Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bogomolov multipliers and retract rationality for semi-direct products   Let $G$ be a finite group. The Bogomolov multiplier $B_0(G)$ is constructed
as an obstruction to the rationality of $\bm{C}(V)^G$ where $G\to GL(V)$ is a
faithful representation over $\bm{C}$. We prove that, for any finite groups
$G_1$ and $G_2$, $B_0(G_1\times G_2)\xrightarrow{\sim} B_0(G_1)\times B_0(G_2)$
under the restriction map. If $G=N\rtimes G_0$ with $\gcd\{|N|,|G_0|\}=1$, then
$B_0(G)\xrightarrow{\sim} B_0(N)^{G_0}\times B_0(G_0)$ under the restriction
map. For any integer $n$, we show that there are non-direct-product $p$-groups
$G_1$ and $G_2$ such that $B_0(G_1)$ and $B_0(G_2)$ contain subgroups
isomorphic to $(\bm{Z}/p \bm{Z})^n$ and $\bm{Z}/p^n \bm{Z}$ respectively. On
the other hand, if $k$ is an infinite field and $G=N\rtimes G_0$ where $N$ is
an abelian normal subgroup of exponent $e$ satisfying that $\zeta_e\in k$, we
will prove that, if $k(G_0)$ is retract $k$-rational, then $k(G)$ is also
retract $k$-rational provided that certain "local" conditions are satisfied;
this result generalizes two previous results of Saltman and Jambor \cite{Ja}.

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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: econ