INVARIANT FIELDS AND LOCALIZED INVARIANT RINGS OF p-GROUPS

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.