On the resolution of index form equations in quartic number fields

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 A 4 and discriminant < 10 6 (31 fields), (ii) in the 50 totally real fields with smallest discriminant and Galois group S 4, (iii) in the 50 quartic fields with mixed signature and smallest absolute discriminant, (iv) and in all totally complex quartic fields with discriminant < 10 6 and Galois group A 4 (90 fields) or S 4 (44122 fields).