Resumen de Inductive valuations and defectless polynomials over henselian fields
Nathalia Moraes de Oliveira
Let (K,v) be a discrete rank-one valued field. In a pioneering work, S. MacLane studied and characterized the extensions of the valuation v to the rational function field K(x). M. Vaquié generalized his work for an arbitrary valued field (K,v), not necessarily rank-one nor discrete. A more constructive contribution for the theory was given in the case where v is discrete of rank-one, where J. Fernández, J. Guàrdia, J. Montes and E. Nart provided a computation of generators of the graded algebras and introduced some residual polynomial operators. In this memoir we extend these results to a valued field (K,v), not necessarily rank-one nor discrete. We also establish a connection between inductive valuations and irreducible polynomials with coefficients in the henselization of K, precisely, we construct a bijective mapping between the MacLane space of (K,v) (considered as the set of strong types) and a certain quotient of the subset of defectless polynomials with coefficients in the henselian field K with degree greater than one. Finally, as an application of the techniques presented in this work we reobtain some results on the computation of invariants of tame algebraic elements over henselian fields.
