Galois module structure of Milnor K-theory in characteristic p
Ganesh Bhandari, Nicole Lemire, Jan Minac, John Swallow
Abstract
Let E be a cyclic extension of degree pn of a field F of characteristic p. Using arithmetic invariants of E/F we determine kmE, the Milnor K-groups KmE modulo p, as Fp[Gal(E/F)]-modules for all m in N. In particular, we show that each indecomposable summand of kmE 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 [MSS2], where additionally the main result of this paper becomes an essential induction step in the determination of KmE/psKmE as (Z/psZ)[Gal(E/F)]-modules for all m, s in N.
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