Norm relations and computational problems in number fields
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866917975939874816 |
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| author | Biasse, Jean-François Fieker, Claus Hofmann, Tommy Page, Aurel |
| author_facet | Biasse, Jean-François Fieker, Claus Hofmann, Tommy Page, Aurel |
| contents | 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 a normal extension of algebraic number fields with Galois group $G$. On the algorithmic 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2002_12332 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Norm relations and computational problems in number fields Biasse, Jean-François Fieker, Claus Hofmann, Tommy Page, Aurel Number Theory Primary: 11Y16, 20C05, 11R32, Secondary 11R29, 11R04, 11Y40, 11R18, 11R27 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 a normal extension of algebraic number fields with Galois group $G$. On the algorithmic 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. |
| title | Norm relations and computational problems in number fields |
| topic | Number Theory Primary: 11Y16, 20C05, 11R32, Secondary 11R29, 11R04, 11Y40, 11R18, 11R27 |
| url | https://arxiv.org/abs/2002.12332 |