Hermite-Lindemann-Weierstrass Theorem
(Redirected from Lindemann-Weierstrass Theorem)
Jump to navigation
Jump to search
Theorem
Let $a_1, \cdots, a_n$ be algebraic numbers (possibly complex) that are linearly independent over the rational numbers $\Q$.
Then:
- $e^{a_1}, \cdots, e^{a_n}$ are algebraically independent.
where $e$ is Euler's number.
Weaker
Let $a$ be a non-zero algebraic number (possibly complex).
Then:
- $e^a$ is transcendental
where $e$ is Euler's number.
Proof
![]() | This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also known as
Also known as:
- The Lindemann-Weierstrass Theorem
- The Hermite-Lindemann Theorem
Source of Name
This entry was named for Charles Hermite, Carl Louis Ferdinand von Lindemann and Karl Theodor Wilhelm Weierstrass.