Artin Reciprocity is a central theorem in class field theory, establishing a profound connection between the abelian extensions of a number field and the field's idele class group, providing a generalization of quadratic reciprocity. It serves as a cornerstone for understanding the behavior of primes in extensions and lays the groundwork for further developments in algebraic number theory and the Langlands program.

Artin Reciprocity

Class field theory is a major branch of algebraic number theory that describes the abelian extensions of a number field in terms of the field's arithmetic properties, particularly its ideal class group. It provides a profound connection between field theory and group theory, serving as a foundation for understanding more complex non-abelian extensions in modern number theory.

Class Field Theory

Abelian extensions are a class of field extensions where the Galois group is an abelian group, meaning the group operation is commutative. These extensions are central to class field theory, which provides a comprehensive understanding of abelian extensions of number fields and their relation to ideal class groups and unit groups.

Abelian Extensions

A number field is a finite degree field extension of the rational numbers, providing a rich structure for studying algebraic numbers and their properties. Number fields are fundamental in algebraic number theory, as they generalize the rational numbers and allow for the exploration of Diophantine equations and prime factorization in more complex settings.

Number Field

Idele Class Group

Quadratic Reciprocity is a fundamental theorem in number theory that provides criteria to determine the solvability of quadratic equations modulo prime numbers. It reveals a surprising and profound symmetry in the behavior of quadratic residues, linking the solvability of one prime modulo another to the reverse scenario.

Quadratic Reciprocity

Behavior of primes

Algebraic number theory is a branch of number theory that studies the properties of algebraic numbers, which are roots of non-zero polynomial equations with rational coefficients. It connects number theory with abstract algebra, particularly through the use of field theory, Galois theory, and ring theory to solve problems related to integers and their generalizations.

Algebraic Number Theory

The Langlands program is a set of conjectures and theories that connect number theory and representation theory, proposing a grand unifying framework for understanding the relationships between Galois groups and automorphic forms. It has profound implications across mathematics, linking areas such as algebraic geometry, arithmetic geometry, and mathematical physics, and has led to significant advancements in these fields.

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