scholarly journals CLASS NUMBER ONE CRITERION FOR SOME NON-NORMAL TOTALLY REAL CUBIC FIELDS

2013 ◽  
Vol 17 (3) ◽  
pp. 981-989 ◽  
Author(s):  
Jun Ho Lee
Keyword(s):  
Class Number ◽  
Cubic Fields ◽  
Totally Real

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.

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.

scholarly journals Computing Stark units for totally real cubic fields

1997 ◽  
Vol 66 (219) ◽  
pp. 1239-1268 ◽  
Author(s):  
David S. Dummit ◽  
Jonathan W. Sands ◽  
Brett A. Tangedal
Keyword(s):  
Cubic Fields ◽  
Stark Units ◽  
Totally Real

On Totally Real Cubic Fields with Discriminant D < 10 7

1988 ◽  
Vol 50 (182) ◽  
pp. 581
Author(s):  
Pascual Llorente ◽  
Jordi Quer
Keyword(s):  
Cubic Fields ◽  
Totally Real

scholarly journals A note on class-number one in pure cubic fields

1979 ◽  
Vol 33 (148) ◽  
pp. 1317-1317
Author(s):  
H. C. Williams ◽  
Daniel Shanks
Keyword(s):  
Class Number ◽  
Cubic Fields

scholarly journals Congruence relations of Ankeny-Artin-Chowla type for pure cubic fields

1984 ◽  
Vol 96 ◽  
pp. 95-112 ◽  
Author(s):  
Hiroshi Ito
Keyword(s):  
Elliptic Curve ◽  
Class Number ◽  
Congruence Relation ◽  
Bernoulli Numbers ◽  
Quadratic Fields ◽  
Hurwitz Numbers ◽  
Real Quadratic Fields ◽  
Congruence Relations ◽  
Cubic Fields ◽  
Real Quadratic

Ankeny, Artin and Chowla [1] proved a congruence relation among the class number, the fundamental unit of real quadratic fields, and the Bernoulli numbers. Our aim of this paper is to prove similar congruence relations for pure cubic fields. For this purpose, we use the Hurwitz numbers associated with the elliptic curve defined by y2 = 4x3 — 1 instead of the Bernoulli numbers (§ 3).

Sign in / Sign up

Export Citation Format

Share Document