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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Class numbers of totally imaginary quadratic extensions of totally real fields


Author: Judith S. Sunley
Journal: Trans. Amer. Math. Soc. 175 (1973), 209-232
MSC: Primary 12A50
DOI: https://doi.org/10.1090/S0002-9947-1973-0311622-9
MathSciNet review: 0311622
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Abstract: Let K be a totally real algebraic number field. This paper provides an effective constant $ C(K,h)$ such that every totally imaginary quadratic extension L of K with $ {h_L} = h$ satisfies $ \vert{d_L}\vert < C(K,h)$ with at most one possible exception.

This bound is obtained through the determination of a lower bound for $ L(1,\chi )$ where $ \chi $ is the ideal character of K associated to L. Results of Rademacher concerning estimation of L-functions near $ s = 1$ are used to determine this lower bound. The techniques of Tatuzawa are used in the proof of the main theorem.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0311622-9
Keywords: Number field, class number, imaginary quadratic extension
Article copyright: © Copyright 1973 American Mathematical Society