A remark on the class number of quadratic fields. (English) Zbl 0177.07403

Means of resolving the long-standing conjecture that there are only nine imaginary quadratic fields with class number 1 where discovered independently by H. M. Stark [Mich. Math. J. 14, 1–27 (1967; Zbl 0148.27802)] and the author [Mathematika 13, 204–216 (1966; Zbl 0161.05201]. The work of the author was motivated by an earlier paper of A. O. Gel’fond and Yu. V. Linnik [Dokl. Akad. Nauk SSSR, n. Ser. 61, 773–776 (1948; Zbl 0040.31001)] in which the conjecture was reduced to a well-known problem from the theory of transcendental numbers. Here it is shown that results from the latter theory can also be employed to obtain a complete determination of all imaginary quadratic fields \(\mathbb Q(\sqrt{-d})\) with class number 2, provided that \(d\equiv 3\pmod 8\); more precisely, it is proved that all such square-free positive integers \(d\) are less than \(10^{500}\). The proof utilizes a formula for the product of \(L\)-functions analogous to the classical Kronecker Grenzformel.
Reviewer: Alan Baker


11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
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