Class numbers of totally imaginary quadratic extensions of totally real fields

Sunley, Judith S.

doi:10.1090/S0002-9947-1973-0311622-9

Class numbers of totally imaginary quadratic extensions of totally real fields
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by Judith S. Sunley

Trans. Amer. Math. Soc. 175 (1973), 209-232

DOI: https://doi.org/10.1090/S0002-9947-1973-0311622-9

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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 $|{d_L}| < C(K,h)$ 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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