Advanced Search

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 problem...

Given a number field $K$ Galois over the rational field $\mathbb{Q}$, and a positive integer $n$ prime to the class number of $K$, there exists an abelian extension $L/K$ (of exponent $n$) such that the $n$-torsion subgroup of the Brauer group of $K$ is equal to the relative Brauer group of $L/K$.