About Bredon motivic cohomology of a field

We prove that, over a perfect field, Bredon motivic cohomology can be computed by Suslin-Friedlander complexes of equivariant equidimensional cycles. Partly based on this result we completely identify Bredon motivic cohomology of a quadratically closed field and of a euclidian field in weights 1 and $\sigma$. We also prove that Bredon motivic cohomology of an arbitrary field in weight 0 with integer coefficients coincides (as abstract groups) with Bredon cohomology of a point.