Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-22T20:23:20.353Z Has data issue: false hasContentIssue false
  • English
  • Français

ELEMENTS OF ORDER FOUR IN THE NARROW CLASS GROUP OF REAL QUADRATIC FIELDS

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  28 September 2015

ELLIOT BENJAMIN  and
C. SNYDER
ELLIOT BENJAMIN*
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USA email [email protected]
C. SNYDER
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USA email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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.

Using the elements of order four in the narrow ideal class group, we construct generators of the maximal elementary $2$-class group of real quadratic number fields with even discriminant which is a sum of two squares and with fundamental unit of positive norm. We then give a characterization of when two of these generators are equal in the narrow sense in terms of norms of Gaussian integers.

Keywords

quadratic number fieldideal class groupnarrow class group

MSC classification

Primary: 11R11: Quadratic extensions
Secondary: 11R29: Class numbers, class groups, discriminants
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 1 , February 2016 , pp. 21 - 32
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Cohen, H., A Course in Computational Algebraic Number Theory (Springer, New York, 1996).Google Scholar
Cohn, H., Advanced Number Theory (Dover, New York, 1980).Google Scholar
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 4th edn (Oxford University Press, London, 1960).Google Scholar
Lemmermeyer, F., Reciprocity Laws (Springer, New York, 2000).CrossRefGoogle Scholar
Lemmermeyer, F., ‘Relations in the 2-class group of quadratic number fields’, J. Aust. Math. Soc. 93 (2012), 115–120.CrossRefGoogle Scholar