Hermite's constant for quadratic number fields

Abstract

We develop a method to compute the Hermite-Humbert constants gamma(K,n) of a real quadratic number field K, the analogue of the classical Hermite constant gamma(n) when Q is replaced by a quadratic extension. In the case n = 2, the problem is equivalent to the determination of lowest points of fundamental domains in H-2 for the Hilbert modular group over K, that had been studied experimentally by H. Cohn. We establish the results he conjectured for the fields Q(root2), Q(root3) and Q(root5). The method relies on the characterization of extreme forms in terms of perfection and eutaxy given by the second author in an earlier paper.