Algebraic tori revisited

Let $K/k$ be a finite Galois extension and \pi = \fn{Gal}(K/k). An algebraic torus $T$ defined over $k$ is called a $\pi$-torus if T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n for some integer $n$. The set of all algebraic $\pi$-tori defined over $k$ under the stably isomorphism form a semigroup, denoted by $T(\pi)$. We will give a complete proof of the following theorem due to Endo and Miyata \cite{EM5}. Theorem. Let $\pi$ be a finite group. Then $T(\pi)\simeq C(\Omega_{\bm{Z}\pi})$ where $\Omega_{\bm{Z}\pi}$ is a maximal $\bm{Z}$-order in $\bm{Q}\pi$ containing $\bm{Z}\pi$ and $C(\Omega_{\bm{Z}\pi})$ is the locally free class group of $\Omega_{\bm{Z}\pi}$, provided that $\pi$ is isomorphic to the following four types of groups : $C_n$ ($n$ is any positive integer), $D_m$ ($m$ is any odd integer $\ge 3$), $C_{q^f}\times D_m$ ($m$ is any odd integer $\ge 3$, $q$ is an odd prime number not dividing $m$, $f\ge 1$, and $(\bm{Z}/q^f\bm{Z})^{\times}=\langle \bar{p}\rangle$ for any prime divisor $p$ of $m$), $Q_{4m}$ ($m$ is any odd integer $\ge 3$, $p\equiv 3 \pmod{4}$ for any prime divisor $p$ of $m$).Comment: To appear in Asian J. Math. ; the title is change

Retract Rational Fields

Let $k$ be an infinite field. The notion of retract $k$-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 $k\subset K\subset L$ be fields. If $K$ is retract $k$-rational and $L$ is retract $K$-rational, then $L$ is retract $k$-rational. Theorem 2. For any finite group $G$ containing an abelian normal subgroup $H$ such that $G/H$ is a cyclic group, for any complex representation $G \to GL(V)$, the fixed field $\bm{C}(V)^G$ is retract $\bm{C}$-rational. Theorem 3. If $G$ is a finite group, then all the Sylow subgroups of $G$ are cyclic if and only if $\bm{C}_{\alpha}(M)^G$ is retract $\bm{C}$-rational for all $G$-lattices $M$, for all short exact sequences $\alpha : 0 \to \bm{C}^{\times} \to M_{\alpha} \to M \to 0$. Because the unramified Brauer group of a retract $\bm{C}$-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