Chistov, A. L. On the birational equivalence of tori with a cyclic decomposition field. (Russian) Zbl 0358.14017 Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 64, 153-158 (1976). Let \(K\) be an arbitrary field, and \(L\) a cyclic finite extension of \(K\) with \(G=\)Gal\((L/K)\) of order \(n\). Let \(C(L/K)\) be the category of algebraic tori over \(K\) with decomposition field \(L\). If \(T\in C(L/K)\), then the group of characters \(\hat T\) of \(T\) is naturally a module of finite type over the group ring \(\mathbb Z[G]\cong\mathbb Z [X]/(X^n-1)\). For every polynomial \(f\in \mathbb Z[X]\) dividing \(X^n -1\), one can define a functor \(T\to T^f\) on \(C(L/K)\) as follows: \(\widehat{T^f}=\)Hom\((T,G_m)\) is nothing but Hom\(_G (\mathbb Z[X]/(f(X)), \hat T)\) \(T\) is said to be rational over \(K\) if the function field \(K(T)\) is a pure transcendental extension of \(K\), and stable rational, if \(T\times G_m^r\) is rational for a suitable natural number \(r\). Finally, let \(\Phi_d (d|n)\) denote the cyclotomic polynomial. It is known by V. E. Voskresenskiĭ [Usp. Mat. Nauk 28, No. 4 (172), 77–102 (1973; Zbl 0283.14004)] that the tori \(T\) such that \(\hat T=\mathbb Z[X]/(\Phi_d)\) are stable rational. If moreover \(n=p^t\) (\(p\) prime), then every stable rational torus over \(K\) is rational, every \(T\in C(L/K)\) is birationally equivalent over \(K\) to the product \(\prod_{d|n}T^{\Phi_d}\), and \(T\) is stable rational over \(K\) if and only if \(\widehat{T^{\Phi_d}}\) is a free \(\mathbb Z[X]/(\Phi_d )-d|n\) module for every \(d|n\). In general these statements are false for an arbitrary natural number \(n\), and the aim of this paper is to give some analogous results for arbitrary \(n\) by changing convenably the functors \(T\to T^{\Phi_d}\). Reviewer: Lucian Bădescu (Genova) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 2 ReviewsCited in 3 Documents MSC: 14J10 Families, moduli, classification: algebraic theory 14E05 Rational and birational maps 14J25 Special surfaces 14L10 Group varieties Citations:Zbl 0283.14004 × Cite Format Result Cite Review PDF Full Text: EuDML