The number of octic D4-fields with absolute discriminant below X is asymptotic to an explicit constant times X^{1/4} (log X)^2.
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Malle's Conjecture for Galois octic fields over $\mathbb Q$
The number of octic D4-fields with absolute discriminant below X is asymptotic to an explicit constant times X^{1/4} (log X)^2.