THE TWENTIETH COLLOQUIUMFEST

Speaker: Arno Fehm

Title: Denseness of algebraic fields in real and p-adic closures

Abstract: 
Every number field is dense in all its real closures and all its p-adic closures, and the same holds for every algebraic field, i.e. every algebraic extension of the rationals. Works from the 80's (Prestel, Grob, Pop, Ershov, ...) exhibit other classes of fields for which this property holds, like PRC and PpC fields. We show that this property holds for arbitrary models of the theory of algebraic fields, which subsumes at least some of the classes mentioned above. The proof is nontrivial already in the case of elementary extensions of the rationals and offers some surprises in the case of p-adic closures of higher p-rank. Joint work with Sylvy Anscombe and Philip Dittmann.