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'Gelfond's theorem'
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| Title of object: |
Gelfond's theorem |
| Canonical Name: |
GelfondsTheorem |
| Type: |
Theorem |
| Created on: |
2003-01-31 15:23:27 |
| Modified on: |
2003-09-12 15:36:11 |
| Classification: |
msc:11J81 |
Revision comment (for changes between this and next version):
| Changes for correction #2631 ('Typesetting'). |
Preamble:
% 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 |
Content:
Let $\alpha$ and $\beta$ be algebraic over $\mathbb{Q}$, with $\beta$ irrational and $\alpha$ not equal to 0 or 1.
Then
\gamma = {\alpha}^{\beta}
is transcendental over $\mathbb{Q}$.
This is perhaps the most useful result in determining whether a number is algebraic or transcendental. |
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