Motivic Eilenberg--Mac Lane spaces at the characteristic

We provide a novel, homotopy theoretic model for the motivic Eilenberg--Mac Lane spaces $K(Z_{(p)}(n), 2n) \in Spc(k)$ over fields of characteristic p. Using this, we determine their motives and, consequently, determine the motivic Steenrod algebra at the characteristic. Applications include the Hopkins--Morel isomorphism over arbitrary base schemes as well as the computation of the p-completed motivic stable stems and algebraic cobordism groups of fields of characteristic p. Along the way we expand the theory of normed algebraic structures in motivic homotopy theory, compare them to derived algebra at the characteristic, and produce a functor computing the motivic homology of the $P^1$-loop spaces of (certain) motivic spaces in terms of the motives.