Finding fundamental units in cubic fields

1982 ◽  
Vol 92 (3) ◽  
pp. 385-389 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Computational Problem ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real ◽  
Fundamental Units

This paper improves a method of Godwin (4) for finding a pair of fundamental units in a totally real cubic field. The determination of such a unit pair is a well known computational problem. There is an old algorithm (circa 1896) of Voronoi which solves this problem, but the algorithm is quite complicated (an account of it is given in the book of Delone and Faddeev ((3), chapter IV, part A)). The method of Godwin is, in principle, much simpler. However, this method also has its drawbacks (more is said about this in Section 4 below). Indeed, when Godwin's student Angell produced his large table (see (1)) of totally real cubic fields some 15 years after (4) appeared, Voronoi's algorithm was used to compute the pairs of fundamental units.

The determination of units in totally real cubic fields

1960 ◽  
Vol 56 (4) ◽  
pp. 318-321 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Trial And Error ◽  
Integral Lattice ◽  
Cubic Field ◽  
Cubic Fields ◽  
Trial And Error Method ◽  
Totally Real ◽  
Rational Integral ◽  
Fundamental Units ◽  
Error Method

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

A note on Cusick's theorem on units in totally real cubic fields

1984 ◽  
Vol 95 (1) ◽  
pp. 1-2 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Present Note ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real ◽  
Fundamental Units

Let ε = ε1, with conjugates ε2, ε3, be a unit in a totally real cubic field, and let . Let ε be a unit for which T (ε) is least and let η be a unit, not a power of ε, for which T(η) is least. It was shown by Cusick[l] that ε,η form a pair of fundamental units under certain conditions. The purpose of the present note is to show that these conditions are unnecessary and that ε, η form a pair of fundamental units in all cases.

The determination of the class-numbers of totally real cubic fields

1961 ◽  
Vol 57 (4) ◽  
pp. 728-730 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Class Number ◽  
Sum Of Squares ◽  
Class Numbers ◽  
Cubic Field ◽  
Cubic Fields ◽  
Totally Real

In a previous paper (2) it was shown how the work of finding the units of a totally real cubic field could be facilitated by consideration of the sum of squares of differences between a number and its conjugates. In the present paper it is shown that the same ideas can be helpful in the calculation of class-numbers, and a list of the fields with class-number greater than unity and discriminant less than 20,000 is given.

Finding fundamental units in totally real fields

1984 ◽  
Vol 96 (2) ◽  
pp. 191-194 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Cubic Field ◽  
Image Position ◽  
Totally Real ◽  
Totally Real Fields ◽  
Fundamental Units

Let F a totally real cubic field. For any number α in F, let α, α′, α″ denote the conjugates of α. Define the function T(α) by.

scholarly journals Colmez cones for fundamental units of totally real cubic fields

2012 ◽  
Vol 132 (8) ◽  
pp. 1653-1663 ◽  
Author(s):  
Francisco Diaz y Diaz ◽  
Eduardo Friedman
Keyword(s):  
Cubic Fields ◽  
Totally Real ◽  
Fundamental Units

scholarly journals Torsion of rational elliptic curves over different types of cubic fields

2020 ◽  
Vol 16 (06) ◽  
pp. 1307-1323
Author(s):  
Daeyeol Jeon ◽  
Andreas Schweizer
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Torsion Group ◽  
Number Fields ◽  
Cubic Field ◽  
Different Types ◽  
Cubic Fields ◽  
Cubic Number Fields ◽  
Totally Real

Let [Formula: see text] be an elliptic curve defined over [Formula: see text], and let [Formula: see text] be the torsion group [Formula: see text] for some cubic field [Formula: see text] which does not occur over [Formula: see text]. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) [Formula: see text] can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves [Formula: see text] together with cubic fields [Formula: see text] so that [Formula: see text].

Fundamental units of cubic fields of positive discriminant

1981 ◽  
Vol 29 (6) ◽  
pp. 407-413
Author(s):  
�. T. Avanesov ◽  
K. K. Billevich
Keyword(s):  
Cubic Fields ◽  
Fundamental Units ◽  
Positive Discriminant

scholarly journals Determination of the Mean Number of Bonds per snow grain And of the Dependence of the Tensile Strength of Snow on Stereological Parameters

1978 ◽  
Vol 20 (83) ◽  
pp. 329-341 ◽  
Author(s):  
H. Gubler
Keyword(s):  
Tensile Strength ◽  
Density Distribution ◽  
Structural Parameters ◽  
Theoretical Density ◽  
Thin Sections ◽  
Distribution Parameters ◽  
The Mean ◽  
Fundamental Units ◽  
Effective Distribution

AbstractA theoretical density distribution of the Dumber of bonds per grain as a function of the mean number of bonds per grain is derived from the assumption of randomness and isotropy of grain and grain-bond location and orientation. The knowledge of the theoretical density distribution allows the determination of the effective distribution parameters from section planes or thin sections. The concept of the fundamental unit to describe the strength of snow is introduced. Structural parameters developed on the basis of the fundamental units show improved correlations with the tensile strength.

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