I was reading a proof on the fact that $\pi$ is transcendental and the author mentioned that
The $\alpha_i$ satisfy a polynomial equation over $\mathbb{Q}$ so their elementary symmetric functions are rational. Hence the elementary symmetric functions of the sums of pairs are symmetric functions of the $\alpha_i$ and are also rational.
I'm not quite sure why the elementary symmetric functions of the sums of pairs are rational given that the elementary symmetric functions of the $\alpha_i$ are rational. For instance, the 2nd elementary symmetric function of the sum of pairs contains $\alpha_i^2$, and how can we deal with these terms?
EDIT: The proof can be found here: https://sixthform.info/maths/files/pitrans.pdf (Theorem 4).
The author is using the fundamental theorem of symmetric polynomials, according to which the $A$-subalgebra of $A[X_1,\cdots,X_n]$ generated by the $n$ elementary symmetric polynomials is the ring of symmetric polynomials with coefficients in $A$ (in this case, it's $A=\Bbb Z$ or, at worst, $\Bbb Q$).