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Cubic Fields

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Cubic Fields with Geometry

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Abstract

A cubic irrationality is a root of an irreducible (over the rationals) polynomial of degree 3 with rational coefficients. In this chapter we discuss many of the well-known properties of these numbers. In particular, we describe the cubic polynomial and develop many of the attributes of the cubic field generated by such a polynomial. This involves examining orders, the maximal order, integral bases of an order, the discriminant, and the performance of arithmetic in these structures. We also discuss the various types of cubic fields and the properties of the units and regulator. We conclude with a collection of results concerning the development of the simple continued fraction of a cubic irrationality.

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Notes

  1. 1.

    Eisenstein [70, 71] discussed \(\mathcal{Q}\) and \(\mathcal{F}\) obtained from \(\mathcal{C} = (a, 3b, 3c,d)\).

  2. 2.

    (1.47) was known to Eisenstein [70, p. 97] in relation to binary cubic forms.

  3. 3.

    Cubic hyperbolas and even elliptic curves in Weierstrass form (without their arithmetic) were considered by Newton in the first edition of Optiks [149, Tables I, II, IV, VI].

  4. 4.

    Eisenstein [72] in 1844 considered the ternary cubic form u 3 + pp 1 y 3 + pp 2 z 3 − 3puyz, equal to the product \(\prod _{j=1}^{3}\left (u +\rho ^{j}\eta y +\rho ^{-j}\theta z\right )\), where the rational prime \(p \equiv 1\pmod 3\), p = p 1 p 2 in the quadratic field \(\mathbb{Q}(\rho )\), where ρ is a primitive cube root of unity, and \(\eta = \root{3}\of{pp_{1}}\), \(\theta = \root{3}\of{pp_{2}}\). Dickson [66, p. 259] begins the chapter on ternary cubic forms discussing Eisenstein’s work, where it is noted that the change of variables y = v + ρw, z = v + ρ 2 w produces a ternary cubic form with rational integer coefficients.

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Authors and Affiliations

  1. School of Mathematics and Physics, The University of Queensland, St. Lucia, Brisbane, QLD, Australia

    Samuel A. Hambleton

  2. Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada

    Hugh C. Williams

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  1. Samuel A. Hambleton
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Hambleton, S.A., Williams, H.C. (2018). Cubic Fields. In: Cubic Fields with Geometry. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-01404-9_1

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