On dp-minimal fields

We classify dp-minimal pure fields up to elementary equivalence. Most are equivalent to Hahn series fields $K((t^{\mathrm{\Gamma }}))$ where $\mathrm{\Gamma }$ satisfies some divisibility conditions and $K$ is ${\mathbb{F}}_p^{alg}$ or a local field of characteristic zero. We show that dp-small fields (including VC-minimal fields) are algebraically closed or real closed.