The student
- has good knowledge of the main theorems of class field theory: the Artin Reciprocity, the Existence Theorem and the Classification Theorem
- has a good understanding of the main ingredients of the proofs: L-series, completion and Galois cohomology
- can derive classical reciprocity laws from the Artin Reciprocity Law
- can apply the theorems of class field theory to various concrete abelian field extensions of number fields
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Class field theory is the study of abelian extensions of global and local fields. The emphasis in this course is on number fields. The central theme is a correspondence between abelian extensions of a number field and generalized ideal class groups of that field. The way these structures are connected leads to the Artin Reciprocity Law. It subsumes a variety of reciprocity laws in number theory, the simplest and most well-known of these being the Law of Quadratic Reciprocity.
Class field theory has a rich history and by now there are many routes to the main theorems of class field theory. In this course the theory is presented along the classical lines of Tagaki, Hecke and Artin. This is done using modern language and techniques. The main ingredients are Dirichlet series, the completion of number fields, and the Galois cohomology of cyclic groups. A good understanding of this approach is helpful in the study of later developments in class field theory.
Instructional Modes
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Algebraic Number Theory up to the level of decomposition and inertia groups of primes in a Galois number field extension
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Student presentations and two homework assignments |
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