## 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).

### MSC:

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

Zbl 1130.11066
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### References:

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