Abstract
In this chapter, we provide an overview of Voronoi’s continued fraction algorithm that forms the basis for finding the fundamental unit(s) of a cubic field. We begin with a discussion of how Voronoi extended the idea of a simple continued fraction of a quadratic irrationality to that of a cubic irrationality. Next, we provide an account of relative minima in cubic lattices, reduced lattices (lattices in which 1 is a relative minimum), and chains of relative minima in these lattices in order to show how the automorphisms of these lattices can be detected. This can then be specialized to the determination of the fundamental unit of a cubic field of negative discriminant or of a fundamental pair of units of a cubic field of positive discriminant. These problems reduce to the task of finding a particular relative minimum adjacent to 1 in a reduced lattice which we will discuss in the next chapter.
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Hambleton, S.A., Williams, H.C. (2018). Voronoi’s Theory of Continued Fractions. In: Cubic Fields with Geometry. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-01404-9_7
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