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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 struc- ture of pure cubic fields are also given. The paper closes with some conjectures which were inspired by the computer results. 1. Introduction. The theory of pure cubic fields QiDxl3),D rational, was founded in 1892 by Markov (12) ; in his paper he gives some class numbers and fundamental units, not always in an explicit form. In (7) Dedekind describes a method for deter- mining the class number of a pure cubic field Q{DX^3). He also gives a short table of class numbers for some small values of D. Cohn (6) implemented Dedekind's method on a computer and obtained class numbers for some fields in which he could easily determine the regulator. Cohn's technique was modified somewhat by Beach, Williams, and Zarnke in (4), and class numbers were obtained for QXP1 /3) for D = 2, 3, . . . , 999. Other tables of class numbers have been calculated by hand by Cassels (5) and Selmer (14). It should also be mentioned that Angeli (1) has recently given a list of class numbers for all cubic fields with negative discriminant greater than -20, 000. The purpose of this paper is to present a new technique for determining the class number of Q{D1'3). This method is much faster than the computational technique of (6) and (4). The algorithm was implemented on a computer and the class numbers for Q{Dl/3) obtained for D = 2, 3.9999. The total number of these fields is 8122. We also describe here some of the results of these calculations. 2. Some Properties of Pure Cubic Fields. Let K be any cubic field with discrim-

A computational technique for determining the class number of a pure cubic field