Generators for extensions of valuation rings

For a finite valued field extension $(L\mathrm{/}K,v)$ we describe the problem of find sets of generators for the corresponding extension ${\mathcal{O}}_L\mathrm{/}{\mathcal{O}}_K$ of valuation rings. The main tool to obtain such sets are complete sets of (key) polynomials. We show that when the initial index coincide with the ramification index, sequences of key polynomials naturally give rise to sets of generators. We use this to prove Knaf’s conjecture for pure extensions.