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The Quarterly Journal of Mathematics Advance Access originally published online on May 14, 2007
The Quarterly Journal of Mathematics 2007 58(2):151-157; doi:10.1093/qmath/ham011
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

INVARIANT FIELDS AND LOCALIZED INVARIANT RINGS OF p-GROUPS

H. E. A. Campbell1,{dagger} and J. Chuai2 {ddagger}

1 Mathematics and Statistics Department, Memorial University of Newfoundland, St John's, NL, Canada A1A 5S7
2 Département de mathématiques et de statistique, Université de Montréal, Montréal, QC, H3C 3J7 Canada

{dagger} Corresponding author. E-mail: eddy{at}mun.ca

Received 17 May 2005; revised 8 November 2006
  Abstract

It is well known that for a p-group, the invariant field is purely transcendental (T. Miyata, Invariants of certain groups I, Nagoya Math. J. 41 (1971), 69–73). In this note, we show that a minimal generating set of this field can be chosen as homogeneous invariants from the invariant ring. As a result, we show that the invariant ring localized at one suitable invariant is the localization of a polynomial subring at this same invariant. This second result is a generalization of a recent result of the first author for cyclic groups of order p (H. E. A. Campbell, Rings of invariants of representations of Cp in characteristic p, preprint, 2006). As well, we specialize these results to this latter case.

{ddagger} E-mail: chuai{at}DMS.UMontreal.CA


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