Malle's Conjecture for Galois octic fields over Q

We compute the asymptotic number of octic number fields whose Galois groups over Q are isomorphic to 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 D₄-fields and obtain the constant of proportionality. Our result answers the question of whether a positive proportion of Galois octic extensions of Q have non-abelian Galois group in the negative. 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 D₄-fields. This is the first instance of asymptotics being recovered for a non-concentrated family of number fields of Galois group neither abelian nor symmetric. Previously, this was only known for abelian fields, degree-n Sₙ-fields for n=3, 4, 5, and degree-6 S₃-fields.