Infinite products with algebraic numbers

Abstract

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