Upper bounds for Euclidean minima of algebraic number fields. (English) Zbl 1130.11066

Let \(K\) be an algebraic number field with ring of integers \({\mathcal O}_K\), and let \(N\) denote the absolute value of the norm of elements. If for every \(x \in K\) there is a \(c\in {\mathcal O}_K\) such that \(N(x-c) < 1\), then \(K\) is said to be Euclidean (with respect to the norm). Thus we are led to study the real number \[ M(K) = \sup_{x \in K} \inf_{c \in {\mathcal O}_K} N(x-c) \] called the Euclidean minimum of \(K\). Clearly \(K\) is Euclidean if \(M(K) < 1\). In this article, it is shown that for a number field of degree \(n\) and with discriminant \(d_K\), we always have \(M(K) \leq 2^{-n}d_K\). This is a modest step towards Minkowski’s conjecture (now proved for all \(n \leq 6\)), which claims that \(M(K) \leq 2^{-n}\sqrt{d_K}\) for totally real fields. The author proves Minkowski’s conjecture for the maximal real subfields of the fields of \(p^r\)th roots of unity, and she also proves the bound \(M(K) \leq 2^{-n}\sqrt{| d_K| }\) for all fields of roots of unity. This is accomplished by the theory of ideal lattices in number fields and appealing to the theory of sphere packings.


11R47 Other analytic theory
11R04 Algebraic numbers; rings of algebraic integers
Full Text: DOI


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