Artins

Artins is a term most often encountered in mathematics, used as a plural form to refer to objects and results named after Emil Artin, a German-born mathematician who became a leading figure in 20th‑century algebra and number theory. The surname is associated with several foundational concepts and theories that bear his name and are frequently cited in mathematical literature.

In particular, Artins contributions include several widely studied constructs. Artin reciprocity is a central theorem in global class field theory, linking abelian extensions of number fields to idèle class groups and Galois groups. Artin L-functions generalize Dirichlet L-functions by attaching complex analytic functions to representations of Galois groups, playing a key role in analytic number theory. The Artin symbol, or reciprocity symbol, generalizes Frobenius elements in Galois theory and appears in formulations of reciprocity laws. Artin groups, also known as Artin–Tits groups, are a broad class of groups defined by generators and relations that generalize braid groups and connect to Coxeter groups. Artin’s conjecture on primitive roots proposes a density statement about primes for which a given integer is a primitive root, a notable open problem in number theory with partial results known under additional hypotheses.

The term Artins thus functions as a shorthand for these and related ideas collectively associated with Emil Artin’s legacy. In literature and reference works, “Artins” may be used to indicate multiple concepts derived from or connected to Artin’s work, as well as discussions of the mathematician himself. See also Artin reciprocity, Artin L-functions, Artin symbol, and Artin groups.