Upper bounds for Euclidean minima of algebraic number fields (Q863293)

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.