Sharply k-Transitive Permutation Groups Viewed as Galois Groups

The classification of finite sharply k-transitive groups was achieved by the efforts of Jordan (1873), Dickson (1905), and Zassenhaus (1936). Likewise for other families of finite groups, one expects that they are realizable as Galois groups over the field of rational numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}$$\end{document}. In this article, we study some properties of the polynomials \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f \in \mathbb{Q}[x]}$$\end{document} such that the Galois group Gal(f) acts sharply k-transitively on its roots.