Abstract
Choose any two rational numbers and get an element of norm 1 of a cubic field. We give formulas for doing this in terms of the generic letters a, b, c, d used to define the cubic field generated by the polynomial with these coefficients. This is related to Hilbert’s theorem 90 but accomplishes the goal of parametrization without requiring the cubic field to be cyclic. It is well known that this is possible due to a result of Voskresenskiĭ, but we discuss this work with very little of the language or tools of algebraic geometry, with the exception of some projective geometry. We also discuss conics and singular elliptic curves as they are significantly easier to parameterize.
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Notes
- 1.
If we chose D −1 = 1, then 1 ∘ (−1) would not be defined in \(\mathbb{P}^{1}(\mathbb{Q})\).
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Hambleton, S.A., Williams, H.C. (2018). Parametrization of Norm 1 Elements of \(\mathbb{K}\) . In: Cubic Fields with Geometry. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-01404-9_9
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