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 …

At the end of the paper we present numerical tables. We computed mininnai indices and all
elements of minimai iridex (i) in all totally real quartic fields with Galois group A and
discriminant< 10 (31 fields),(ii) in the 50 totally real fiełds with smallest discriminant and
Galois group S4,(iii) in the 5G quartic fields with mixed signature and smallest absolute
discriminart,(iv) and in ail totally complex quartic fields with discriminant< 10 and Galois
group A4 (90 fields) or S4 (44122 fields),