Sidney A. Morris – författare

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.

This book offers a gentle yet rigorous introduction to probability theory, with a special focus on finite probability spaces. Drawing inspiration from card games, casino games, mahjong, and two-up, it also delves into real-world applications such as weather forecasting, lotteries, hereditary diseases, and PCR virus testing. Discover which casino game gives you the best chance of winning and which one offers the worst odds.

Assuming only a high school mathematics background, this book is an excellent resource for both students and teachers, providing clear explanations and engaging examples. The technical material is lightened with entertaining stories, such as how someone became a millionaire by spotting a flaw in a national lottery and how another person helped fund a war using winnings from a well-known card game he invented.

Engaging and informative, this book is perfect for anyone looking to deepen their understanding of probability theory while enjoying some fascinating anecdotes along the way.

Dealing with subject matter of compact groups that is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics, this book – now in its third revised and augmented edition – has been conceived with the dual purpose of providing a text book for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups. After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups. However, the thrust of book points in the direction of the structure theory of infinite dimensional, not necessarily commutative compact groups, unfettered by weight restrictions or dimensional bounds. In the process it utilizes infinite dimensional Lie algebras and the exponential function of arbitrary compact groups.