Galois module structure of Milnor K-theory in characteristic p

摘要

Let E be a cyclic extension of degree pn of a field F of characteristic p. Using arithmetic invariants of E/F we determine km E, the Milnor K-groups Km E modulo p, as Fp[Gal(E/F)]-modules for all m ∈ N. In particular, we show that each indecomposable summand of km E has Fp-dimension a power of p. That all powers pi, i=0, 1, ..., n, occur for suitable examples is shown in a subsequent paper, Mináč, Schultz and Swallow, 2008, where additionally the main result of this paper becomes an essential induction step in the determination of Km E/ps Km E as (Z/psZ)[Gal(E/F)]-modules for all m, s ∈ N.