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Computing Generators of Free Modules over Orders in Group Algebras
Zusammenfassung
Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] \cong \oplus_xM_x is explicitly computable and each M_x is in fact a matrix ring over a field, this leads to an algorithm that either gives elements \alpha_1,...,\alpha_d \in X such that X = A\alpha_1 \oplus ... \oplusA\alpha_d or determines that no such elements exist. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A(E[G];O_L) \subseteq E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = \IQ.
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@article{urn:nbn:de:hebis:34-2008022620483,
author={Bley, Werner and Johnston, Henri},
title={Computing Generators of Free Modules over Orders in Group Algebras},
year={2007}
}
0500 Oax 0501 Text $btxt$2rdacontent 0502 Computermedien $bc$2rdacarrier 1100 2007$n2007 1500 1/eng 2050 ##0##urn:nbn:de:hebis:34-2008022620483 3000 Bley, Werner 3010 Johnston, Henri 4000 Computing Generators of Free Modules over Orders in Group Algebras / Bley, Werner 4030 4060 Online-Ressource 4085 ##0##=u http://nbn-resolving.de/urn:nbn:de:hebis:34-2008022620483=x R 4204 \$dPreprint 4170 Mathematische Schriften Kassel ;; 07, 07 7136 ##0##urn:nbn:de:hebis:34-2008022620483
2008-02-26T10:53:02Z 2008-02-26T10:53:02Z 2007 urn:nbn:de:hebis:34-2008022620483 http://hdl.handle.net/123456789/2008022620483 274689 bytes 140309 bytes application/pdf application/pdf eng Urheberrechtlich geschützt https://rightsstatements.org/page/InC/1.0/ Galois module structure of rings of integers associated orders algebraic number theory computations 510 Computing Generators of Free Modules over Orders in Group Algebras Preprint Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] \cong \oplus_xM_x is explicitly computable and each M_x is in fact a matrix ring over a field, this leads to an algorithm that either gives elements \alpha_1,...,\alpha_d \in X such that X = A\alpha_1 \oplus ... \oplusA\alpha_d or determines that no such elements exist. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A(E[G];O_L) \subseteq E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = \IQ. open access Bley, Werner Johnston, Henri Mathematische Schriften Kassel ;; 07, 07 11R33 11Y40 Mathematische Schriften Kassel 07, 07
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