Lindemann’s proof that pi is transcendental was published in 1882, nine years after Hermite’s proof that e is transcendental. Lindemann used Hermite’s methods, but generalised them considerably.
Lindemann showed that if z is complex and algebraic (the root of a polynomial with integer coefficients) then ez+1 cannot be zero. Since by Euler’s identity eiπ+1 is zero, iπ is not algebraic. From this it is easy to show that pi is not algebraic.
The University of St Andrews mathematical biography series says of this work of Lindemann:
‘Many historians of science regret that Hermite, despite doing most of the hard work, failed to make the final step to prove the result concerning which would have brought him fame outside the world of mathematics. This fame was instead heaped on Lindemann but many feel that he was a mathematician clearly inferior to Hermite who, by good luck, stumbled on a famous result. Although there is some truth in this, it is still true that many people make their own luck and in Lindemann’s case one has to give him much credit for spotting the trick which Hermite had failed to see.’
I feel this opinion is a little unjust. To achieve his result, Lindemann exploited the symmetry properties of the roots of integral algebraic equations in an extremely subtle way, and his results were far more general.
Rather than provide a proof within this blog, I have attached a pdf:
This proof follows the concise and elegant proof given by Baker in his book ‘Transcendental Number Theory’ but enlarges on some aspects, in particular those to do with the properties of symmetric functions. The proof of the ‘Fundamental Theorem of Symmetric Functions’, given as Lemma 3 in the pdf, closely follows that in Tignol’s book ‘Galois’ Theory of Algebraic Equations’. The reference to H&W is of course to Hardy and Wright’s book on Number Theory.