Residual ideals of MacLane valuations

Abstract

Let $K$ be a field equipped with a discrete valuation $v$. In a pioneering work, S. MacLane determined all extensions of $v$ to discrete valuations on $K(x)$. His work was recently reviewed and generalized by M. Vaquié, by using the graded algebra of a valuation. We extend Vaquié's approach by studying residual ideals of the graded algebra of a valuation as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of the theory. As a consequence, we determine the structure of the graded algebra of any discrete valuation on $K(x)$ and we show how these valuations may be used to parameterize irreducible polynomials over local fields up to Okutsu equivalence.