A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

Abstract

We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the way, we also prove that a positive proportion of quartic rings of integers do not arise as the invariant order of an integral binary quartic form despite having no local obstruction.

The scheme of monogenic generators I: representability

“…Using elliptic curves, [1] shows that a positive proportion of cubic number fields are not monogenic despite having no local obstructions. More recently, the trio have undertaken a similar investigation for quartic fields [2]. For quartic orders, Bhargava [8] also establishes a new upper bound on the number of essentially different monogenerators.…”

“…Iterating through the four possible values of (b, c) ∈ (Z/2Z) 2 shows that the index form always to reduces to 0.…”

“…By contrast, the extension of their rings of integers Z L /Z K may require up to log 2 ([L : K ]) elements of Z L to generate Z L as a Z K -algebra [57]. This paper is motivated by the observation that monogenicity 2 of an algebra can be restated geometrically:…”

“…10 [A purely inseparable extension] For F 3 (α)[β]/(β 3 − α) over F 3 (α), write a, b, c for the universal coefficients of the basis 1, β, β 2 . In other words, θ = a + bβ + cβ2 . One computes that the index form is thenb 3 − c 3 α.To find the monogenic generators of this extension, we look for a, b, c ∈ F 3 (α) so that b 3 − c 3 α = 0.…”

Quadrics in arithmetic statistics

“…The first example we work out is a generalization of the main theorem of Ruth's Princeton Ph.D. thesis [11] (see Theorem 1.1.2 of his [11], which amounts to a proof of the upper bound when B = Z − {0}) that was important for [2] (and thus [3]).…”