I know that it hasn't been shown that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$. However it must be easier to show that the fields $\mathbb{Q}(\pi)$ and $\mathbb{Q}(e)$ are not the same. Yet, I have no ideas on how to approach this problem. Does anyone can show a proof or give any useful hint?
2026-03-28 19:39:56.1774726796
How to prove $\mathbb{Q}(\pi)$ and $\mathbb{Q}(e)$ are not the same field?
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This is an open problem. For all we know, it is possible that $e+\pi$ is a rational in which case the fields would clearly be the same.