Analytic proof of the Lindemann - Weierstrass Theorem

[GoutamTmv]

Hey guys,

I would like to know whether there exists a proof for the Lindemann-Weierstrass Theorem that uses only the tools and techniques of elementary analysis.

If such a proof does not exist, I would like to know what would be the mathematical knowledge required to understand the proof.

 

[DonAntonio]

Hey guys,

I would like to know whether there exists a proof for the Lindemann-Weierstrass Theorem that uses only the tools and techniques of elementary analysis.

If such a proof does not exist, I would like to know what would be the mathematical knowledge required to understand the proof.

In this case I think that's nearly impossible: the very wording of the L-W theorem includes the notion of transcendence and fields extensions.

DonAntonio

 

[GoutamTmv]

Well then, what would be the minimum knowledge required to understand the proof?

Thanks in advance

 

[mathwonk]

I would suggest warming up to the proof by learning the proof in spivak's calculus that pi is irrational.

then consult the little carus mathematical monograph: irrational numbers, by ivan niven, chapter IX. in this book complete statements of all needed prerequisites are either proved, or given with references to places where they are proved, such as the carus monograph on algebraic numbers by harry pollard.

 

[blue_raver22]

Thank you for your question. The Lindemann-Weierstrass Theorem, also known as the Lindemann's Transcendence Theorem, is a fundamental result in complex analysis that states that if α is a non-zero algebraic number, then e^α is transcendental. This theorem was first proved by Ferdinand von Lindemann in 1882 and later generalized by Karl Weierstrass in 1885.

To answer your first question, yes, there does exist an analytic proof of the Lindemann-Weierstrass Theorem. In fact, the original proof by Lindemann himself was an analytic proof using the tools of elementary analysis. The proof relies on the concept of Liouville's Theorem, which states that if a function is entire and bounded, then it must be constant. Lindemann used this result to show that if e^α were algebraic, then it would contradict Liouville's Theorem, thus proving the transcendence of e^α.

As for your second question, the mathematical knowledge required to understand the proof of the Lindemann-Weierstrass Theorem is a strong understanding of complex analysis and the properties of entire functions. This includes topics such as power series, Cauchy's Integral Theorem, and the residue theorem. A solid foundation in real analysis and algebraic number theory would also be helpful in understanding the proof.

I hope this answers your questions and provides some insight into the proof of the Lindemann-Weierstrass Theorem. This theorem has important implications in number theory and has been used to solve long-standing problems, making it a significant result in mathematics.