Inhomogeneous and Euclidean spectra of number fields with unit rank strictly greater than 1

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Let K be a number field with unit rank r > 1. In this article we show that the inhomogeneous minimum of K is attained by at least one rational point. In particular, if M ( K ) is the Euclidean minimum of K , we have . This phenomenon has consequences on the decidability of the Euclidean nature of such a field. Moreover, in case K is not a CM-field, we prove that is attained, isolated, and that the inhomogeneous minimum function takes discrete rational values.