A characterization of Eisenstein polynomials generating extensions of degree $p^2$ and cyclic of degree $p^3$ over an unramified $𝔭$-adic field

Monge, Maurizio. "A characterization of Eisenstein polynomials generating extensions of degree $p^2$ and cyclic of degree $p^3$ over an unramified $\mathfrak{p}$-adic field." Journal de Théorie des Nombres de Bordeaux 26.1 (2014): 201-231. <http://eudml.org/doc/275781>.

@article{Monge2014,
abstract = {Let $p\ne 2$ be a prime. We derive a technique based on local class field theory and on the expansions of certain resultants allowing to recover very easily Lbekkouri’s characterization of Eisenstein polynomials generating cyclic wild extensions of degree $p^2$ over $\mathbb\{Q\}_p$, and extend it to when the base fields $K$ is an unramified extension of $\mathbb\{Q\}_p$.When a polynomial satisfies a subset of such conditions the first unsatisfied condition characterizes the Galois group of the normal closure. We derive a complete classification of Eisenstein polynomials of degree $p^2$ whose splitting field is a $p$-extension, providing a full description of the Galois group and its higher ramification subgroups.The same methods are used to give a characterization of Eisenstein polynomials of degree $p^3$ generating a cyclic extension.In the last section, we deduce a combinatorial interpretation of monomial symmetric functions evaluated in the roots of the unity, which appear in certain expansions.},
affiliation = {Instituto de Matemática da UFRJ, Av. Athos da Silveira Ramos 149, Centro de Tecnologia Bloco C Cidade Universitária Ilha do Fundão Caixa Postal 68530 21941-909 Rio de Janeiro - RJ - Brasil},
author = {Monge, Maurizio},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {4},
number = {1},
pages = {201-231},
publisher = {Société Arithmétique de Bordeaux},
title = {A characterization of Eisenstein polynomials generating extensions of degree $p^2$ and cyclic of degree $p^3$ over an unramified $\mathfrak\{p\}$-adic field},
url = {http://eudml.org/doc/275781},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Monge, Maurizio
TI - A characterization of Eisenstein polynomials generating extensions of degree $p^2$ and cyclic of degree $p^3$ over an unramified $\mathfrak{p}$-adic field
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/4//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 1
SP - 201
EP - 231
AB - Let $p\ne 2$ be a prime. We derive a technique based on local class field theory and on the expansions of certain resultants allowing to recover very easily Lbekkouri’s characterization of Eisenstein polynomials generating cyclic wild extensions of degree $p^2$ over $\mathbb{Q}_p$, and extend it to when the base fields $K$ is an unramified extension of $\mathbb{Q}_p$.When a polynomial satisfies a subset of such conditions the first unsatisfied condition characterizes the Galois group of the normal closure. We derive a complete classification of Eisenstein polynomials of degree $p^2$ whose splitting field is a $p$-extension, providing a full description of the Galois group and its higher ramification subgroups.The same methods are used to give a characterization of Eisenstein polynomials of degree $p^3$ generating a cyclic extension.In the last section, we deduce a combinatorial interpretation of monomial symmetric functions evaluated in the roots of the unity, which appear in certain expansions.
LA - eng
UR - http://eudml.org/doc/275781
ER -