Invariants of Defectless Irreducible Polynomials

Defectless irreducible polynomials over a valued field (F, v) have been studied by means of strict systems of polynomial extensions and complete distinguished chains. Strong connections are developed here between these two approaches and applications made to both. In the tame case where a root α of an irreducible polynomial f generates a tamely ramified extension of (F, v), simple formulas are given for the Krasner constant, the Brink separant and the diameter of f . Applications are made to analyze situations where f has an approximate root in an extension field, where a polynomial is close to f , and where α is the first coordinate of a minimal pair. A key technical result is a computation in the tame case of the Newton polygon of f(x + α). Invariants of defectless polynomials are discussed and the existence of defectless polynomials with given invariants is proven. Khanduja’s characterization of the tame polynomials whose Krasner constants equal their diameters is generalized to arbitrary defectless polynomials over Henselian fields. 2000 Mathematics Subject Classification. Primary: 12J20, 12E05 ; Secondary: 12J10 .