Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T15:33:02.205Z Has data issue: false hasContentIssue false

Bicyclic Bicubic Fields

Published online by Cambridge University Press:  20 November 2018

Charles J. Parry*
Affiliation:
Virgina Polytechnic Institute and State University, Blacksburg, Virginia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

There is an extensive body of literature on the bicyclic biquadratic fields. These fields provide the simplest examples of abelian noncyclic extensions of Q. In sharp contrast, there is a dearth of literature on the bicyclic bicubic extensions of the rational numbers. These fields together with the abelian noncyclic octic extensions provide the next simplest abelian noncyclic extensions.

In this article, we shall study abelian bicyclic bicubic extensions of Q of degree 9. Hasse [4, v-ix] has stated as important objectives: the computation of an integral basis, the determination of class number and the calculation of fundamental units for abelian fields. In this article, we will solve the first problem completely, and show that the solution to the unit problem leads to a solution of the class number problem. Moreover, we shall give a method for determining the unit group up to a subgroup which has index 1 or 3 and so determine the class number up to a factor of 3.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Gras, M.N., Méthodes et algorithmes pour le calcul numérique du nombre de classes et desunité des extensions cubiques cycliques de Q, J. Reine Angew. Math. 277(1975), 89116.Google Scholar
2. Hasse, H., Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischenund biquadratischen Zahlkôrpern, Abh. Deutsch. Akad. Wiss. Berlin 2 (1948), 195.Google Scholar
3. Hasse, H., Bericht über neuere Untersuchungen und Problème aus der Théorie der algebraischenZahlkörper, Würzburg, Wien (1970).Google Scholar
4. Hasse, H., Über die Klassenzahl abelscher Zahlkörper, Berlin (1952).Google Scholar
5. Maki, S., The determination of units in real cyclic sextic fields (Berlin, Heidelberg, New York, 1980).Google Scholar
6. Der Waerden, B.L. Van, Modern algebra (New York, 1964).Google Scholar
7. Walter, C.D., Kuroda s class number relation, Acta Arithmetica 35 (1979), 4151.Google Scholar