On degree bounds for invariant rings of finite groups over finite fields
Peter Fleischmann, Wolfgang Lempken · Contemporary mathematics - American Mathematical Society · 1999
. Let G be a finite group acting as algebra automorphisms on A := F[X1 ; : : : ; Xn ]. If F is a field of characteristic zero, then, due to classical results of Emmy Noether one knows that the invariant ring A G can be generated in degrees less or equal to jGj. If F an arbitrary commutative ring (e.g. a finite field), the situation is much less satisfying. In this paper we give an outline on known results (with some new proofs) on degree bounds for arbitrary rings and arbitary finite groups. It turns out that for the finite field Fq with q = p s , A G can be constructed explicitly from A P , if P is a Sylow - p group of G. We also present some new results on the range of finite fields over which Noether's bound holds for subgroups of classical groups. 1. Introduction Let R be a commutative ring, A a finitely generated (unitary) commutative R -algebra and G a group acting on A as R - algebra automorphisms. Then the invariant ring A G is defined to be the R - algebra ...