|
|
|
Viewing Version
7
of
'Lindemann-Weierstrass theorem'
|
[ view 'Lindemann-Weierstrass theorem'
|
back to history
]
| Title of object: |
Lindemann-Weierstrass theorem |
| Canonical Name: |
LindemannWeierstrassTheorem |
| Type: |
Theorem |
| Created on: |
2004-04-21 18:33:11 |
| Modified on: |
2006-03-28 10:22:38 |
| Classification: |
msc:12D99, msc:11J85 |
| Synonyms: |
Lindemann-Weierstrass theorem=Lindemann's theorem |
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:
If $\alpha_1,\ldots,\alpha_n$ are linearly independent algebraic numbers over $\mathbb{Q}$, then $e^{\alpha_1},\ldots,e^{\alpha_n}$ are algebraically independent over $\mathbb{Q}$.
An equivalent version of the theorem \PMlinkescapetext{states} that if $\alpha_1,\ldots,\alpha_n$ are distinct algebraic numbers over $\mathbb{Q}$, then $e^{\alpha_1},\ldots,e^{\alpha_n}$ are linearly independent over $\mathbb{Q}$.
Some immediate consequences of this theorem:
\begin{itemize}
\item
If $\alpha$ is a non-zero algebraic number over $\mathbb{Q}$, then $e^{\alpha}$ is transcendental over $\mathbb{Q}$.
\item
$e$ is transcendental over $\mathbb{Q}$.
\item
$\pi$ is transcendental over $\mathbb{Q}$. As a result, it is impossible to ``square the circle''!
\end{itemize}
It is easy to see that $\pi$ is transcendental over $\mathbb{Q}(e)$ iff $e$ is transcendental over $\mathbb{Q}(\pi)$ iff $\pi$ and $e$ are algebraically independent. However, this is still an open question today.
Schanuel's conjecture is a generalization of the Lindemann-Weierstrass theorem. If Schanuel's conjecture were proven to be true, then the algebraic independence of $e$ and $\pi$ over $\mathbb{Q}$ can be shown. |
|
|
|
|
|
