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Zbl 1161.11032
Bayer-Fluckiger, Eva; Nebe, Gabriele
On the Euclidean minimum of some real number fields. (English)
[J] J. Théor. Nombres Bordx. 17, No. 2, 437-454 (2005). ISSN 1246-7405

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 $|N(x-y)| \le \mu$. 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 $M(K) \le 2^{-n} \sqrt{D_K}$, where $n$ is the degree and $D_K$ the discriminant of $K$. This conjecture is known to hold for all $n \le 6$. \par By studying ideal lattices in number fields and their invariants, the authors can show that $M(K) \le |D_K|$ for all number fields. For real quadratic number fields, they prove $M(K) \le \sqrt{D_K}/4$, and derive better bounds in special cases. For totally real cyclotomic fields of prime power conductor $p^m$ they prove $M(K) \le 2^{-n} \sqrt{D_K}$, 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).
[Franz Lemmermeyer (Jagstzell)]

MSC 2000:: *11R20 Other abelian and metabelian extensions
11H31 Lattice packing and covering (number-theoretic results)

Keywords: Euclidean rings; thin fields; lattices; Minkowski's conjecture; real quadratic fields; cyclotomic fields

Citations: Zbl 1130.11066

Cited in: Zbl 1186.11040 Zbl 1137.11046

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