Daniel's Exotic Secret

Mysterious pure cubic fields

The kataklysmic event:

In the time between August 1987 and January 1989 I discovered [1] 5 exotic radicands, missing from the listing of 16 radicands D of exotic pure cubic fields Q(D^1/3) of principal factorization type BETA on page 273 in the paper [2] of H. C. WILLIAMS.

These radicands are:

1430

6370

9922

11284

12673

Concepts and details:

A description of the concepts "principal factorization type BETA" resp. "exotic" you will find on the page Principal Factors in Pure Cubic Fields resp. Algorithm of VORONOI in Pure Cubic Fields.

A correct table of all 21 exotic pure cubic fields Q(D^1/3) with normalized radicands D < 15000 showing the discriminant d, the radicand D = mn² with its quadratic part n, the Dedekind type, 2 minimal norms of principal factors (if they exist), the period length, the regulator R, and the class number h you get by linking to my database Exotic Pure Cubic Fields. For the most recent results see my series Joint Research 2002.

Bibliography:

[1] D. C. Mayer, Differential principal factors and units in pure cubic number fields, 1989, Dept. of Math., Univ. of Graz
[2] H. C. Williams, Determination of principal factors in Q(D^1/2) and Q(D^1/3), Math. Comp. 38 (1982), no. 157, 261-274

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