INVARIANT FIELDS AND LOCALIZED INVARIANT RINGS OF p-GROUPS

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