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Circulant Graphs and 4-Ranks of Ideal Class Groups

Published online by Cambridge University Press:  20 November 2018

Jurgen Hurrelbrink
Jurgen Hurrelbrink*
Affiliation:
Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803 U.S.A.
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Abstract

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This is about results on certain regular graphs that yield information about the structure of the ideal class group of quadratic number fields associated with these graphs. Some of the results can be formulated in terms of the quadratic forms x2 + 27y2, x2 + 32y2, x2 + 64y2.

Keywords

05C9011R1111R29
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 1 , 01 February 1994 , pp. 169 - 183
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Barrucand, P. and Cohn, H., Note on primes of type x2 + 32y2, class number and residuacity, J. reine angew. Math. 238(1969), 67–70.Google Scholar
2. Biggs, N. L., Algebraic Graph Theory, Cambridge Tracts in Math. 67, Cambridge Univ. Press, 1974.Google Scholar
3. Biggs, N. L. and White, A. T., Permutation Groups and Combinatorical Structures, London Math. Soc. Lecture Note Ser. 33, Cambridge Univ. Press, 1979.Google Scholar
4. Bollobâs, B., Random Graphs, Academic Press, London, 1985.Google Scholar
5. Brauckmann, B., 4-ranks ofS-ideal class groups, preprint, (1990).Google Scholar
6. Conner, P. E. and J. Hurrelbrink, Class Number Parity, Ser. Pure Math. 8, World Scientific, Singapore, 1988.Google Scholar
7. Gerth, F. III, The 4-class ranks of quadratic fields, Invent. Math. 77(1984), 489–515.Google Scholar
8. Gerth, F. III, The 4-class ranks of quadratic extensions of certain real quadratic fields, J. Number Theory 33(1989), 18–31.Google Scholar
9. Gras, G., Sur la norme du groupe des unites d'extensions quadratiques relatives, preprint, (1990).Google Scholar
10. Halter-Koch, F., Uber den 4-Rang der Klassengruppe quadratischerZahlkorper, J. Number Theory, ( 1984), 219–227.Google Scholar
11. Hurrelbrink, J., On the norm of the fundamental unit, preprint, (1990).Google Scholar
12. Lagarias, J. C., On determining the 4-rank of the ideal class group of a quadratic field, J. Number Theory 12(1980), 191–196.Google Scholar
13. Morton, P., Density results for the 2-ciass groups of imaginary quadratic fields, J. reine angew. Math. 332(1982), 156–187.Google Scholar
14. Rédei, L. and Reichardt, H., Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppen eines beliebigen quadratischen Zahlkorpers, J. reine angew. Math. 170(1934), 69–74.Google Scholar
15. Rédei, L., Arithmetischer Beweis desSatzes uber die Anzahl der durch 4 teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkorper, J. reine angew. Math. 171(1935), 55–60.Google Scholar
16. Rédei, L., Uber einige Mittelwertfragen im quadratischen Zahlkorper, J. reine angew. Math. 174(1936), 131–148.Google Scholar
17. Rose, H. E., A Course in Number Theory, Oxford Science Publ., Clarendon Press, Oxford, 1988.Google Scholar
18. Stevenhagen, P., Rédei-matrices and the structure of quadratic 2-ciass groups, preprint, (1991 ).Google Scholar
19 Stevenhagen, P., On the 2-power divisibility of certain quadratic class numbers, preprint, (1991).Google Scholar
20. Uehara, T. Y., On the 4-rank of the narrow ideal class group of a quadratic field, J. Number Theory 31( 1989), 167–173.Google Scholar