A question of the proof of Kronecker’s Theorem regarding Cyclotomic Polynomials Theorem: ${f \in \Bbb{Z}[X]}$ is irreducible, monic and all roots of ${f}$ have absolute value at most 1. Then $f$ is a cyclotomic polynomial.
Here is a part of the proof I feel confused:
 It suffices to prove that every root of $f$ is a root of unity. If $f(X)=(X-\alpha_1)...(X-\alpha_r)$consider the family of polynomials
$f_n(X) =(X-\alpha_1^n)...(X-\alpha_r^n)$.The coefficients of ${f_n}$ are algebraic integers, since they are calculated using multiplications and additions starting from ${\alpha_1,...,\alpha_r}$. On the other hand, the coefficients of ${f_n}$ are symmetric polynomials in ${(\alpha_i)}$, so they are rational, and therefore integers.
I don't understand why "the coefficients of ${f_n}$ are symmetric polynomials in ${(\alpha_i)}$, so they are rational, and therefore integers."? Why is it rational? And why therefore integer?
 A: As it is written right now, the claim is false. Indeed, $f=X$ is an irreducible monic polynomial with integers coefficients and whose roots lie in the unit disk. However, it is not a cyclotomic polynomial. The result will hold if you assume that $0$ is not a root of the given polynomial! Note that I am being picky, this is the only counterexample. In fact, one has:

Theorem. An irreducible monic polynomial with integers coefficients is either $X$ or a cyclotomic polynomial. 

Regarding your question, notice that for $\ell\in\{0,\cdots,r\}$, the coefficient of $X^{r-\ell}$ in $f_n$ is: $$(-1)^\ell\sigma_{\ell}({\alpha_1}^n,\ldots,{\alpha_r}^n),$$
where $\sigma_{\ell}$ is the $\ell$-th elementary symmetric polynomial. Furtheremore, notice that $\sigma_{\ell}({X_1}^n,\ldots,{X_r}^n)$ is a symmetric polynomial and therefore there exists $S_{\ell}\in\mathbb{Z}[X_1,\cdots,X_n]$ such that: 
$$\sigma_{\ell}({X_1}^n,\ldots,{X_r}^n)=S_\ell(\sigma_1(X_1,\ldots,X_r),\ldots,\sigma_r(X_1,\ldots,X_r)).$$
In particular, one has:
$$\sigma_{\ell}({\alpha_1}^n,\ldots,{\alpha_r}^n)=S_\ell(\sigma_1(\alpha_1,\ldots,\alpha_r),\ldots,\sigma_r(\alpha_1,\ldots,\alpha_r)).$$
However, for all $k\in\{0,\cdots,r\}$, $\sigma_k(\alpha_1,\cdots,\alpha_r)$ is itself an integer as it is the the coefficient of $X^{n-k}$ in $f\in\mathbb{Z}[X]$ multiply by $(-1)^r$. Whence the result, since $\mathbb{Z}$ is closed under addition and multiplication.
The result you are trying to prove is a consequence of:

Theorem. (Kronecker) Let $f$ be a monic polynomial with integers coefficients whose complex roots are nonzero and lie in the unit disk, then the roots of $f$ are roots of unity.

Remark. The key point is that if $A$ is a commutative ring with a units, then $A[X_1,\cdots,X_n]^{S_n}$ is a finitely generated $A$-module. Moreover, the elementary symmetric polynomials are generators: $$\sum_{\substack{I\subseteq\{0,\cdots,n\}\\|I|=k}}\prod_{i\in I}X_i$$
