By Stephan Körner

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But r log(αx/βy) = log(α /βσ ) + (xj − yj ) log ηj , j=1 where the logarithms on the right can be assumed to have their principal Y . It values. 5 implies that log | | −log Y . Thus Y log Y and so Y is bounded. Hence also X is bounded and so there are only ﬁnitely many x, y as asserted. Plainly we can begin with an equation conjugate to αx + βy = 1 and thus it remains only to prove that log y Y for some conjugate y of y. Accordingly let α (1) , . . , α (d ) be the conjugates of any element α of K with the usual ordering with respect to real and complex conjugates.

This indeed is the technique used in establishing a generalised version of Siegel’s lemma to number ﬁelds [25, Ch. 1. Then √ we would need only the √ condition N > M and we could take L = 2hk instead of L = 2dhk . In any event, our √ k whence N = (L + 1)2 M and choice of L ensures that L LR k. The latter is applied in the bound for the integral and it is a critical point in the proof. One may be curious as to the origin of the integral equation. In fact it arises from the classical Newton interpolation formula.

It follows that d is bounded, that is there are only ﬁnitely many imaginary quadratic ﬁelds with class number 1. In practice we obtain an explicit bound for d (originally 10500 , now about 1020 ) and it is then easy to show by computing the continued fraction for log α1 / log α2 that there are only the nine cases listed by Gauss. 8. It has thus been established that there are precisely eighteen ﬁelds in question (see Baker [21]; Stark [236]). In 1976 Goldfeld [114] found a new approach to the topic based on the theory of elliptic curves and Gross and Zagier√[121] succeeded in this way to show that the class number h(d ) of Q( −d ) satisﬁes h(d ) (log d )1−ε for any ε > 0 where the implied constant depends only on ε and is effectively computable; thus all imaginary quadratic ﬁelds with any given class number can now be determined in principle.