Back to Search
Start Over
Noether's problem for p_groups with a cyclic subgroup of index p^2
- Publication Year :
- 2009
-
Abstract
- Let $K$ be any field and $G$ be a finite group. Let $G$ act on the rational function field $K(x_g:g\in G)$ by $K$-automorphisms defined by $g\cdot x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field $K(G)=K(x_g:g\in G)^G$ is rational (=purely transcendental) over $K$. We will prove that if $G$ is a non-abelian $p$-group of order $p^n$ ($n\ge 3$) containing a cyclic subgroup of index $p^2$ and $K$ is any field containing a primitive $p^{n-2}$-th root of unity, then $K(G)$ is rational over $K$. As a corollary, if $G$ is a non-abelian $p$-group of order $p^3$ and $K$ is a field containing a primitive $p$-th root of unity, then $K(G)$ is rational.
- Subjects :
- Mathematics - Algebraic Geometry
Mathematics - Rings and Algebras
13A50, 14E08
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.0911.1162
- Document Type :
- Working Paper