Density of composite places in function fields and applications to real holomorphy rings

Given an algebraic function field F|K$F|K$ and a place ℘ on K, we prove that the places that are composite with extensions of ℘ to finite extensions of K lie dense in the space of all places of F, in a strong sense. We apply the result to the case of K=R$K=R$ any real closed field and the fixed place on R being its natural (finest) real place. This leads to a new description of the real holomorphy ring of F, which can be seen as an analog to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space M(F)$M(F)$ of all R$\mathbb {R}$‐places of F (places with residue field contained in R$\mathbb {R}$), its subspace of all R$\mathbb {R}$‐places of F that are composite with the natural R$\mathbb {R}$‐place of R, and the topological space of all R‐rational places. Further results about these spaces as well as various classes of relative real holomorphy rings are proven. At the conclusion of the paper, the theory of real spectra of rings will be applied to interpret basic concepts from that angle and to show that the space M(F)$M(F)$ has only finitely many topological components.