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On utilise les méthodes de Neukirch et Poitou pour écrire les conditions locales et globales des problèmes de plongement. Le cas étudié ici est celui du plongement d’une extension diédrale dans une extension diédrale ou quaternionienne, le corps de base étant un corps de nombres.

Polynômes à groupe de Galois diédral

Dominique Martinais, Leila Schneps (1992)

Journal de théorie des nombres de Bordeaux

Soit $K$ un corps et $K_{1}$ une extension quadratique de $K$ . Étant donné un polynôme $P$ de $K_{1} [X]$ à groupe de Galois cyclique, nous donnons une méthode pour construire un polynôme $Q$ de $K [X]$ à groupe de Galois diédral, à partir des racines de $P$ . Cette méthode est tout à fait explicite : nous donnons de nombreux exemples de polynômes à groupe de Galois diédral sur le corps $ℚ$ .

Polynomial Imaginary Decompositions for Finite Separable Extensions

Adam Grygiel (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Let K be a field and let L = K[ξ] be a finite field extension of K of degree m > 1. If f ∈ L[Z] is a polynomial, then there exist unique polynomials $u ₀, . . ., u_{m - 1} \in K [X ₀, . . ., X_{m - 1}]$ such that $f (\sum_{j = 0}^{m - 1} ξ^{j} X_{j}) = \sum_{j = 0}^{m - 1} ξ^{j} u_{j}$ . A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then $u ₀, . . ., u_{m - 1}$ have no common divisor in $K [X ₀, . . ., X_{m - 1}]$ of positive degree. We extend this result to the case when L is a separable extension of a field K of arbitrary characteristic. We also show that the same is true for a formal power series in several variables....

Polynomial subfields of k (x).

Alan McConnell (1974)

Journal für die reine und angewandte Mathematik

Polynomials of the form $g (x^{k})$ and pseudoprimes with respect to linear recurring sequences

František Marko (1992)

Mathematica Slovaca

Polynomials over $Q$ solving an embedding problem

Nuria Vila (1985)

Annales de l'institut Fourier

The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_{n}$ , can be embedded in any central extension of $A_{n}$ if and only if $n \equiv 0 (m o d 8)$ , or $n \equiv 2 (m o d 8)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$ , every central extension of $A_{n}$ occurs as a Galois group over $Q$ .

Polynomials whose Galois groups are Frobenius groups with prime order complement

Leonardo Cangelmi (1994)

Journal de théorie des nombres de Bordeaux

We give an effective characterization theorem for integral monic irreducible polynomials $f$ of degree $n$ whose Galois groups over $ℚ$ are Frobenius groups with kernel of order $n$ and complement of prime order.

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$p$ -extensions abéliennes

Parallélogrammes galoisiens infinis

Pfister's conjecture on quadratic Co-fields.

Plongeants d'extensions galoisiennes

Plongement d'une extension diédrale dans une extension diédrale ou quaternionienne

Polynômes à groupe de Galois diédral

Polynomial Imaginary Decompositions for Finite Separable Extensions

Polynomial subfields of k (x).

Polynomials of the form $g (x^{k})$ and pseudoprimes with respect to linear recurring sequences

Polynomials over $Q$ solving an embedding problem

Polynomials whose Galois groups are Frobenius groups with prime order complement

Polynomials with nontrivial relations between their roots

Precise study of some number fields and Galois actions occurring in conformal field theory

Prehomogeneous vector spaces and field extensions.

Primitive elements in rings of holomorphic functions.

Pro-2-Demuskin groups of rank ...0 as Galois groups of maximal 2-extensions of fields.

Problème de plongement et contraintes galoisiennes sur le groupe des classes

Prolongement de la valeur absolue de Gauss et problème de Skolem

Pro-p Galois groups of rank ... 4.

Propriété Ci revue et généralisée

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