A Course in complex analysis & Riemann Surfaces by Wilhelm Schlag ISBN 9781470454722
Gauss famously referred to mathematics as the “queen of the sciences” and to number theory as the “queen o mat emancs . this book is an introduction to algebraic number theory, meaning the study of arthmetic in finite extensions of the rational number field Q. Onginating m the work of Gauss, the foundations of modern algebraic number theory are due to Dirichlet, Dedekind, Kronecker, Kummer, and others. This b0ok lays out basic results, including the three “fundamental theorems”: un!que factorization_ of ideals, finiteness of the class number, arid 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 highways, the book reveals remarkable vistas by exploring scenic sides 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.