A classification of the extensions of degree p^{2} over \mathbb{Q}_{p} whose normal closure is a p-extension

Caputo, Luca

doi:10.5802/jtnb.590

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“…While some condition are necessary to force the splitting field to be a p-extension, the remaining conditions can be tested in order, and the first which fails gives information on the Galois group of the splitting field. Taking into account another family of polynomial which can never provide a cyclic extension of degree p 2 , we give a full classification of the polynomials of degree p 2 whose normal closure is a p-extension, providing a complete description of the Galois group of the normal closure with its ramification filtration, see [Cap07] for an abstract classification of all such extensions when the base field is Q p .…”

Section: Introductionmentioning

confidence: 99%

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

Monge

2011

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Let p = 2 be a prime. We show a technique based on local class field theory and on the expansions of certain resultants which allows to recover very easily Lbekkouri's characterization of Eisenstein polynomials generating cyclic wild extensions of degree p 2 over Qp, and to extend it to the case of the base field K being an unramified extension of Qp.Furthermore, when a polynomial satisfies only some of the stated conditions, we show that the first unsatisfied condition gives information about the Galois group of the normal closure. This permits to give 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.We then apply the same methods 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 the monomial symmetric function evaluated in the roots of the unity which appear in certain expansions.

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Section: Introductionmentioning

confidence: 99%

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

Monge

2011

Preprint

View full text Add to dashboard Cite

show abstract

A classification of the extensions of degree p^{2} over \mathbb{Q}_{p} whose normal closure is a p-extension

Cited by 1 publication

References 3 publications

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

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

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