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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
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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

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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
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[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