Gelfond's theorem GelfondsTheorem 2003-01-31 15:23:27 2004-02-12 11:53:34 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $\alpha$ and $\beta$ be algebraic over $\mathbb{Q}$, with $\beta$ irrational and $\alpha$ not equal to 0 or 1. Then $\alpha^{\beta}$ is transcendental over $\mathbb{Q}$. This is perhaps the most useful result in determining whether a number is algebraic or transcendental. The theorem is also known as the Gelfond-Schneider Theorem.