Function fields of algebraic tori revisited
The present article may be regarded as a supplement to the papers [\textit{S. Endo} and \textit{T. Miyata}, Nagoya Math. J. 56, 85--104 (1975; Zbl 0301.14008)] and [\textit{R. G. Swan}, AMS/IP Stud. Adv. Math. 48, 259--269 (2010; Zbl 1241.20008)]. It contains the complete proof of \((1')\Rightarrow (2)\) [Zbl 1241.20008, Theorem 3.3]. In [Zbl 1241.20008], Endo and Miyata investigated the classification of the function fields of algebraic tori. An algebraic torus \(T\) defined over a field k is an affine algebraic group defined over \(k\) such that \(T\times_{\mathrm{Spec}(k)} \mathrm{Spec}(\bar k)\simeq \mathbb G_{m,\bar k}^n\) where \(n\in \mathbb Z, \bar k\) is the algebraic closure of \(k\) and \(\mathbb G_{m,K}^n\) is the 1-dimensional multiplicative group defined over a field \(K\) (containing the base field \(k\)). Let \(K/k\) be a finite Galois extension and \(\pi = \mathrm{Gal}(K/k)\). An algebraic torus \(T\) defined over \(k\) is called a \(\pi\)-torus if \(T\times_{\mathrm{Spec}(k)} \mathrm{Spec}(K)\simeq \mathbb G_{m,K}^n\) for some integer \(n\). The set of all algebraic \(\pi\)-tori defined over \(k\) under the stably birational equivalence forms a semigroup, denoted by \(T(\pi)\). The purpose of this article is to give a proof of Theorem 1.4 supplementing the proof outlined in [Zbl 1241.20008]: {Theorem 1.4} ([Zbl 1241.20008, page 95, Theorem 3.3; page 187]). Let \(\pi\) be a finite group. Then the following statements are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] \(\pi\) is isomorphic to {\parindent=12mm \begin{itemize}\item[(i)] a cyclic group \(C_n\) where \(n\) is any positive integer, or \item[(ii)] a dihedral group \(D_m\) of order \(2m\) where \(m\) is an odd integer \(\geq 3\), or \item[(iii)] a direct product \(C_{q^f}\times D_m\) where \(q\) is an odd prime number, \(f\geq 1, m\) an odd integer \(\geq 3\), gcd\(\{q,m\} = 1\) and for any prime divisor \(p\) of \(m\), \((\mathbb Z/q^f\mathbb Z)^\times=\langle\bar p\rangle\), or \item[(iv)] \(Q_{4m} =\langle\sigma,\tau : \sigma^{2m} = \tau^4 = 1, \sigma^m = \tau^2, \tau^{-1}\sigma\tau = \sigma^{-1}\rangle\), the generalized quaternion group of order \(4m\), where \(m \geq 3\) is an odd integer and \(p\equiv 3 \pmod 4\) for any prime divisor \(p\) of \(m\). \end{itemize}} \item[(2)] \(T(\pi)\simeq C(\mathbb Z\pi)/C^q(\mathbb Z\pi)\simeq C(\Omega_{\mathbb Z\pi})\) where \(\Omega_{\mathbb Z\pi}\) is a maximal \(\mathbb Z\)-order in \(\mathbb Q\pi\) containing \(\mathbb Z\pi\). \item[(3)] \(T(\pi)\) is a finite group. \end{itemize}}
scientific article; zbMATH DE number 6822973
scientific article