  We establish an explicit upper bound for the Euclidean minimum of a number
field which depends, in a precise manner, only on its discriminant and the
number of real and complex embeddings. Such bounds were shown to exist by
Davenport and Swinnerton-Dyer. In the case of totally real fields, an optimal
bound was conjectured by Minkowski and it is proved for fields of small degree.
In this note we develop methods of McMullen in the case of mixed signature in
order to get explicit bounds for the Euclidean minimum.
