In the following linked proof for the transcendence of pi https://fermatslibrary.com/s/the-transcendence-of-pi, the author uses the notation $\theta_1(x)=(e^{\alpha_1}+1)(e^{\alpha_2}+1)(e^{\alpha_3}+1)...(e^{\alpha_n}+1)=0$ where $\alpha_1=i\pi, \alpha_2, \alpha_3,...,\alpha_n$.
I'm unsure what the subscript $1$ in $\theta_1(x)$ is referring to here, how would it differ from $\theta_2(x)$ and how exactly $\theta(x)$ is a function of $x$?
Lastly, can someone just explain how $\theta_1(x)=(e^{\alpha_1}+1)(e^{\alpha_2}+1)(e^{\alpha_3}+1)...(e^{\alpha_n}+1)=0$ implies that the elementary symmetric polynomials of $\alpha_1, \alpha_2, \alpha_3,...,\alpha_n$ are all rational numbers because from my understanding, the line $\alpha_1=i\pi, \alpha_2, \alpha_3,...,\alpha_n$ means that a symmetric polynomial such as $\displaystyle \sum_{1≤j<i≤n}\alpha_i\alpha_j=\binom{n}{2}\pi^{2}$, which is clearly not rational.