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| Artin map | |
|---|---|
| Name | Artin map |
| Field | Algebraic number theory |
| Introduced by | Emil Artin |
| Introduced in | 1927 |
| Related | Class field theory, Frobenius element, Galois group, Idèle class group |
Artin map
The Artin map is a central homomorphism in algebraic number theory connecting ideal-theoretic and Galois-theoretic objects. It appears in the formulation of Class field theory and links the arithmetic of number fields and local fields to the structure of Galois groups via Frobenius elements, idèles, and reciprocity laws. The map underlies major results associated with Emil Artin, Helmut Hasse, Claude Chevalley, and the development of global reciprocity linking splitting of primes to automorphisms.
The Artin map originates in the study of splitting behavior of prime ideals in extensions of Number fields and in the reciprocity laws generalizing the Law of quadratic reciprocity, the Kronecker–Weber theorem, and work of Évariste Galois. It provides a homomorphism from ray class groups, idèle class groups, or relative ideal groups to abelian Galois groups of finite extensions, encapsulating the action of Frobenius conjugacy classes such as those identified in Chebotarev's density theorem and in the formulation of Frobenius by Ferdinand Frobenius.
For a finite abelian extension L/K of Number fields, the Artin map is defined on the group of fractional ideals of K prime to the conductor or modulus and sends a prime ideal p of K to the Frobenius automorphism in Gal(L/K) determined by any prime P of L above p. The construction uses decomposition and inertia groups in the extension, along with Frobenius elements arising in unramified primes, mirroring concepts from Dedekind's ideal theory and the structure of Hilbert class field. Global constructions employ the idèle group of K as developed by Claude Chevalley and link to idèle class groups, while local constructions use local fields such as completions at places like p-adic numbers and rely on local reciprocity maps introduced by Helmut Hasse.
The Artin map is surjective with kernel equal to the norm group of ideals or idèles from L to K in the abelian setting, yielding an isomorphism between quotient class groups and Gal(L/K). It is compatible with restriction and corestriction maps between Galois groups in towers of fields, reflecting functoriality under field inclusion and under passage to subextensions as in the behavior of transfer maps and corestriction in Galois cohomology. Compatibility with the Frobenius substitution at unramified primes is central, and the map intertwines with characters such as Hecke characters and L-series studied by Erich Hecke and John Tate.
Locally, for a finite abelian extension of local fields F/E, the local Artin map gives an isomorphism from E^×/N_{F/E}(F^×) to Gal(F/E), connected to local class field theory as developed by Claude Chevalley, Helmut Hasse, John Tate, and Serre. Globally, the Artin map is defined on the idèle class group of a global field and surjects onto the abelianized Galois group of the maximal abelian extension, synthesizing results from the Takagi existence theorem and the Artin reciprocity law formulated by Emil Artin and refined by Teiji Takagi and Helmut Hasse.
In the cyclotomic setting of the Kronecker–Weber theorem, the Artin map identifies the ray class group modulo n with Gal(Q(ζ_n)/Q), sending rational primes p not dividing n to the automorphism ζ_n ↦ ζ_n^p. For quadratic extensions K = Q(√d), the Artin map reproduces classical quadratic reciprocity by associating Legendre or Kronecker symbols to Frobenius elements. Computations in imaginary quadratic fields relate to the Hilbert class field and complex multiplication studied by Kronecker and Weber, while explicit reciprocity in local fields employs the formalism used in computations by Iwasawa and in explicit class field constructions over p-adic numbers.
The Artin map provides the principal isomorphism of global class field theory: the idèle class group modulo the norm group maps onto Gal(L/K) for each finite abelian extension L/K, underpinning the classification of abelian extensions over global fields, including Number fields and Function fields over finite fields. It is essential in formulating L-functions and reciprocity laws for automorphic forms as developed in the work of Langlands, and it connects to explicit class field theory in contexts studied by Shimura, Tate, Iwasawa, and Weil.
The Artin map is named after Emil Artin, who articulated the reciprocity law now bearing his name in the 1920s, building on earlier contributions by Richard Dedekind, Ernst Kummer, Leopold Kronecker, and David Hilbert. Subsequent formalization and proofs involved Teiji Takagi's existence theorem, Helmut Hasse's work on local-global principles, and the idèlic reformulation by Claude Chevalley. Later conceptual frameworks were provided by John Tate, Jean-Pierre Serre, and André Weil in modern class field theory and by Robert Langlands in the context of non-abelian generalizations.