Let K be a field equipped with a discrete valuation v. In a pioneering work,
MacLane determined all valuations on K(x) extending v. His work was recently reviewed
and generalized by Vaqui´e, by using the graded algebra of a valuation. We extend Vaqui´e’s approach by studying residual ideals of the graded algebra 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 the
discrete valuations on K(x) and we show how these valuations may be used to parameterize
irreducible polynomials over local fields up to Okutsu equivalence
CitacióFernández, J. [et al.]. Residual ideals of MacLane valuations. "Journal of algebra", 2015, vol. 427, p. 30-75.