# A characterization of Eisenstein polynomials generating extensions of degree ${p}^{2}$ and cyclic of degree ${p}^{3}$ over an unramified $\U0001d52d$-adic field

Maurizio Monge^{[1]}

- [1] 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

Journal de Théorie des Nombres de Bordeaux (2014)

- Volume: 26, Issue: 1, page 201-231
- ISSN: 1246-7405

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topMonge, 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 -

## References

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