ABSTRACT
GuΓ rdia, Montes and Nart generalized the well-known method of Ore to find complete factorization of polynomials with coeο¬cients in finite extensions of p-adic numbers using Newton polygons of higher order (cf. [Trans. Amer. Math. Soc. 364 (2012), 361β416]). In this paper, we develop the theory of higher order Newton polygons for polynomials with coeο¬cients in henselian valued fields of arbitrary rank and use it to obtain factorization of such polynomials. Our approach is different from the one followed by GuΓ rdia et al. Some preliminary results needed for proving the main results are also obtained which are of independent interest.
Acknowledgment
The authors are highly thankful to the referee for valuable suggestions.
Notes
1A prolongation W of V0 to an overfield of K is called residually transcendental if the residue field of W is a transcendental extension of the residue field of V0.
2On dividing by successive powers of Ο(x), every polynomial f(x)βK[x] can be uniquely written as a finite sum with deg(fi(x))<deg(Ο(x)), called the Ο-expansion of f(x).