Residual ideals of MacLane valuations

Let (K,v)(K,v) be a discrete valued field and let x   be an indeterminate. In 1936, MacLane determined all valuations on K(x)K(x) extending v  . His work has been reviewed and generalized by Vaquié, by using the graded algebra of a valuation. We extend Vaquié's approach by studying residual ideals of the graded algebra as an abstract counterpart of certain residual polynomials having a key role in the computational applications of the theory. This enables us to determine the structure of the graded algebra of the discrete valuations on K(x)K(x). Also, let PP be the set of monic irreducible polynomials with coefficients in the completion OvOv of the valuation ring of v  . The constructive methods of the paper yield a canonical bijection P/≈→MP/≈→M, between the set of Okutsu equivalence classes of prime polynomials and a certain MacLane   space MM, whose points may be described in terms of discrete parameters associated with valuations on K(x)K(x).