Computing generators of free modules over orders in group algebras
<p>Let <em>E</em> be a number field and <em>G</em> be a finite group. Let <em>A</em> be any <em>O<sub>E</sub></em>-order of full rank in the group algebra <em>E</em>[<em>G</em>] and <em>X</em> be a...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Elsevier
2008
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| _version_ | 1797073838314356736 |
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| author | Bley, W Johnston, H |
| author_facet | Bley, W Johnston, H |
| author_sort | Bley, W |
| collection | OXFORD |
| description | <p>Let <em>E</em> be a number field and <em>G</em> be a finite group. Let <em>A</em> be any <em>O<sub>E</sub></em>-order of full rank in the group algebra <em>E</em>[<em>G</em>] and <em>X</em> be a (left) <em>A</em>-lattice. We give a necessary and sufficient condition for <em>X</em> to be free of given rank <em>d</em> over <em>A</em>. In the case that the Wedderburn decomposition <em>E</em>[<em>G</em>]and#8773;and#8853;<sub>χ</sub><em>M</em>χ is explicitly computable and each <em>M</em><sub>χ</sub> is in fact a matrix ring over a field, this leads to an algorithm that either gives elements α<sub>1</sub>,…,α<sub>d</sub>∈<em>X</em> such that <em>X</em>=<em>A</em>α<sub>1</sub>and#8853;<sup>...</sup>and#8853;<em>A</em>α<sub>d</sub> or determines that no such elements exist.</p> <p>Let <em>L/K</em> be a finite Galois extension of number fields with Galois group <em>G</em> such that <em>E</em> is a subfield of <em>K</em> and put <em>d</em>=[<em>K</em>:<em>E</em>]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take <em>X</em> to be <em>O</em><sub>L</sub>, the ring of algebraic integers of <em>L</em>, and <em>A</em> to be the associated order <em>A</em>(<em>E</em>[<em>G</em>];<em>O</em><sub>L</sub>)⊆<em>E</em>[<em>G</em>]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when <em>K</em>=<em>E</em>=<em>Q</em>.</p> |
| first_indexed | 2024-03-06T23:27:46Z |
| format | Journal article |
| id | oxford-uuid:6afd47cf-07e0-4ee2-bc9d-9419cf69f010 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T23:27:46Z |
| publishDate | 2008 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:6afd47cf-07e0-4ee2-bc9d-9419cf69f0102022-03-26T19:00:53ZComputing generators of free modules over orders in group algebrasJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:6afd47cf-07e0-4ee2-bc9d-9419cf69f010EnglishSymplectic Elements at OxfordElsevier2008Bley, WJohnston, H<p>Let <em>E</em> be a number field and <em>G</em> be a finite group. Let <em>A</em> be any <em>O<sub>E</sub></em>-order of full rank in the group algebra <em>E</em>[<em>G</em>] and <em>X</em> be a (left) <em>A</em>-lattice. We give a necessary and sufficient condition for <em>X</em> to be free of given rank <em>d</em> over <em>A</em>. In the case that the Wedderburn decomposition <em>E</em>[<em>G</em>]and#8773;and#8853;<sub>χ</sub><em>M</em>χ is explicitly computable and each <em>M</em><sub>χ</sub> is in fact a matrix ring over a field, this leads to an algorithm that either gives elements α<sub>1</sub>,…,α<sub>d</sub>∈<em>X</em> such that <em>X</em>=<em>A</em>α<sub>1</sub>and#8853;<sup>...</sup>and#8853;<em>A</em>α<sub>d</sub> or determines that no such elements exist.</p> <p>Let <em>L/K</em> be a finite Galois extension of number fields with Galois group <em>G</em> such that <em>E</em> is a subfield of <em>K</em> and put <em>d</em>=[<em>K</em>:<em>E</em>]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take <em>X</em> to be <em>O</em><sub>L</sub>, the ring of algebraic integers of <em>L</em>, and <em>A</em> to be the associated order <em>A</em>(<em>E</em>[<em>G</em>];<em>O</em><sub>L</sub>)⊆<em>E</em>[<em>G</em>]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when <em>K</em>=<em>E</em>=<em>Q</em>.</p> |
| spellingShingle | Bley, W Johnston, H Computing generators of free modules over orders in group algebras |
| title | Computing generators of free modules over orders in group algebras |
| title_full | Computing generators of free modules over orders in group algebras |
| title_fullStr | Computing generators of free modules over orders in group algebras |
| title_full_unstemmed | Computing generators of free modules over orders in group algebras |
| title_short | Computing generators of free modules over orders in group algebras |
| title_sort | computing generators of free modules over orders in group algebras |
| work_keys_str_mv | AT bleyw computinggeneratorsoffreemodulesoverordersingroupalgebras AT johnstonh computinggeneratorsoffreemodulesoverordersingroupalgebras |