Noether's problem for p-groups of order p^{5}

Abstract

Let $k$ be any field, $p>3$ be any prime number and $G$ be a nonabelian $p$-group of order $p^5$. Consider the action of $G$ on the rational function field $k(x_h:h\in G)$ by $g\cdot x_h=x_{gh}$ for all $g,h\in G$. Let $e$ be the exponent of $G$. Noether's problem asks whether the fixed field $k(G)=k(x_h:h\in G{)}^G$ is rational (i.e., purely transcendental) over $k$. In this paper, we will prove that if $G$ does not belong to the isoclinic family ${\Phi }_{10}$ in James's classification \cite{Jam1980} and $k$ contains a primitive $e$th root of unity, then $k(G)$ is rational over $k$. As a corollary, if $k=\text{C}$ is the field of complex numbers, then $\text{C}(G)$ is rational over $\text{C}$ if and only if $G$ is not in the family ${\Phi }_{10}$. This refines a recent result of Hoshi, Kang and Kunyavskii (\cite{HKK2012}, Theorem 1.12).

This paper has been withdrawn by the author due to a crucial Lemma 3.3

References in corpus (1)

• Noether's problem and unramified Brauer groups