On Newton polygons techniques and factorization of polynomials over Henselian fields

Abstract

Let [Formula: see text] be a valued field, where [Formula: see text] is a rank-one discrete valuation, with valuation ring [Formula: see text]. The goal of this paper is to investigate some basic concepts of Newton polygon techniques of a monic polynomial [Formula: see text]; namely, theorem of the product, of the polygon, and of the residual polynomial, in such way that improves that given in [D. Cohen, A. Movahhedi and A. Salinier, Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika 47 (2000) 173–196] and generalizes that given in [J. Guardia, J. Montes and E. Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364(1) (2012) 361–416] to any rank-one valued field.

On power integral bases of certain pure number fields defined by $x^{3^r\cdot 7^s}-m$

“…Now we recall some fundamental facts about Newton polygons. For more details, we refer to [14] and [22]. For any prime integer p and for any monic…”

On power integral bases of certain pure number fields defined by $$x^{42} - m$$

“…We start by recalling some fundamental facts on Newton polygon techniques. For more details, we refer to [10,11,21]. Let p be a rational prime, m p the discrete valuation of Q p ðxÞ defined on Z p ½x by m p ð P r i¼0 a i x i Þ ¼ minfm p ða i Þ; 0 i rg, and / 2 Z½x be a monic polynomial whose reduction modulo p is irreducible.…”

On power integral bases for certain pure number fields defined by $x^{2\cdot 3^k}-m$

“…In 1928, Ore developed an alternative method for obtaining the index (Z K : Z[α]), the absolute discriminant, and the prime ideal factorization of the rational primes of Z K by using Newton polygons (see [24,26]). For more details on Newton polygon techniques, we refer to [13,21].…”