Thirty years ago, I heard a talk by Lou in Princeton on the theory of the algebraic integers. They are essentially bi-interpretable with the theory of the algebraic closure $Q^{alg}$ of $Q$ as a valued field, where the valuation is not fixed but ranges in a certain (Boolean-valued) way over the space of all possible valuations.

Since then I have been wondering how to add the prime at infinity to the picture; the main difficulty, and interest, is in that a local-global principle is no longer valid and the theory must directly confront purely global phenomena. I will discuss some progress on this, joint with Itay Ben Yaacov, for the function field case. In addition I will answer a question raised (at least implicitly) in the paper of Macintyre-Van den Dries of that time, as to the possibility of replacing $F_p(t)$ by $Cc(t)$.