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Gauss famously referred to mathematics as the "queen of the sciences" and to number theory as the "queen of mathematics". This book is an introduction to algebraic number theory, meaning the study of arithmetic in finite extensions of the rational number field Q. Originating in the work of Gauss, the foundations of modern algebraic number theory are due to Dirichlet, Dedekind, Kronecker, Kummer, and others. This book lays out basic results, including the three "fundamental theorems" : unique factorization of ideals, finiteness of the class number, and Dirichlet's unit theorem. While these theorems are by now quite classical, both the text and the exercises allude frequently to more recent developments. In addition to traversing the main highways, the book reveals some remarkable vistas by exploring scenic side roads. Several topics appear that are not present in the usual introductory texts. One example is the inclusion of an extensive discussion of the theory of elasticity, which provides a precise way of measuring the failure of unique factorization. The book is based on the author's notes from a course delivered at the University of Georgia ; pains have been taken to preserve the conversational style of the original lectures.
|DATE DE PUBLICATION||2017-Aug-01|
|TAILLE DU FICHIER||4,77 MB|
|NOM DE FICHIER||A Conversational Introduction to Algebraic Number Theory - Arithmetic Beyond Z.pdf|
to Algebraic Number Theory Arithmetic Beyond Z Paul Pollack, University of Georgia, Athens Written in a conversational style, this introduction to algebraic number theory lays out the three classical "fundamental theorems": unique factorization of ideals, finiteness of the class number, and Dirichlet's unit theorem, while also
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields.These properties, such as whether a ring admits ...