Defining \(\mathbb{Z}\) using unit groups (Q6595585)

In this astonishing work, the authors solve a very old problem of A. Tarski: the first-order theory of the field of constructible numbers is undecidable, meaning that there is no algorithm to determine whether an arbitrary arithmetic statement made about this field is true or false. Constructible numbers are those that correspond to the length of a segment that can be drawn from a unit segment using only compass and straightedge, or equivalently they are those numbers that can be built from rational numbers using only the four basic operations and the extraction of square roots of positive numbers.\N\NCorollaries of their work also include the undecidability of, for instance, the union of a tower made of extensions of degree at most some \(d>1\), or the compositum of all the extensions of a number field that have degree at most some \(d>1\).\N\NAll these results are immediate consequences of their main theorem: There is a first-order definition of the integers in any non-big field of algebraic numbers (a field is \textit{non-big} if there is a positive integer \(n\) for which it does not contain any subfield of degree \(n\) over the field of rational numbers). Moreover, their first-order definition is uniform among all non-big fields -- indeed, uniformity seems to have been the initial motivation to this work. Their astute definition is based on some basic properties of units, following a path started by Julia Robinson in the sixties.