On the Euclidean minimum of some real number fields. (English) Zbl 1161.11032

Let \(K\) be a number field; the Euclidean minimum is the infimum of the set of all real numbers \(\mu\) with the following property: for all \(x \in K\) there is a \(y \in O_K\) (its ring of integers) such that \(|N(x-y)| \leq \mu\). A number field \(K\) is Euclidean with respect to the norm if \(M(K) < 1\). For totally real number fields, a conjecture going back to Minkowski predicts that \(M(K) \leq 2^{-n} \sqrt{D_K}\), where \(n\) is the degree and \(D_K\) the discriminant of \(K\). This conjecture is known to hold for all \(n \leq 6\).
By studying ideal lattices in number fields and their invariants, the authors can show that \(M(K) \leq |D_K|\) for all number fields. For real quadratic number fields, they prove \(M(K) \leq \sqrt{D_K}/4\), and derive better bounds in special cases. For totally real cyclotomic fields of prime power conductor \(p^m\) they prove \(M(K) \leq 2^{-n} \sqrt{D_K}\), and they get stronger bounds if \(p > 2\). In the last section, they classify all totally real fields that are thin (a somewhat technical but natural property of a number field which implies that it is Euclidean).


11R20 Other abelian and metabelian extensions
11H31 Lattice packing and covering (number-theoretic aspects)


Zbl 1130.11066
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