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Norm relations and computational problems in number fields

Abstract : 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 quasi-polynomial-time reduction to the corresponding problem in subfields. For families of numbers field where the $2$-valuation of $|G|$ is bounded, it is a polynomial-time reduction.
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https://hal.inria.fr/hal-02497890
Contributor : Aurel Page <>
Submitted on : Tuesday, March 3, 2020 - 10:39:45 PM
Last modification on : Thursday, March 5, 2020 - 1:18:21 AM

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  • HAL Id : hal-02497890, version 1
  • ARXIV : 2002.12332

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Jean-François Biasse, Claus Fieker, Tommy Hofmann, Aurel Page. Norm relations and computational problems in number fields. 2020. ⟨hal-02497890⟩

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