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Class field theory, the study of abelian extensions of algebraic number fields, is one of the largest branches of algebraic number theory. It brings together the quadratic and higher reciprocity laws of Gauss, Legendre, and others, and vastly generalizes them. Some of its consequences (e.g., the Chebotarev density theorem) apply even to nonabelian extensions.
This book is an accessible introduction to class field theory. It takes a traditional approach in that it attempts to present the material using the original techniques of proof (global to local), but in a fashion which is cleaner and more streamlined than most other books on this topic. It could be used for a graduate course on algebraic number theory, as well as for students who are interested in self-study. The book has been class-tested, and the author has included exercises throughout the text.
List of contents
Preface.- A Brief Review.- Dirichlet's Theorem on Primes in Arithmetic Progressions.- Ray Class Groups.- The Idelic Theory.- Artin Reciprocity.- The Existence Theorem, Consequences and Applications.- Local Class Field Theory.- Bibliography.- Index.
Summary
Class field theory, the study of abelian extensions of algebraic number fields, is one of the largest branches of algebraic number theory. It brings together the quadratic and higher reciprocity laws of Gauss, Legendre, and others, and vastly generalizes them. Some of its consequences (e.g., the Chebotarev density theorem) apply even to nonabelian extensions.
This book is an accessible introduction to class field theory. It takes a traditional approach in that it attempts to present the material using the original techniques of proof (global to local), but in a fashion which is cleaner and more streamlined than most other books on this topic. It could be used for a graduate course on algebraic number theory, as well as for students who are interested in self-study. The book has been class-tested, and the author has included exercises throughout the text.
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From the reviews:
"This book has grown out of lectures on the subject by the author. ... it must have been fun for both the author presenting these courses on class field theory and the students taking them and eager to learn the subject. ... list of contents may give a good impression of how class field theory is developed in this book. ... each chapter is commenced by a short introduction describing what is going on next. I enjoyed seeing explicit examples and nice applications ... ." (Jürgen Ritter, Mathematical Reviews, Issue 2009 i)
"Class field theory studies abelian extensions of number fields and their completions. ... The clarity of the exposition and the many exercises ranging from routine to quite challenging problems make the book perfect for a first introduction to class field theory." (Franz Lemmermeyer, Zentralblatt MATH, Vol. 1165, 2009)
"In a fast ride, running half the length of many competing volumes, Childress (Arizona State) employs a balanced mix of standard tools for a remarkably honed introduction ... . Good to read alongside fleshier accounts; probably more accessible ... . Summing Up: Recommended. Upper-division undergraduates through researchers/faculty." (D. V. Feldman, Choice, Vol. 47 (3), November, 2009)
"This is a first introduction to class field theory. ... The author succeeds in making the material accessible by proceeding at a moderate pace. This relatively slim book is a good choice for anyone who wants to get an idea of what class field theory is about before tackling a more comprehensive textbook or monograph." (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)
