Infinite products with algebraic numbers

TL;DR

This paper establishes criteria to determine lower bounds on the degree of certain infinite products involving algebraic integers, based on growth conditions of their components.

Contribution

It introduces general criteria for bounding the degree of algebraic numbers formed by infinite products with algebraic integer factors, depending on growth conditions.

Findings

Provides lower bounds on degrees of infinite product numbers.

Applies to products with algebraic integer factors and natural numbers.

Depends on growth conditions of the involved sequences.

Abstract

We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{\alpha_n}\right)$ 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.