Emil Artin

Emil Artin (1898–1962) was an Austrian-American mathematician whose work shaped modern algebra and number theory, especially class field theory, algebraic number theory, and the theory of braids.

Group theory: The study of algebraic structures called groups, which encode symmetry and reversible operations. Group theory supplies the language behind Artin’s work on braids, representations, and reciprocity.
Artin reciprocity law: A theorem relating ideal classes in a number field to automorphisms of its abelian extensions. It is Artin’s defining contribution to class field theory.
Richard Brauer: A German-American mathematician who founded modern modular representation theory. Brauer and Artin were leading figures in the development of representation theory.
Langlands program: A web of conjectures linking representations of Galois groups with automorphic forms and related structures. Artin L-functions and reciprocity provide important precedents for its arithmetic correspondences.
Kronecker–Weber theorem: The theorem that every finite abelian extension of the rational numbers lies in a cyclotomic field. It is the rational-field case that Artin’s reciprocity theory generalizes.
Field theory: The study of fields and their extensions, central to algebra and the arithmetic of polynomial roots. Artin’s algebraic number theory treats number fields as extensions of the rational numbers.
Artin symbol: A Galois group element associated with an unramified prime ideal in an abelian extension. The symbol encodes how prime ideals act in the reciprocity law.
Helmut Hasse: A German mathematician who made foundational contributions to algebraic number theory and class field theory. Hasse’s work and Artin’s reciprocity law jointly shaped class field theory.
Class field theory of local fields: The theory describing abelian extensions of a local field through its multiplicative group. Artin reciprocity was extended into the local theory that underpins modern formulations.
Quadratic reciprocity: A theorem describing when one odd prime is a square modulo another. Artin reciprocity extends the pattern of exchanging arithmetic data seen in this classical law.

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