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Noether's problem asks whether the fixed field $k(G)=k(x_g:g\\in G)^G$ is rational (i.e. purely transcendental) over $k$. Theorem 1. If $G$ is a group of order $2^n$ ($n\\ge 4$) and of exponent $2^e$ such that (i) $e\\ge n-2$ and (ii) $\\zeta_{2^{e-1}} \\in k$, then $k(G)$ is $k$-rational. Theorem 2. Let $G$ be a group of order $4n$ where $n$ is any positive integer (it is unnecessary to assume that $n$ is a power of 2). 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Michailov, Jian Zhou, Ming-chang Kang","submitted_at":"2011-08-17T01:50:48Z","abstract_excerpt":"Let $G$ be a finite group and $k$ be a field. 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 (i.e. purely transcendental) over $k$. Theorem 1. If $G$ is a group of order $2^n$ ($n\\ge 4$) and of exponent $2^e$ such that (i) $e\\ge n-2$ and (ii) $\\zeta_{2^{e-1}} \\in k$, then $k(G)$ is $k$-rational. Theorem 2. Let $G$ be a group of order $4n$ where $n$ is any positive integer (it is unnecessary to assume that $n$ is a power of 2). 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