Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Birational classification of fields of invariants for groups of order
  $128$   Let $G$ be a finite group acting on the rational function field
$\mathbb{C}(x_g : g\in G)$ by $\mathbb{C}$-automorphisms $h(x_g)=x_{hg}$ for
any $g,h\in G$. Noether's problem asks whether the invariant field
$\mathbb{C}(G)=k(x_g : g\in G)^G$ is rational (i.e. purely transcendental) over
$\mathbb{C}$. Saltman and Bogomolov, respectively, showed that for any prime
$p$ there exist groups $G$ of order $p^9$ and of order $p^6$ such that
$\mathbb{C}(G)$ is not rational over $\mathbb{C}$ by showing the non-vanishing
of the unramified Brauer group: $Br_{nr}(\mathbb{C}(G))\neq 0$. For $p=2$, Chu,
Hu, Kang and Prokhorov proved that if $G$ is a 2-group of order $\leq 32$, then
$\mathbb{C}(G)$ is rational over $\mathbb{C}$. Chu, Hu, Kang and Kunyavskii
showed that if $G$ is of order 64, then $\mathbb{C}(G)$ is rational over
$\mathbb{C}$ except for the groups $G$ belonging to the two isoclinism families
$\Phi_{13}$ and $\Phi_{16}$. Bogomolov and B\"ohning's theorem claims that if
$G_1$ and $G_2$ belong to the same isoclinism family, then $\mathbb{C}(G_1)$
and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic. We investigate the
birational classification of $\mathbb{C}(G)$ for groups $G$ of order 128 with
$Br_{nr}(\mathbb{C}(G))\neq 0$. Moravec showed that there exist exactly 220
groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$ forming 11
isoclinism families $\Phi_j$. We show that if $G_1$ and $G_2$ belong to
$\Phi_{16}, \Phi_{31}, \Phi_{37}, \Phi_{39}, \Phi_{43}, \Phi_{58}, \Phi_{60}$
or $\Phi_{80}$ (resp. $\Phi_{106}$ or $\Phi_{114}$), then $\mathbb{C}(G_1)$ and
$\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic with
$Br_{nr}(\mathbb{C}(G_i))\simeq C_2$. Explicit structures of non-rational
fields $\mathbb{C}(G)$ are given for each cases including also the case
$\Phi_{30}$ with $Br_{nr}(\mathbb{C}(G))\simeq C_2\times C_2$.

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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex