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LindemannWeierstrass 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 nonzero algebraic number over $\mathbb{Q}$, then $e^{{\alpha}}$ is transcendental over $\mathbb{Q}$.

$e$ is transcendental over $\mathbb{Q}$.
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, whether $\pi$ and $e$ are algebraically independent is still an open question today.
Schanuel’s conjecture is a generalization of the LindemannWeierstrass theorem. If Schanuel’s conjecture were proven to be true, then the algebraic independence of $e$ and $\pi$ over $\mathbb{Q}$ can be shown.
Mathematics Subject Classification
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