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.

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.

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.

Upper Bounds on |L(1, χ)| and Applications

1998 ◽  
Vol 50 (4) ◽  
pp. 794-815 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Finite Field ◽  
Galois Group ◽  
Linear Group ◽  
Class Number ◽  
Upper Bounds ◽  
Special Linear Group ◽  
Class Numbers ◽  
Special Linear ◽  
Abelian Extensions

AbstractWe give upper bounds on the modulus of the values at s = 1 of Artin L-functions of abelian extensions unramified at all the infinite places.We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for CM-fields. For example, we will reduce the determination of all the non-abelian normal CM-fields of degree 24 with Galois group SL2(F3) (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of 23 such CM-fields.

scholarly journals Values of zeta functions and class number 1 criterion for the simplest cubic fields

2000 ◽  
Vol 160 ◽  
pp. 161-180 ◽  
Author(s):  
Hyun Kwang Kim ◽  
Hyung Ju Hwang
Keyword(s):  
Zeta Function ◽  
Class Number ◽  
Irreducible Polynomial ◽  
Zeta Functions ◽  
Rational Integer ◽  
Dedekind Zeta Function ◽  
Cubic Field ◽  
Cubic Fields

AbstractLet K be the simplest cubic field defined by the irreducible polynomial where m is a nonnegative rational integer such that m2 + 3m + 9 is square-free. We estimate the value of the Dedekind zeta function ζK(s) at s = −1 and get class number 1 criterion for the simplest cubic fields.

MODULAR LATTICES OVER CM FIELDS

2009 ◽  
Vol 05 (05) ◽  
pp. 859-869
Author(s):  
IVAN SUAREZ
Keyword(s):  
Modular Lattice ◽  
Number Fields ◽  
Class Numbers ◽  
Modular Lattices ◽  
Fixed Level ◽  
Ideal Lattices ◽  
Totally Real

We study some properties of Arakelov-modular lattices, which are particular modular ideal lattices over CM fields. There are two main results in this paper. The first one is the determination of the number of Arakelov-modular lattices of fixed level over a given CM field provided that an Arakelov-modular lattice is already known. This number depends on the class numbers of the CM field and its maximal totally real subfield. The first part gives also a way to compute all these Arakelov-modular lattices. In the second part, we describe the levels that can occur for some multiquadratic CM number fields.

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].

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