ON IRREDUCIBLE FACTORS OF POLYNOMIALS OVER COMPLETE FIELDS

Let (K, v) be a complete rank-1 valued field. In this paper, we extend classical Hensel's Lemma to residually transcendental prolongations of v to a simple transcendental extension K(x) and apply it to prove a generalization of Dedekind's theorem regarding splitting of primes in algebraic number fields. We also deduce an irreducibility criterion for polynomials over rank-1 valued fields which extends already known generalizations of Schonemann Irreducibility Criterion for such fields. A refinement of Generalized Akira criterion proved in Khanduja and Khassa [Manuscripta Math. 134(1-2) (2010) 215-224] is also obtained as a corollary of the main result.