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Galois module structure of Milnor K-theory in characteristic p

Ganesh Bhandari·Nicole Lemire·Jan Minac·John Swallow
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Abstract

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.

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Ganesh Bhandari, Nicole Lemire, Jan Minac, John Swallow. 2007-09-07. Galois module structure of Milnor K-theory in characteristic p. https://arxiv.org/abs/math/0405503

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