On the resolution of index form equations in quartic number fields

István Gaál, Attila Petho, Michael Pohst

Research output: Contribution to journalArticle

27 Citations (Scopus)

Abstract

In this paper we reduce the problem of solving index form equations in quartic number fields K to the resolution of a cubic equation F (u, v) = i and a corresponding system of quadratic equations Q1 (x, y, z) = u, Q2(x, y, z) = v, where F is a binary cubic form and Q1, Q2 are ternary quadratic forms. This enables us to develop a fast algorithm for calculating “small” solutions of index form equations in any quartic number field. If, additionally, the field K is totally complex we can combine the two forms to get an equation T(x, y, z) = To with a positive definite quadratic form T(x, y, z). Hence, in that case we obtain a fast method for the complete resolution of index form equations. At the end of the paper we present numerical tables. We computed minimal indices and all elements of minimal index. (i) in all totally real quartic fields with Galois group A4 and discriminant < 106 (31 fields),. (ii) in the 50 totally real fields with smallest discriminant and Galois group S4,. (iii) in the 50 quartic fields with mixed signature and smallest absolute discriminant,. (iv) and in all totally complex quartic fields with discriminant < 106 and Galois group A4 (90 fields) or S4 (44122 fields).

Original languageEnglish
Pages (from-to)563-584
Number of pages22
JournalJournal of Symbolic Computation
Volume16
Issue number6
DOIs
Publication statusPublished - Dec 1993

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics

Fingerprint Dive into the research topics of 'On the resolution of index form equations in quartic number fields'. Together they form a unique fingerprint.

  • Cite this