Galois module structure of Milnor $K$-theory in characteristic $p$

Ganesh Bhandari, Nicole Lemire, Jan Minac and John Swallow

\( \newcommand{\F}{\mathbb{F}} \newcommand{\Fp}{\F_p} \newcommand{\N}{\mathbb{N}} \) Let $E$ be a cyclic extension of degree $p^n$ of a field $F$ of characteristic $p$. We determine $k_mE$, the Milnor $K$-groups modulo $p$, as $\Fp[$Gal$(E/F)]$-modules for all $m\in \N$.