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Central extensions and rational quadratic forms

Published online by Cambridge University Press:  22 January 2016

Yoshiomi Furuta  and
Tomio Kubota
Yoshiomi Furuta
Affiliation:
Department of Mathematics, Faculty of Science, Kanazawa UniversityKanazawa 920, Japan
Tomio Kubota
Affiliation:
Department of Mathematics, School of Science, Nagoya UniversityChikusa-ku, Nagoya 464-01, Japan, Department of Mathematics, Meijo University, Nagoya, 468, Japan
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The purpose of this paper is to characterize by means of simple quadratic forms the set of rational primes that are decomposed completely in a non-abelian central extension which is abelian over a quadratic field. More precisely, let L = Q be a bicyclic biquadratic field, and let K = Q. Denote by the ray class field mod m of K in narrow sense for a large rational integer m. Let be the maximal abelian extension over Q contained in and be the maximal extension contained in such that Gal(/L) is contained in the center of Gal(/Q). Then we shall show in Theorem 2.1 that any rational prime p not dividing d1d2m is decomposed completely in /Q if and only if p is representable by rational integers x and y such that x ≡ 1 and y ≡ 0 mod m as follows

where a, b, c are rational integers such that b2 − 4ac is equal to the discriminant of K and (a) is a norm of a representative of the ray class group of K mod m.

Moreover is decomposed completely in if and only if .

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 130 , June 1993 , pp. 177 - 182
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

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