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The exact formula for the number of fields is of course $$F(n) = \pi(n) + \pi(n^{1/2}) + \pi(n^{1/3}) + \cdots$$ There are $O(\log n)$ nonzero lower order terms each of which is $O(\sqrt{n})$. So the leading term $\pi(n)$ dominates and the Prime Number Theorem gives the asymptotic $F(n) \sim \mathrm{Li}(n) \sim n/\log(n)$.