Computing subfields of number fields and applications to Galois group computations. (July 2019)

• Record Type: Journal Article
• Authors: Elsenhans, Andreas-Stephan; Klüners, Jürgen
• Is Part Of: Journal of symbolic computation. Volume 93(2019)
• Page Start: 1
• Page End: 20
• Publication Date: 2019-07
• Subjects: Subfields -- Galois group -- Number fields -- LLL; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2018.04.013
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: A polynomial time algorithm to find generators of the lattice of all subfields of a given number field was given invan Hoeij et al. (2013) . This article reports on a massive speedup of this algorithm. This is primary achieved by our new concept of Galois-generating subfields . In general this is a very small set of subfields that determine all other subfields in a group-theoretic way. We compute them by targeted calls to the method fromvan Hoeij et al. (2013) . For an early termination of these calls, we give a list of criteria that imply that further calls will not result in additional subfields. Finally, we explain how we use subfields to get a good starting group for the computation of Galois groups.