Computing with quadratic forms over number fields. (November 2018)

• Record Type: Journal Article
• Authors: Koprowski, Przemysław; Czogała, Alfred
• Is Part Of: Journal of symbolic computation. Volume 89(2018)
• Page Start: 129
• Page End: 145
• Publication Date: 2018-11
• Subjects: Algorithms -- Quadratic forms -- Number fields -- Level -- Pythagoras number -- Witt equivalence; 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.2017.11.009
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: This paper presents fundamental algorithms for the computational theory of quadratic forms over number fields. In the first part of the paper, we present algorithms for checking if a given non-degenerate quadratic form over a fixed number field is either isotropic (respectively locally isotropic) or hyperbolic (respectively locally hyperbolic). Next we give a method of computing the dimension of an anisotropic part of a quadratic form. The second part of the paper is devoted to algorithms computing two field invariants: the level and the Pythagoras number. Ultimately we present an algorithm verifying whether two number fields have isomorphic Witt rings (i.e. are Witt equivalent).