Valuation rings in simple algebraic extensions of valued fields

2025/03/12 by Novacoski, Josnei, Spivakovsky, Mark
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.09096

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.

Related

• Generators for extensions of valuation rings
• Extensions of a valuation from K to K[x]
• Key Polynomials
• Key polynomials for simple extensions of valued fields
• Ramification theory and key polynomials