Class field theory
Applications

Class field theory is used to prove Artin-Verdier duality. Very explicit class field theory is used in many subareas of algebraic number theory such as Iwasawa theory and Galois modules theory.

Most main achievements toward the Langlands correspondence for number fields, the BSD conjecture for number fields, and Iwasawa theory for number fields use very explicit but narrow class field theory methods or their generalizations. The open question is therefore to use generalizations of general class field theory in these three directions.
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