Kummer theory for norm algebraic tori. (English) Zbl 1146.14304
From the text: A generalization of Kummer theory using algebraic groups was initiated by [S. Lang and J. Tate, Am. J. Math. 80, 659–684 (1958; Zbl 0097.36203)] and further developed by T. Honda [Jap. J. Math. 30, 84–101 (1960; Zbl 0109.39602)], M. I. Bashmakov [Russ. Math. Surv. 27(1972), 25–70 (1973); translation from Usp. Mat. Nauk 27, No. 6(168), 25–66 (1972; Zbl 0256.14016)] and K. Ribet [Duke Math. J. 46, 745–761 (1979; Zbl 0428.14018)], from which we can derive the classical Kummer theory as Kummer theory for the multiplicative group \(\mathbb G_m\). Recently this theory was successfully applied to solve the so-called ‘support problem’ [see, for example, E. Kowalski, Manuscr. Math. 111, No. 1, 105–139 (2003; Zbl 1089.11031)]. On the other hand, H. Ogawa [RIMS Kokyuroku 1324, 217–224 (2003; Zbl 1040.11503)] and T. Komatsu [Manuscr. Math. 114, No. 3, 265–279 (2004; Zbl 1093.11068)] independently construct a theory classifying certain cyclic extensions over the maximal real subfields of cyclotomic fields, using Kummer theory for one-dimensional norm algebraic tori. The aim of this paper is to generalize their result to higher-dimensional norm algebraic tori and obtain a description of cyclic extensions over certain subfields of cyclotomic fields.
Citations:
Zbl 0097.36203; Zbl 0109.39602; Zbl 0256.14016; Zbl 0428.14018; Zbl 1089.11031; Zbl 1040.11503; Zbl 1093.11068References:
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