathbf{Z}[t]$ we can say that $e_k(x) \in \mathbf{Z}$. Since $\sigma_1(x),\dots,\sigma_n(x)$ are bounded, so are $e_0(x),\dots,e_n(x)$.