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Capitulation and Transfer Triples
Author(s) -
Gruenberg K. W.,
Weiss A.
Publication year - 2003
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611503014199
Subject(s) - mathematics , homomorphism , isomorphism (crystallography) , abelian group , transfer (computing) , galois group , combinatorics , algebraic number field , kernel (algebra) , ideal (ethics) , group (periodic table) , discrete mathematics , extension (predicate logic) , pure mathematics , physics , philosophy , chemistry , epistemology , quantum mechanics , parallel computing , computer science , crystal structure , crystallography , programming language
Let K / k be a finite unramified Galois extension of number fields with Galois group G . This determines two homomorphisms from the ideal class group Cl k of k : the capitulation map Cl k → C K and the Artin map C k ↠ G ab onto the abelianization G ab of G . We call (ker (capitulation), Cl k , Artin) the capitulation triple of K / k . Artin's transition to group theory shows that any triple ( X, Y , ζ) which arises in this way satisfies the group‐theoretic property of being a transfer triple for G , defined as follows: there exist a group extension A ↣ H ↠ G with A finite abelian and an isomorphism η : Y→ ∼H a bsuch that η( X ) is the kernel of the transfer homomorphism H ab → A , and ζ is the composite of η with H ab → G ab . When G is abelian, we show that a triple ( X, Y , ζ) is a transfer triple for G if and only if | G | X = 0 and | G | divides | X |. Whether all transfer triples for G can be realized arithmetically remains an unsolved problem. 2000 Mathematics Subject Classification 11R33, 20J99.