scholarly journals Class number 3 problem for the simplest cubic fields

1999 ◽  
Vol 128 (5) ◽  
pp. 1319-1323 ◽  
Author(s):  
Dongho Byeon
Keyword(s):  
Class Number ◽  
Cubic Fields

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).

scholarly journals The Burgess inequality and the least kth power non-residue

2015 ◽  
Vol 11 (05) ◽  
pp. 1653-1678 ◽  
Author(s):  
Enrique Treviño
Keyword(s):  
Upper Bound ◽  
Class Number ◽  
Character Sums ◽  
Explicit Estimate ◽  
Dirichlet Characters ◽  
Cubic Fields ◽  
Incomplete Character Sums

The Burgess inequality is the best upper bound we have for incomplete character sums of Dirichlet characters. In 2006, Booker gave an explicit estimate for quadratic Dirichlet characters which he used to calculate the class number of a 32-digit discriminant. McGown used an explicit estimate to show that there are no norm-Euclidean Galois cubic fields with discriminant greater than 10140. Both of their explicit estimates are on restricted ranges. In this paper, we prove an explicit estimate that works for any range. We also improve McGown's estimates in a slightly narrower range, getting explicit estimates for characters of any order. We apply the estimates to the question of how large must a prime p be to ensure that there is a kth power non-residue less than p1/6.

scholarly journals A note on class-number one in certain real quadratic and pure cubic fields

1986 ◽  
Vol 46 (173) ◽  
pp. 333-333
Author(s):  
M. Tennenhouse ◽  
H. C. Williams
Keyword(s):  
Class Number ◽  
Cubic Fields ◽  
Real Quadratic

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.

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.

scholarly journals Certain pure cubic fields with class-number one

1977 ◽  
Vol 31 (138) ◽  
pp. 578-578 ◽  
Author(s):  
H. C. Williams
Keyword(s):  
Class Number ◽  
Cubic Fields
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