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Über dieses Buch

This exposition is primarily a survey of the elementary yet subtle innovations of several mathematicians between 1929 and 1934 that led to partial and then complete solutions to Hilbert’s Seventh Problem (from the International Congress of Mathematicians in Paris, 1900). This volume is suitable for both mathematics students, wishing to experience how different mathematical ideas can come together to establish results, and for research mathematicians interested in the fascinating progression of mathematical ideas that solved Hilbert’s problem and established a modern theory of transcendental numbers.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Hilbert’s seventh problem: Its statement and origins

Abstract
At the second International Congress of Mathematicians in Paris, in 1900, the mathematician David Hilbert was invited to deliver a keynote address, just as Henri Poincaré had been invited to do at the first International Congress of Mathematicians in Zurich in 1896.
Robert Tubbs

Chapter 2. The transcendence of e, π and

Abstract
The fantasy calculation at the end of the last chapter, a fantasy because the linear combination of the intermediate sums.
Robert Tubbs

Chapter 3. Three partial solutions

Abstract
We recall from Hilbert’s address that he considered it to be a very difficultproblem to prove that the expression α β , for an algebraic base and an irrational algebraic exponent.
Robert Tubbs

Chapter 4. Gelfond’s solution

Abstract
Before we consider Gelfond’s and Schneider’s complete solutions to Hilbert’s seventh problem let’s look back and see what common elements we can find in Fourier’s demonstration of the irrationality of e, the Hermite/Hurwitz demonstration of the transcendence of e, and Gelfond’s proof of the transcendence of e π .
Robert Tubbs

Chapter 5. Schneider’s solution

Abstract
In this chapter we will briey examine Schneider’s solution [25] to Hilbert’s seventh problem, which appeared within a few months of Gelfond’s. (The story goes that Schneider learned of Gelfond’s solution as he was submitting his own paper for publication.) Like Gelfond’s proof, Schneider’s depended on an application of the pigeonhole principle, elementary complex analysis, and the fundamental fact that the algebraic norm of a nonzero algebraic integer is a nonzero rational integer.
Robert Tubbs

Chapter 6. Hilbert’s seventh problem and transcendental functions

Abstract
So far we have not said much about an important portion of Hilbert’s seventh problem.
Robert Tubbs

Chapter 7. Variants and generalizations

Abstract
There are several variants and/or extensions of the original Gelfond-Schneider Theorem that led to many significant advances in transcendental number theory in the twentieth century.
Robert Tubbs

Backmatter

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