Small extensions of abelian ordered groups

Abstract

Let $\Gamma$ be a totally ordered group. We use Hahn's embedding theorem to construct a totally ordered set $\Gamma \subset {\Gamma }_{\mathrm{sme}}$ which classifies small extensions of $\Gamma$. This small-extensions closure ${\Gamma }_{\mathrm{sme}}$ is complete and plays a crucial role in the description of equivalence classes of valuations on the polynomial ring $K[x]$ over a field $K$.

References in corpus (1)

• MacLane-Vaquié chains of valuations on a polynomial ring