Function fields of algebraic tori revisited
type: Item
instance of: scholarly article
title: Function fields of algebraic tori revisited
zbMATH Open document ID: 1388.14048
author: Shizuo Endo
author: Ming-Chang Kang
published in: The Asian Journal of Mathematics
publication date: 2018-01-08T00:00:00Z
review text: 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 \it…
reviewed by: Ivo M. Michailov
Mathematics Subject Classification ID: 14E08
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 6822973
zbMATH Keywords: algebraic torus
zbMATH Keywords: rationality problem
zbMATH Keywords: locally free class groups
zbMATH Keywords: class numbers
zbMATH Keywords: maximal orders
zbMATH Keywords: twisted group rings
MaRDI profile type: Publication
DOI: 10.4310/AJM.2017.V21.N2.A1
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description: scientific article; zbMATH DE number 6822973