Generators for extensions of valuation rings

For a finite valued field extension (L/K, v) we describe the problem of finding sets of generators for the corresponding extension OL/OK of valuation rings. The main tool to obtain such sets is complete sets of (key) polynomials. We show that when the initial index coincides 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. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.