Mathematics

Let k be an algebraic number field which is a Galois
extension of the rational field. A Minkowski unit in k is
a unit with the additional property that its conjugates under
the Galois action generate a subgroup of units with maximum
possible rank. Minkowski proved long ago that such units always
exist. I will outline a new proof that establishes the existence
of a Minkowski unit \beta such that the Weil height of \beta
is comparable to the sum of the heights of a basis for the
group of units.
If time permits I will describe an analogous but more
difficult result for relative units. This is joint work with
Shabnam Akhtari that appeared recently in the European Journal
of Mathematics.