On Ramification Polygons and Galois Groups of Eisenstein Polynomials
نویسنده
چکیده
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.
منابع مشابه
Galois Groups of Eisenstein Polynomials Whose Ramification Polygon Has One Side
Let φ(x) be an Eisenstein polynomial over a local field and α be a root of φ(x). We describe the Galois group of φ(x) in the case where the Newton polygon of φ(αx+ α)/x has only one side.
متن کاملTotally Ramified Primes and Eisenstein Polynomials
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...
متن کاملp-adic Galois Representations Math 162b Winter 2012 Lecture Notes
1 Local Class Field Theory 2 1.1 Local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Newton polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Ramification of local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Main results of local class field ...
متن کاملMulti-parameter Polynomials with Given Galois Group
The non-Abelian finite simple groups and their automorphism groups play a crucial role in an inductive approach to the inverse problem of Galois theory. The rigidity method (see, for example, Malle and Matzat, 1999) has proved very efficient for deducing the existence of Galois extensions with such groups, as well as for the construction of polynomials generating such extensions. Nevertheless, ...
متن کاملMathematics Surrounding Applications of Riemann’s Existence Theorem
In these notes we collect in one place the various background material involved in applications of Riemann’s existence theorem to questions about polynomials or curves. This will eventually include material on Riemann surfaces, algebraic topology, Galois theory, group theory, ramification, elliptic curves, rigidity,
متن کامل