Primes and fields in stable motivic homotopy theory

Let F be a field of characteristic different from 2. We establish surjectivity of Balmer’s comparison map ρ•∙: Spc(SH^A1(F)^c) → Spec^h(K_∗^MW(F)) from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor-Witt K-theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.