Ganesh Bhandari, Nicole Lemire, Jan Minac, John Swallow
Published 2004-05-26, updated 2007-09-07Version 5
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.
Comments: v5 (10 pages); revised introduction and shortened paper; incorporated and acknowledged suggestions by readers of earlier versions
Journal: New York J. Math. 14 (2008), 215--224
Galois module structure of Milnor K-theory mod p^s in characteristic p
Galois scaffolds and Galois module structure in extensions of characteristic $p$ local fields of degree $p^2$
Galois Module Structure of $p$th-Power Classes of Extensions of Degree p
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