Algebraically independent

Algebraic independence of known constants

• the numbers ${\displaystyle \pi }$, ${\displaystyle e^{\pi }}$, and ${\displaystyle \Gamma (1/4)}$, where ${\displaystyle \Gamma }$ is the gamma function, are algebraically independent over ${\displaystyle \mathbb {Q} }$.
• the numbers ${\displaystyle \pi ,e^{\pi {\sqrt {3}}}}$, and ${\displaystyle \Gamma (1/3)}$ are algebraically independent over ${\displaystyle \mathbb {Q} .}$
• for all positive integers n, the numbers ${\displaystyle \pi }$ and ${\displaystyle e^{\pi {\sqrt {n}}}}$ are algebraically independent over ${\displaystyle \mathbb {Q} .}$