Infinite products with algebraic numbers

We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{\alpha_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty \frac{b_{n,m}}{\alpha_{n,m}}\right)$, where the $\alpha_n$ and $\alpha_{n,m}$ are assumed to be algebraic integers, and the $b_n$ and $b_{n,m}$ are natural numbers. In each case, we give a lower bound of the degree over the smallest extension of $\mathbb{Q}$ containing all algebraic numbers in the expression. The criteria obtained depend on growth conditions on the involved quantities.