This paper considers some refined versions of the Inverse Galois Problem. We study the local or global behavior of rational specializations of some finite Galois covers of ${\mathbb{P}}_{\mathbb{Q}}^1$.

An effective construction of homogeneous linear differential equations of order 2 with Galois group $2A_4,2S_4$ or $2A_5$ is presented.

The problem of the construction of number fields with Galois group over Q a given finite groups has made considerable progress in the recent years. The aim of this paper is to survey the current state of this problem, giving the most significant methods developed in connection with it.

Let ${\mathbb{Z}}_d\wr$m$bethegeneralizedalternatinggroup.Weprovethatalldoublecoversof$ℤd ≀ m$occurasGaloisgroupsoveranyalgebraicnumberfield.WefurtherrealizesomeofthesedoublecoversastheGaloisgroupsofregularextensionsof\mathbb{Q}\left(T\right).Ifdisoddandm>7,theneverycentralextensionof$ℤd ≀ m$occursastheGaloisgroupofaregularextensionof\mathbb{Q}\left(T\right).Wefurtherimprovesomeofourearlierresultsconcerningdoublecoversofthegeneralizedsymmetricgroup$ℤd ≀ m$.$

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 $n\equiv 0\left(mod8\right)$, or $n\equiv 2\left(mod8\right)$ 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 $\mathbf{Q}$.

A (monic) polynomial $f\left(x\right)\in \mathbb{Z}\left[x\right]$ is called if the congruence $f\left(x\right)\equiv 0$ mod $m$ has a solution for all positive integers $m$. Call $f\left(x\right)$ if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group $G$ can be realized as the Galois group over $\mathbb{Q}$ of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable $G$ reduces to the ordinary inverse Galois...