is a powerful tool in algebraic number theory. It describes abelian extensions of global and local fields, providing insights into field arithmetic and solving complex problems in number theory and arithmetic geometry.
This theory enables us to construct and analyze important objects like the . It also has practical applications in solving , computing , and studying in number fields.
Class Field Theory Applications
Solving Problems in Algebraic Number Theory and Arithmetic Geometry
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algebraic number theory - On Hilbert Class Polynomial - Mathematics Stack Exchange View original
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elliptic curves - Solving Quadratic equations in Galois Field (2^163) - Cryptography Stack Exchange View original
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Class field theory comprehensively describes abelian extensions of global and local fields
Serves as a powerful tool for solving various problems in algebraic number theory and arithmetic geometry
establishes correspondence between abelian extensions of a number field and quotients of its idele class group
Enables study of the of the maximal of a number field
Provides insights into the field's arithmetic properties
Hilbert class field constructed and analyzed using class field theory techniques
Maximal unramified abelian extension of a number field
Applications in arithmetic geometry include:
Study of rational points on
Investigation of
Provides tools for understanding behavior of prime ideals in abelian extensions
Theory of combines class field theory with elliptic curve theory
Significant applications in constructing abelian extensions of imaginary quadratic fields
Practical Applications and Problem-Solving
Solves Diophantine equations using abelian extensions
Determines solvability of certain algebraic equations over number fields
Computes class numbers and unit groups of number fields
Constructs number fields with specific Galois groups
Analyzes the distribution of prime ideals in number fields