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On cubic rings and quaternion rings. (English) Zbl 1166.11034

In their book on “The theory of irrationalities of the third degree” [Translations of Mathematical Monographs. 10. American Mathematical Society (1964; Zbl 0133.30202)] B. N. Delone and D. K. Fadeev established a correspondence between three-dimensional \(\mathbb Z\)-orders and binary cubic forms which is essentially one-to one. This remarkable observation was reinspected and generalized recently by W. T. Gan, B. H. Gross, and G. Savin [Duke Math. J. 115, No. 1, 105–169 (2002; Zbl 1165.11315)]. As is well-known, a similar correspondence holds in dimension 4. The authors extend both correspondences to coefficient rings which are either local or PID.

MSC:

11R16 Cubic and quartic extensions
11E76 Forms of degree higher than two
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