zbMATH — the first resource for mathematics

Class numbers of the simplest cubic fields. (English) Zbl 0613.12002
The “simplest” cubic fields were defined by D. Shanks [ibid. 28, 1137–1152 (1974; Zbl 0307.12005)] as those generated by equations \(x^ 3+mx^ 2-(m+3)x+1=0\). In the present paper it is shown how such fields can be used to construct cyclic cubic fields with class number divisible by \(n\), for any given \(n\). K. Uchida [J. Math. Soc. Japan 26, 447–453 (1974; Zbl 0281.12007)] had already shown, using a special case of the present work, that there are infinitely many such fields.)
A connection is shown between the 2-part of the class group of the simplest cubic fields and elliptic curves, and these curves are used to construct the quartic fields associated with the cubic field, whose existence H. Heilbronn [Stud. pure Math., Papers presented to Richard Rado on the Occasion of his sixty-fifth Birthday, 117–119 (1971; Zbl 0249.12012)] had proved.
Reviewer: H.J.Godwin

11R29 Class numbers, class groups, discriminants
11R16 Cubic and quartic extensions
11G05 Elliptic curves over global fields
11R20 Other abelian and metabelian extensions
Full Text: DOI