Norm relations and computational problems in number fields
| Autor: | Biasse, Jean-François, Fieker, Claus, Hofmann, Tommy, Page, Aurel |
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| Přispěvatelé: | University of South Florida [Tampa] (USF), Technische Universität Kaiserslautern (TU Kaiserslautern), Technical University of Kaiserslautern (TU Kaiserslautern), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Lithe and fast algorithmic number theory (LFANT), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-19-CE48-0008,CIAO,Cryptographie, isogenies et variété abéliennes surpuissantes(2019), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest |
| Rok vydání: | 2022 |
| Předmět: |
[INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC]
Primary: 11Y16 20C05 11R32 Secondary 11R29 11R04 11Y40 11R18 11R27 Mathematics - Number Theory [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT] Mathematics::Number Theory General Mathematics FOS: Mathematics Number Theory (math.NT) [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] |
| Zdroj: | Journal of the London Mathematical Society Journal of the London Mathematical Society, In press, ⟨10.1112/jlms.12563⟩ |
| ISSN: | 1469-7750 0024-6107 |
| Popis: | International audience; For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb{Q}[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the arithmetic invariants of the subfields of an algebraic number field with Galois group $G$. On the algorithm side this leads to subfield based algorithms for computing rings of integers, $S$-unit groups and class groups. For the $S$-unit group computation this yields a polynomial-time reduction to the corresponding problem in subfields. We compute class groups of large number fields under GRH, and new unconditional values of class numbers of cyclotomic fields. |
| Databáze: | OpenAIRE |
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