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| Artin induction | |
|---|---|
| Name | Artin induction |
| Field | Algebraic number theory; Representation theory |
| Introduced | 1920s |
| Introduced by | Emil Artin |
| Related | Frobenius reciprocity; Brauer induction; Class field theory |
Artin induction is a central theorem in representation theory and algebraic number theory giving an expression of certain characters as integer combinations of induced characters from cyclic subgroups. It connects work on Ferdinand Frobenius-type characters, Emil Artin's reciprocity ideas, and the structure of Galois group representations arising in the study of number field extensions and L-functions. The theorem provides both conceptual insight and practical tools used in proofs concerning conductors, ramification, and non-vanishing of Artin L-series.
The statement of the theorem asserts that for a finite group G and any complex-valued character χ that is trivial on a normal subgroup with abelian quotient, there exists an integer N > 0 such that N·χ is a Z-linear combination of characters induced from 1-dimensional characters of cyclic subgroups of G. The conclusion is often phrased: each rational-valued character of G is an integral combination of characters induced from linear characters of cyclic subgroups. This relates to the classification of irreducible representations considered by William Burnside and the study of characters that appear in the context of Galois group actions on field extensions studied by Richard Dedekind and David Hilbert.
Artin induction grew out of efforts in the early 20th century to understand reciprocity laws and the representation-theoretic content of field extensions studied by Emil Artin and contemporaries. The theorem followed development in character theory initiated by Ferdinand Frobenius and consolidated by Issai Schur and F.E. Molien; it responded to needs in analytic studies of L-functions and their factorization into Euler products for non-abelian extensions, a problem central to work by John Tate and later contributors such as Haruzo Hida and Andrew Wiles. The result also complemented Richard Brauer's later induction theorem, and its influence permeates applications in class field theory as explored by Henri Poincaré-era successors and modern researchers in Iwasawa theory.
Proofs of the theorem use elementary character-theoretic manipulations, induction and restriction functors, and averaging over conjugacy classes, employing tools developed by Ferdinand Frobenius and formalized by Isaac Jacob Schoenberg in character orthogonality techniques. A key idea is to construct idempotent combinations of induced characters from cyclic subgroups to approximate given rational characters; passing to a common multiple N clears denominators arising from character inner products. The argument frequently invokes Frobenius reciprocity and properties of characters under induction from index-subgroups studied by William Burnside and refined in systematic treatments by George Mackey and Jacobson. Alternative proofs leverage cohomological perspectives tied to the cohomology of groups as developed by Samuel Eilenberg and Saunders Mac Lane.
Artin induction has multiple consequences across representation theory and algebraic number theory. It allows reduction of questions about arbitrary characters to questions about 1-dimensional characters of cyclic subgroups, streamlining proofs concerning conductor-discriminant relations in the study of Artin L-series and facilitating analytic continuation arguments as used in work by Erich Hecke and Atle Selberg. The theorem underpins Brauer’s approach to expressing zeta and L-functions of non-abelian extensions via abelian pieces, a technique exploited in proofs by Heinrich Weber-inspired investigations and in modularity arguments influenced by Pierre Deligne. In algebraic topology, analogous induction principles influence calculations in equivariant cohomology related to notions introduced by G. W. Mackey and later used in equivariant stable homotopy theory by researchers influenced by J. P. May.
Several variants strengthen or adapt Artin induction. Brauer induction expresses arbitrary characters as Z-linear combinations of characters induced from 1-dimensional characters of elementary subgroups, as developed by Richard Brauer and refined in later expositions by David Benson and J. L. Alperin. There are p-local and completed versions important in local field contexts and Iwasawa-theoretic settings considered by John Coates and Kenkichi Iwasawa. Equivariant generalizations appear in the context of K-theory and equivariant stable homotopy, where induction principles analogous to Artin’s are formulated by authors in the tradition of Daniel Quillen and Graeme Segal.
Concrete examples illustrate the theorem for small groups: for cyclic groups C_n induced characters coincide with linear characters, and Artin induction is trivial in this case; for dihedral groups D_n one expresses 2-dimensional irreducible characters as integer combinations of induced linear characters from cyclic subgroups of order n or 2, methods reminiscent of calculations by Edward T. Whittaker-era analysts. Computational implementations exploit character tables compiled in projects like the Atlas of Finite Groups and algorithms influenced by computational algebra systems developed at institutions such as Max Planck Institute for Mathematics and University of Cambridge research groups. Explicit computations for Galois groups of small degree number fields use databases curated by teams at University of Bordeaux and Princeton University to verify induction decompositions in examples arising from classical extensions studied by Évariste Galois and later cataloged by modern computational number theorists.