Hermite-Lindemann-Weierstrass Theorem

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

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Also known as

The Hermite-Lindemann-Weierstrass Theorem is 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.