Families of Non-Galois Quartic Fields

A family of monogenic S4 quartic fields arising from elliptic curves

“…We apply this to our specific situation here. In particular, we will encounter the sequence R n (1, 2), which begins from n = 1 as follows: 0, 1, 2, 4, 6,9,12,16,20,25,30,36,42, . .…”

“…Consider the field K α = Q(θ). This family of number fields was studied by Fleckinger and Vérant [9]. Let α ≥ 9, α ∈ Z, and α = 24.…”

“…Fleckinger and Vérant also study the family of quartic fields given by T 4 + α 2 T 3 + 6T 2 + α 2 T + 1 of discriminant −4 α 2 2 − 16 3 , which they observe arise from a point of order four on a Fueter model [9]. The authors prove that this family is monogenic whenever (α/2) 2 − 16 is odd and squarefree, and α ≥ 12 [9,Corollary 1.4]. This appears to be a D 8 family.…”

“…Then Fleckinger and Vérant showed that K α is an S 4 quartic field with two real embeddings [9, Proposition 2.10]. They give an explicit basis for the ring of integers in general [9,Proposition 2.11], but it is not a power basis and they do not mention monogenicity. Finally, they remark that when 3 | α, then 1 + α 3 θ + 2θ 2 is a unit.…”

“…In fact, Fleckinger and Vérant studied the number fields of Theorem 1.1, motivated by their status as partial torsion fields [9]. However, as they write, "We note that the arithmetic of elliptic curves is not used once we have these polynomials."…”