PlanetMath: Lindemann-Weierstrass theorem

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Lindemann-Weierstrass theorem

(Theorem)

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 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:

If $\alpha$ is a non-zero algebraic number over $\mathbb{Q}$ , then $e^{\alpha}$ is transcendental over $\mathbb{Q}$ .
is transcendental over $\mathbb{Q}$ .
$\pi$ is transcendental over $\mathbb{Q}$ . As a result, it is impossible to “square the circle”!

It is easy to see that $\pi$ is transcendental over $\mathbb{Q}(e)$ iff is transcendental over $\mathbb{Q}(\pi)$ iff $\pi$ and are algebraically independent. However, whether $\pi$ and are algebraically independent 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 and $\pi$ over $\mathbb{Q}$ can be shown.

"Lindemann-Weierstrass theorem" is owned by CWoo.

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Other names:

Lindemann's theorem

Attachments:: proof of Lindemann-Weierstrass theorem and that e and $\pi$ are transcendental (Proof) by rm50
equivalent statements of Lindemann-Weierstrass theorem (Result) by CWoo

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Cross-references: algebraic independence, Schanuel's conjecture, open question, iff, easy to see, transcendental, consequences, equivalent, algebraically independent, algebraic numbers, linearly independent
There are 4 references to this entry.

This is version 8 of Lindemann-Weierstrass theorem, born on 2004-04-21, modified 2006-03-28.
Object id is 5791, canonical name is LindemannWeierstrassTheorem.
Accessed 3837 times total.

Classification:

AMS MSC:	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
	11J85 (Number theory :: Diophantine approximation, transcendental number theory :: Algebraic independence; Gelfond's method)

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