# Divisible elements in the cohomology of Milnor \$K\$-theory

As a consequence of the strong Tate conjecture over finite fields one can deduce (see here proposition 8.20) for every smooth variety $$X$$ over a finite field:
$$H_{cont}^i(X,mathbb{Q}_l(n))=H_{Zar}^{i-n}(X,mathcal{K}_n^M)otimes mathbb{Q}_loplus H_{Zar}^{i-n-1}(X,mathcal{K}_n^M)otimes mathbb{Q}_l$$
Here $$H_{cont}^i(X,mathbb{Q}_l(n))$$ is the continuous etale cohomology with respect to the inverse system given by $$mu_{l^k}^{otimes n}$$ for different $$k$$s, tensored with rationals. The sheaf $$mathcal{K}_n^M$$ is the $$n$$-th Milnor $$K$$-theory sheaf.

A quick comparison implies that since $$H_{cont}^i(X,mathbb{Z}_l(n))$$ does not contain infinitely $$l$$-divisible elements (see here Cor 4.9), then any infinitely $$l$$-divisible element in $$H_{Zar}^{i}(X,mathcal{K}_n^M)$$ is torsion. I was wondering whether this is something one can prove (without the assumption of Tate’s conjecture)? Even for $$n=1$$ the Milnor $$K$$-theory sheaf seems to coincide with $$mathbb{G}_m$$, I’m not sure how to deduce the result.