Abstract We compute the asymptotic number of octic number fields whose Galois groups over ℚ are isomorphic to D 4 D₄, the symmetries of a square, when ordering such fields by their absolute discriminants. In particular, we verify the strong form of Malle’s conjecture for such octic D 4 D₄ -fields and obtain the constant of proportionality. We further demonstrate that the constant of proportionality satisfies the Malle–Bhargava principle of being a product of local masses, despite the fact that this principle does not hold for discriminants of quartic D 4 D₄ -fields. This is the first instance of asymptotics being recovered for a non-concentrated family (in the sense of Alberts–Lemke Oliver–Wang–Wood) of number fields of Galois group neither abelian nor symmetric. Previously, this was only known for abelian fields, S n S₍ -fields with degree 𝑛 for n = 3, 4, 5 n=3, 4, 5, and S 3 S₃ -fields with degree 6.
Shankar et al. (2026) studied this question.