Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2016 | Book

Hilbert’s seventh problem: Its statement and origins Read chapter Read first chapter

Hilbert's Seventh Problem

Solutions and Extensions

Author: Robert Tubbs

Publisher: Springer Singapore

Book Series : HBA Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

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.

Table of Contents

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
Metadata
Title
Hilbert's Seventh Problem
Author
Robert Tubbs
Copyright Year
2016
Publisher
Springer Singapore
Electronic ISBN
978-981-10-2645-4
Print ISBN
978-981-10-2645-4
DOI
https://doi.org/10.1007/978-981-10-2645-4

Premium Partner

    Image Credits