In G. M. L. Powell's note 'Steenrod operations in motivic cohomology', he stated that if $\mathrm{char}(k)=0$,
$$H^{\*,\*}(k,\mathbb{Z}/2)=K\_\*^M(k)/2[\tau]$$
where $\tau\in H^{0,1}$ is the unique nonzero element.


I wonder whether this result holds when $char(k)>0$?



Motivic cohomology with $\mathbb{Z}/2$ coefficients in positive characteristic
This holds if the characteristic of $k$ is not 2, and it follows from the Milnor conjecture proved by Voevodsky.


Voevodsky ultimately proved the following (Theorem 6.17 in <https://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-s.pdf>):
If $m>0$ and $X$ is smooth over a field $k$ of characteristic prime to $m$, then the map
$$ H^n(X, \mathbb Z/m(i)) \to H^n\_{\mathrm{et}}(X,\mu\_m^{\otimes i}) $$
is an isomorphism provided that $n\leq i$. When $X=\operatorname{Spec}(k)$ the right-hand side is $K^M\_\*(k)/m[\tau^{\pm 1}]$ if there is a primitive $m$th root of unity $\tau\in\mu\_m(k)$ (this follows from the isomorphism for $n=i$), and the left-hand side is zero for $n>i$, so we know everything.


When the characteristic is 2, or more generally when the characteristic is $p$ and the coefficients are $\mathbb Z/p$, we also know everything by Geisser and Levine: in this case the motivic cohomology vanishes when $n\neq i$, so there is only Milnor K-theory.


