Quartic Fields

In the cubic case the index form equation was a cubic Thue equation. In quartic fields, the index form equation has already degree six and three variables. The resolution of such an equation can yield a difficult problem. The main goal of this chapter is to point out that in the quartic case the index form equation can be reduced to a cubic and some corresponding quartic Thue equations (see Sect. 9.1). This means that in fact the index form equations in quartic fields are not much harder to solve than in the cubic case. Index form equation In Sect. 9.2 we consider the infinite parametric family of simplest quartic fields. In Sect. 9.3 we give an application to the monogenity of mixed dihedral quartic fields. The general method of Sect. 9.1 simplifies a lot in totally complex quartic fields, see Sect. 9.4. A delicate case of quartic fields is represented by bicyclic biquadratic fields of type K = ℚ ( m , n ) $$K={\mathbb Q}(\sqrt {m},\sqrt {n})$$ , considered in Sect. 9.5. Finally, in Sect. 9.6 we show that index form equations in pure quartic fields lead to binomial Thue equations. Interesting tables about the distribution of minimal indices and about the average behavior of minimal indices of quartic fields can be found in the tables of Sects. 16.5.1 and 16.5.2 , respectively. In cyclic quartic fields the unit equation corresponding to the index form equation is especially simple, see the data of table of Sect. 16.5.3 .