  Let E be a cyclic extension of degree p^n of a field F of characteristic p.
Using arithmetic invariants of E/F we determine k_mE, the Milnor K-groups K_mE
modulo p, as Fp[Gal(E/F)]-modules for all m in N. In particular, we show that
each indecomposable summand of k_mE has Fp-dimension a power of p. That all
powers p^i, i=0,1,...,n, occur for suitable examples is shown in a subsequent
paper [MSS2], where additionally the main result of this paper becomes an
essential induction step in the determination of K_mE/p^sK_mE as
(Z/p^sZ)[Gal(E/F)]-modules for all m, s in N.
