A Computational Technique for Determining the Class Number of a Pure Cubic Field

Abstract

Abstract.Two different computational techniques for determining the class number of a pure cubic field are discussed. These techniques were implemented on an IBM/370-158 computer, and the class number for each pure cubic field Q(D ' ) for D = 2, 3.9999 was obtained. Several tables are presented which summarize the results of these computations. Some theorems concerning the class group structure of pure cubic fields are also given. The paper closes with some conjectures which were inspired by the computer results.

A note on class-number one in pure cubic fields

“…Of the 617 primes q < 104, the table computed for [1] lists 294 with h = 1, and in the review [2] of this table it was pointed out that this mean density: 294/617 = 0.476 tends to remain quite stable as the upper limit for q increases towards 104. It was suggested [2] that the table be extended for N = q > 104 in order to examine the constancy of this mean density.…”

A Rapid Method of Evaluating the Regulator and Class Number of a Pure Cubic Field

“…f?g, 0g 0g (» 0g) is the relative minimum adjacent to (1,0,1) in a reduced lattice 5L. Table 1 icontinued) 5l", en, an, 6^n) 51, = (1, p, v), where {1, p, v) is any basis of 2 [6].…”

Certain Pure Cubic Fields With Class-Number One

“…In his review of [1] Shanks [7] noted that for primes < 10,000, the fraction of the pure cubic fields Q(\/q) with classnumber one tended to remain about 48%. In this note we present some results obtained from evaluating the class number h of Q(y/q) for each q < 35,100.…”

“…In this note we present some results obtained from evaluating the class number h of Q(y/q) for each q < 35,100. The calculations were performed by using the methods of [1,Section 5]. Our main purpose is to study the constancy of this "about 48%", since it remains unknown whether or not infinitely many algebraic number fields have h = 1.…”

“…In Table 2 we give for the 1880 primes q < 35,100, each class number h that occurs, the frequency f(h) with which this h occurs, the value of 100/(/z)/1880, and the smallest value of q such that h is the class number for Q($q), when this h does not occur in Table 1 of [ 1 ]. pure cubic fields with class-number one Let n(x) be the number of primes q (q = -1 (mod 3)) which are less than or equal to x, and let g(x) be the number of those primes such that the class number of Gftfq) is one.…”