The space of R-places on a rational function field

Abstract

Suppose that F is a field such that the value groups of the R-places on F, i.e., places from F into the real numbers R, are all trivial or countable. The path-connected components of the space M(F(x1,x2,⋯,xn)) of R-places on F(x1,x2,⋯,xn) are shown then to correspond bijectively to those of M(F). For example, the space M(R(x1,x2,⋯,xn)) of R-places on the rational function field R(x1,x2,⋯,xn) is path-connected, and similarly for Q(x1,x2,⋯,xn). A key tool is a homeomorphism in the case that F is a maximal field between the space of R-places on F(x) and a certain space of sequences related to the “signatures” of [1].