Binary quartic forms with bounded invariants and small Galois groups

Abstract

In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms $f(x,y)$ of non-zero discriminant. Then, we shall enumerate the ${\mathrm{GL}}_2(\mathbb{Z})$-equivalence classes of all such forms associated to a fixed $f(x,y)$.

36 pages; revised to include new result distinguishing $D_4$-forms from $C_4$-forms

References in corpus (2)

• On binary cubic and quartic forms
• Quartic rings associated to binary quartic forms

Cited by in corpus (1)

• The number of quartic $D_4$-fields with monogenic cubic resolvent ordered by conductor