Noether's Problem on Semidirect Product Group
Date Issued
2015
Date
2015
Author(s)
Huang, Shang
Abstract
Let $K$ be a field, $G$ a finite group. Let $G$ act on function field $L = K(x_{sigma} : sigma in G)$ by $ au cdot x_{sigma} = x_{ ausigma}$ for any $sigma, au in G$. Denote the fixed field of the action by $K(G) = L^{G} = { frac{f}{g} in L : sigma(frac{f}{g}) = frac{f}{g}, forall sigma in G }$. Noether''s problem asks whether $K(G)$ is rational (purely transcendental) over $K$. It is known that if $G = C_m times C_n is semidirect product of cyclic groups C_m, C_n with mathbb{Z}[zeta_n] a unique factorization domain, and K contains an eth primitive root of unity, where e is the exponent of G. Then K(G) is rational over K. But it is still an open question whether there exists prime pair p, q such that mathbb{C}(C_p times C_q) is not rational over mathbb{C}. In this paper, we show that, under some conditions, K(C_m times C_n) is rational over K.
Subjects
Rationality problem
The inverse Galois problem
Semidirect product group
Multiplicative group action
Type
thesis
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