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Invariants of Finite Groups over Finite Fields: Recent Progress and New Conjectures

In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas

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  • Peter Fleischmann

    (University of Kent at Canterbury, Institute of Mathematics and Statistics)

Abstract

Let G be a finite group acting on a polynomial ring A:= F[x1,…, xn] by graded algebra automorphisms. 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 |G|. If F is a field of positive characteristic p dividing the group order |G|, this is no longer true. The situation in characteristic p not dividing |G| has been clarified recently after being open for several decades. This paper presents an account on these developments, including some related questions and conjectures dealing with constructive and structural properties of modular invariant rings.

Suggested Citation

  • Peter Fleischmann, 2002. "Invariants of Finite Groups over Finite Fields: Recent Progress and New Conjectures," Springer Books, in: Gary L. Mullen & Henning Stichtenoth & Horacio Tapia-Recillas (ed.), Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pages 112-122, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-59435-9_9
    DOI: 10.1007/978-3-642-59435-9_9
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