New York Journal of Mathematics
Volume 14 (2008) 215-224

  

Ganesh Bhandari, Nicole Lemire, Ján Mináč, and John Swallow

Galois module structure of Milnor K-theory in characteristic p


Published: June 23, 2008
Keywords: Milnor K-groups modulo p, cyclic extensions, Galois modules
Subject: 19D45, 12F10

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 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.

Acknowledgements

The second author's research was supported in part by NSERC grant R3276A01.
The third author's research was supported in part by NSERC grant R0370A01, and by a Distinguished Research Professorship at the University of Western Ontario.
The fourth author's research was supported in part by NSA grant MDA904-02-1-0061 and by NSF grant DMS-0600122.


Author information

Ganesh Bhandari:
Department of Mathematics, Physics and Engineering, Mount Royal College, 4825 Mount Royal Gate SW, Calgary, Alberta T3E 6K6, CANADA
gbhandari@mtroyal.ca

Nicole Lemire:
Department of Mathematics, Middlesex College, University of Western Ontario, London, Ontario N6A 5B7, CANADA
nlemire@uwo.ca

Ján Mináč:
Department of Mathematics, Middlesex College, University of Western Ontario, London, Ontario N6A 5B7, CANADA
minac@uwo.ca

John Swallow:
Department of Mathematics, Davidson College, Box 7046, Davidson, North Carolina 28035-7046, USA
joswallow@davidson.edu