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Classes de Steinitz d'extensions quaternioniennes généralisées de degré 4 p r
Author(s) -
Carter James E.,
Sodaïgui Bouchaïb
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm053
Subject(s) - combinatorics , mathematics , order (exchange) , galois group , group (periodic table) , algebraic number field , class number , root of unity , natural number , physics , stereochemistry , discrete mathematics , chemistry , geometry , quadratic equation , finance , quantum mechanics , economics , quantum
Let p be an odd prime number, r a natural number, and H 4 p rthe generalized quaternion group of order 4 p r . Let k be a number field and Cl( k ) its class group. Let R m ( k , H 4 p r) be the subset of Cl( k ) consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of k with Galois group isomorphic to H 4 p r. In this article, we determine R m ( k , H 4 p r) and show that it is a subgroup of Cl( k ). In particular, R m ( k , H 4 p r) is the full group Cl( k ) if p does not divide the class number of k , or if either k contains a primitive p r th root of unity or p is unramified in k .