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.

Soit $K$ un corps et $K_1$ une extension quadratique de $K$. Étant donné un polynôme $P$ de $K_1\left[X\right]$ à groupe de Galois cyclique, nous donnons une méthode pour construire un polynôme $Q$ de $K\left[X\right]$ 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 $\mathbb{Q}$.

We describe the ring of constants of a specific four variable Lotka-Volterra derivation. We investigate the existence of polynomial constants in the other cases of Lotka-Volterra derivations, also in n variables.

Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form $d=x(Cy+z)\frac{\partial }{\partial x}+y(Az+x)\frac{\partial }{\partial y}+z(Bx+y)\frac{\partial }{\partial z}$, called the Lotka-Volterra derivation, where A,B,C ∈ k.

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 $u₀,...,u_{m-1}\in K[X₀,...,X_{m-1}]$ such that $f\left({\sum }_{j=0}^{m-1}{\xi }^j{X}_j\right)={\sum }_{j=0}^{m-1}{\xi }^j{u}_j$. A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then $u₀,...,u_{m-1}$ have no common divisor in $K[X₀,...,X_{m-1}]$ 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....

We consider the equations of the form dy/dx = y²-P(x) where P are polynomials. We characterize the possible algebraic solutions and the class of equations having such solutions. We present formulas for first integrals of rational Riccati equations with an algebraic solution. We also present a relation between the problem of algebraic solutions and the theory of random matrices.

Let $𝒜$ be the algebra of quaternions $ℍ$ or octonions $𝕆$. In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial $f\left(t\right)\in 𝒜\left[t\right]$ has a root in $𝒜$. As a consequence, the Jacobian determinant $\left|J\right(f\left)\right|$ is always non-negative in $𝒜$. Moreover, using the idea of the topological degree we show that a regular polynomial $g\left(t\right)$ over $𝒜$ has also a root in $𝒜$. Finally, utilizing multiplication ($*$) in $𝒜$, we prove various results on the topological degree of products...