Valuation rings in simple algebraic extensions of valued fields

Abstract

Consider a simple algebraic valued field extension $(L/K,v)$ and denote by ${\mathcal{O}}_L$ and ${\mathcal{O}}_K$ the corresponding valuation rings. The main goal of this paper is to present, under certain assumptions, a description of ${\mathcal{O}}_L$ in terms of generators and relations over ${\mathcal{O}}_K$. The main tool used here are complete sequences of key polynomials. It is known that if the ramification index of $(L/K,v)$ is one, then every complete set gives rise to a set of generators of ${\mathcal{O}}_L$ over ${\mathcal{O}}_K$. We show that we can find a sequence of key polynomials for $(L/K,v)$ which satisfies good properties (called neat). Then we present explicit ``neat" relations that generate all the relations between the corresponding generators of ${\mathcal{O}}_L$ over ${\mathcal{O}}_K$.