Cubic fields with geometry
This book is a first modern presentation of the theory of cubic fields, stressing its algorithmic and computational features. The only previous monograph on this subject has been published sixty years ago by [\textit{B. N. Delone} and \textit{D. K. Faddeev}, Tr. Mat. Inst. Steklova 11, 340 p. (1940; Zbl 0061.09001)]. The first chapter provides an introduction to the principal notions of the theory of cubic fields (discriminant, integral basis, orders, units, \dots) and presents explicit formulas for integral bases due to \textit{G. Woronoi} [On the algebraic integers which depend on a root of an equation of third degree. St. Petersburg (1894; JFM 25.0302.02); Collected works. In 3 Vols. Vols. I, II (Russian). Kiev: Verlag der Akademie der Wissenschaften der Ukrainischen SSR (1952; Zbl 0049.02804)] and \textit{K. Belabas} [Math. Comput. 66, No. 219, 1213--1237 (1997; Zbl 0882.11070)]. In chapter 2 one finds the theory of ideals in cubic fields, containing a description of factorization of rational primes into prime ideals based on the results of \textit{P. Llorente} and \textit{E. Nart} [Proc. Am. Math. Soc. 87, 579--585 (1983; Zbl 0514.12003)]. This leads to a formula for the field discriminant. The chapter ends with information about the class-number formula and bounds for \(\kappa\), the residue of the Dedekind zeta-function at \(s=1\). Chapter 3 is devoted to binary cubic forms, including their reduction and composition, based on modern approach to the ideas of \textit{G. Eisenstein} [Mathematische Werke. I, II. New York: Chelsea (1975; Zbl 0339.01018)]. It is shown how the theory of equivalence classes of such forms of a given discriminant under the action of \(\mathrm{SL}(2,\mathbb Z)\) influences various algebraic properties of the corresponding cubic fields, and this is applied to tabulation of such fields. An important technical role play here the notions of Hessian quadratic forms \({\mathcal Q}(x,y)\) and Jacobian cubic forms \({\mathcal F}(x,y)\) associated to cubic fields. A previously unpublished algorithm to construct all cubic fields of a given discriminant, due to D. Shanks (manuscript of 1987) and implemented by G. W.-W. Fung, is presented in details in Chapter 4, written by R. Scheidler. In Chapter 5 the authors study some Diophantine equations related to cubic fields, in particular the equation \[N(x+y\rho_1+z\rho_2)=1,\] where \(1,\rho_1,\rho_2\) is the Belabas's integral basis of the field, calling it the \textit{cubic Pell equation of the first kind}, and the equation \[t^3-3t{\mathcal Q}(x,y)+{\mathcal F}(x,y)=27,\] calling it \textit{cubic Pell equation of the second kind}. Chapter 6 is devoted to the problem of minimal values attained by a binary cubic form, the main tool being a method of [\textit{A. Pethő}, J. Symb. Comput. 4, 103--109 (1987; Zbl 0625.10011)] utilizing continued fractions of cubic irrationalities. This is then used for the search of fundamental units. The last topic is studied also in the two next chapters. In the last chapter tools from algebraic geometry and the theory of elliptic curves are utilized to give a description of elements of norm \(1\) in a cubic field. The presented theory is accompanied by several illuminating examples.
continued fractions of cubic irrationalities
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