Infinite products with algebraic numbers

We obtain general criteria for giving a lower bound on the degree of numbers of the form ${\prod }_{n=1}^{\mathrm{\infty }}(1+\frac{b_n}{{\alpha }_n})$ or of the form ${\prod }_{m=1}^{\mathrm{\infty }}(1+{\sum }_{n=1}^{\mathrm{\infty }}\frac{b_{n,m}}{{\alpha }_{n,m}})$, 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.