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A p-adic proof that pi is transcendental

May 16, 2015

Matt Baker's Math Blog

Lindemann Ferdinand von Lindemann

In my last blog post, I discussed a simple proof of the fact that pi is irrational.  That pi is in fact transcendental was first proved in 1882 by Ferdinand von Lindemann, who showed that if $latex alpha$ is a nonzero complex number and $latex e^alpha$ is algebraic, then $latex alpha$ must be transcendental.  Since $latex e^{i pi} = -1$ is algebraic, this suffices to establish the transcendence of $latex pi$ (and setting $latex alpha = 1$ it shows that $latex e$ is transcendental as well).  Karl Weierstrass proved an important generalization of Lindemann’s theorem in 1885.

The proof by Lindemann that pi is transcendental is one of the crowning achievements of 19th century mathematics.  In this post, I would like to explain a remarkable 20th century proof of the Lindemann-Weierstrass theorem due to Bezivin and Robba [Annals of Mathematics Vol. 129, No. 1 (Jan…

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