On Eisenstein polynomials and zeta polynomials
نویسندگان
چکیده
منابع مشابه
On coefficient valuations of Eisenstein polynomials
Résumé. Soit p ≥ 3 un nombre premier et soient n > m ≥ 1. Soit πn la norme de ζpn − 1 sous Cp−1. Ainsi Z(p)[πn]|Z(p) est une extension purement ramifiée d’anneaux de valuation discrète de degré pn−1. Le polynôme minimal de πn sur Q(πm) est un polynôme de Eisenstein; nous donnons des bornes inférieures pour les πm-valuations de ses coefficients. L’analogue dans le cas d’un corps de fonctions, co...
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is Eisenstein at a prime p when each coefficient ci is divisible by p and the constant term c0 is not divisible by p 2. Such polynomials are irreducible in Q[T ], and this Eisenstein criterion for irreducibility is the way nearly everyone first meets Eisenstein polynomials. Here, we will show Eisenstein polynomials are closely related to total ramification of primes in number fields. Let K be a...
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Let φ(x) be an Eisenstein polynomial of degree n over a local field and α be a root of φ(x). Our main tool is the ramification polygon of φ(x), that is the Newton polygon of ρ(x) = φ(αx+α)/(αx). We present a method for determining the Galois group of φ(x) in the case where the ramification polygon consists of one segment.
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An analogue of the Eisenstein irreducibility criterion is developed for linear differential operators, or, more generally, noncommutative polynomials, and is applied to a few simple examples. Introduction. The question of the irreducibility of an ordinary linear differential operator is of interest in the Picard-Vessiot theory (see Kolchin [1, §22]). Indeed, the operator is irreducible if and o...
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ژورنال
عنوان ژورنال: Journal of Pure and Applied Algebra
سال: 2019
ISSN: 0022-4049
DOI: 10.1016/j.jpaa.2019.01.002