Generators for extensions of valuation rings

2023/12/30 by Josnei Novacoski, Novacoski, Josnei
Mathematics · #Rings, Modules, and Algebras #Commutative Algebra and Its Applications #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2401.00182

Abstract

For a finite valued field extension (L/K,v) we describe the problem of find sets of generators for the corresponding extension \mathcal O_L/\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.

Related

• Valuation rings in simple algebraic extensions of valued fields
• Extensions of a valuation from K to K[x]
• Key Polynomials
• Ramification theory and key polynomials
• On common extensions of valued fields