Class field theory

Class field theory describes abelian extensions of number fields and local fields through arithmetic invariants, especially ideal classes and reciprocity maps.

Artin reciprocity: The reciprocity law identifying abelian Galois groups with quotients of arithmetic groups. It is the central correspondence connecting ideals or idèles to automorphisms of abelian extensions.
Ideal class group: The group of nonzero fractional ideals modulo principal fractional ideals. Its elements measure the obstruction to unique factorization and identify unramified abelian extensions.
Quadratic reciprocity: A theorem relating whether one prime is a square modulo another to a reciprocal residue condition. Its generalizations became the reciprocity laws that class field theory unifies.
Kronecker–Weber theorem: The theorem that every finite abelian extension of the rational numbers lies in a cyclotomic field. It is the explicit classification of abelian extensions over the simplest number field.
Inverse Galois problem: The problem of realizing every finite group as the Galois group of an extension of a given field. Class field theory solves the abelian case, while the general problem concerns nonabelian groups.
Frobenius element: A Galois group element describing how an unramified prime acts on residue fields. Artin reciprocity sends unramified prime ideals to their Frobenius elements.
Galois group: The group of field automorphisms fixing a specified base field. Class field theory characterizes abelian extensions through their Galois groups.
David Hilbert: A German mathematician whose work shaped algebra, number theory, and mathematical physics. His ninth problem asked for a general reciprocity law, helping set the theory's agenda.
Cyclotomic field: A field obtained by adjoining roots of unity to a base field. Cyclotomic fields provide concrete examples of abelian extensions and reciprocity.
Nonabelian class field theory: The search for frameworks describing nonabelian extensions through arithmetic and representation-theoretic data. It extends the ambition of class field theory beyond abelian Galois groups.

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