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Algebraic tori revisited
Let be a finite Galois extension and \pi = \fn{Gal}(K/k). An
algebraic torus defined over is called a -torus if
T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n for some integer
. The set of all algebraic -tori defined over under the stably
isomorphism form a semigroup, denoted by . We will give a complete
proof of the following theorem due to Endo and Miyata \cite{EM5}. Theorem. Let
be a finite group. Then where
is a maximal -order in containing
and is the locally free class group of
, provided that is isomorphic to the following four
types of groups : ( is any positive integer), ( is any odd
integer ), ( is any odd integer , is
an odd prime number not dividing , , and
for any prime divisor
of ), ( is any odd integer , for any
prime divisor of ).Comment: To appear in Asian J. Math. ; the title is change
Retract Rational Fields
Let be an infinite field. The notion of retract -rationality was
introduced by Saltman in the study of Noether's problem and other rationality
problems. We will investigate the retract rationality of a field in this paper.
Theorem 1. Let be fields. If is retract -rational
and is retract -rational, then is retract -rational. Theorem 2.
For any finite group containing an abelian normal subgroup such that
is a cyclic group, for any complex representation , the
fixed field is retract -rational. Theorem 3. If is a
finite group, then all the Sylow subgroups of are cyclic if and only if
is retract -rational for all -lattices ,
for all short exact sequences . Because the unramified Brauer group of a retract
-rational field is trivial, Theorem 2 and Theorem 3 generalize previous
results of Bogomolov and Barge respectively (see Theorem \ref{t5.9} and Theorem
\ref{t6.1}).Comment: Several typos in the previous version were correcte
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