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.


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