Pillars Of Transcendental Number Theory
Download Pillars Of Transcendental Number Theory PDF/ePub or read online books in Mobi eBooks. Click Download or Read Online button to get Pillars Of Transcendental Number Theory book now. This website allows unlimited access to, at the time of writing, more than 1.5 million titles, including hundreds of thousands of titles in various foreign languages.
Pillars of Transcendental Number Theory
This book deals with the development of Diophantine problems starting with Thue's path breaking result and culminating in Roth's theorem with applications. It discusses classical results including Hermite–Lindemann–Weierstrass theorem, Gelfond–Schneider theorem, Schmidt’s subspace theorem and more. It also includes two theorems of Ramachandra which are not widely known and other interesting results derived on the values of Weierstrass elliptic function. Given the constantly growing number of applications of linear forms in logarithms, it is becoming increasingly important for any student wanting to work in this area to know the proofs of Baker’s original results. This book presents Baker’s original results in a format suitable for graduate students, with a focus on presenting the content in an accessible and simple manner. Each student-friendly chapter concludes with selected problems in the form of “Exercises” and interesting information presented as “Notes,” intended to spark readers’ curiosity.
Abstract Algebra and Famous Impossibilities
This textbook develops the abstract algebra necessary to prove the impossibility of four famous mathematical feats: squaring the circle, trisecting the angle, doubling the cube, and solving quintic equations. All the relevant concepts about fields are introduced concretely, with the geometrical questions providing motivation for the algebraic concepts. By focusing on problems that are as easy to approach as they were fiendishly difficult to resolve, the authors provide a uniquely accessible introduction to the power of abstraction. Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach. Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.
Perfect Powers—An Ode to Erdős
The book explores and investigates a long-standing mathematical question whether a product of two or more positive integers in an arithmetic progression can be a square or a higher power. It investigates, more broadly, if a product of two or more positive integers in an arithmetic progression can be a square or a higher power. This seemingly simple question encompasses a wealth of mathematical theory that has intrigued mathematicians for centuries. Notably, Fermat stated that four squares cannot be in arithmetic progression. Euler expanded on this by proving that the product of four terms in an arithmetic progression cannot be a square. In 1724, Goldbach demonstrated that the product of three consecutive positive integers is never square, and Oblath extended this result in 1933 to five consecutive positive integers. The book addresses a conjecture of Erdős involving the corresponding exponential Diophantine equation and discusses various number theory methods used to approach a partial solution to this conjecture. This book discusses diverse ideas and techniques developed to tackle this intriguing problem. It begins with a discussion of a 1939 result by Erdős and Rigge, who independently proved that the product of two or more consecutive positive integers is never a square. Despite extensive efforts by numerous mathematicians and the application of advanced techniques, Erdős' conjecture remains unsolved. This book compiles many methods and results, providing readers with a comprehensive resource to inspire future research and potential solutions. Beyond presenting proofs of significant theorems, the book illustrates the methodologies and their limitations, offering a deep understanding of the complexities involved in this mathematical challenge.