Hostname: page-component-669899f699-chc8l Total loading time: 0 Render date: 2025-04-25T23:36:27.308Z Has data issue: false hasContentIssue false
  • English
  • Français

A generalization of a result of Iwasawa on the capitulation problem

Published online by Cambridge University Press:  24 October 2008

J. E. Cremona  and
R. W. K. Odoni
J. E. Cremona
Affiliation:
Department of Mathematics, University of Exeter, North Park Road, Exeter EX4 4QE
R. W. K. Odoni
Affiliation:
Department of Mathematics, University of Exeter, North Park Road, Exeter EX4 4QE
Get access Rights & Permissions [Opens in a new window]

Extract

The aim of this note is to prove

Theorem 1. Let n ≥ 3, and let p1, p2,…, Pn be primes in ℕ: = {z ∈ ℤ:z > 0}, each congruent to 1 (mod 4), which satisfy both of the following conditions:

(i) every unit in ℚ(√(p1p2)) has norm + 1;

(ii) the graph γ = γ(p1, p2, …, pn) associated with p1, p2, …, pn is odd (in the sense of [1]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 1 , January 1990 , pp. 1 - 3
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Cremona, J. E. and Odoni, R. W. K.. Some density results for negative Pell equations; an application of graph theory. J. London Math. Soc. (to appear).Google Scholar
[2]Iwasawa, K.. A note on the capitulation problem for number fields. Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), 59–61.Google Scholar
[3]Iwasawa, K.. A note on the group of units of an algebraic number field. J. Math. Pures Appl. 35 (1956), 189–192.Google Scholar
[4]Lagarias, J.. On determining the 4-rank of the ideal class group of a quadratic field. J. Number Theory 12 (1980), 191–196.Google Scholar
[5]Odoni, R. W. K.. A note on a recent paper of Iwasawa on the capitulation problem. Proc. Japan Acad. Ser. A Math. Sci. (to appear).Google Scholar
[6]Rédei, L.. Über einige Mittelwertfragen im quadratischen Zahlkörper. J. Reine Angew. Math. 174 (1935), 15–55.Google Scholar