scholarly journals A table of fundamental units of purely cubic fields

1970 ◽  
Vol 46 ◽  
pp. 1135-1140 ◽  
Author(s):  
Hideo Wada
Keyword(s):  
Cubic Fields ◽  
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.

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

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.

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.

Fundamental units in a family of cyclic cubic fields

1993 ◽  
Vol 21 (11) ◽  
pp. 3953-3962 ◽  
Author(s):  
Francisca Cánovas Orvay
Keyword(s):  
Cubic Fields ◽  
Fundamental Units ◽  
Cyclic Cubic Fields

scholarly journals On the calculation of the units of algebraic number fields

1986 ◽  
Vol 101 ◽  
pp. 181-185
Author(s):  
Takashi Azuhata
Keyword(s):  
Algebraic Number ◽  
Direct Calculation ◽  
Number Fields ◽  
The Other ◽  
New Method ◽  
Quadratic Fields ◽  
Algebraic Number Fields ◽  
Special Cases ◽  
Cubic Fields ◽  
Fundamental Units

The method to find a system of fundamental units were given by G. F. Voronoi in the case of purely cubic fields (see B. N. Delone and D. K. Faddeev [1]), and by K. K. Biilebic [2] in general case of algebraic number fields except for quadratic fields. But they are rather complicated for direct calculation. On the other hand, in some special cases, units can be calculated by the Jacobi-Perron algorithm which is a generalization of the continued fractional expansion (see, for example, L. Bernstein [3]). In this paper, we will show the new method to calculate a system of independent units by multiplying several numbers. We will give some examples for purely cubic fields in Section 3.

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