Inverse Galois Problem & Root Clusters

Abstract

The first part of the thesis is on right splitting, Galois correspondence, Galois representations and Inverse Galois problem. The famous ‘Inverse Galois problem’ IGP asks whether every finite group appears as the Galois group of some finite Galois extension of $\mathbb{Q}$. Using Galois representations attached to elliptic curves, Arias-de-Reyna and K{\"o}nig in \cite{arias2022locally} have proved the IGP for $\GL_2(\F_p)$ over $\Q$ for all primes $p\geq 5$. Through Galois correspondence and right splitting of some exact sequences of groups, we obtain some general results and apply these to the case in K{\"o}nig and obtain interesting occurrences of IGP. The IGP for $\PSL_2(\F_p)$ over $\Q$ for all primes $p \geq 5 $ was established by Zywina in \cite{zywina2023modular} using the results of Ribet in \cite{ribet1985adic} about the Deligne’s Galois representations associated to certain newforms. Using algebraic operations on Galois representations and right splitting of some exact sequences of groups, we obtain the main results and then apply these to the case in Zywina and obtain equally interesting occurrences of IGP. The second part of the thesis is on Root Clusters, Magnification, Capacity, Unique chains, Base change and Ascending Index. Inspired from the the work of M Krithika and P Vanchinathan in \cite{krithika2023root} on Cluster Magnification and the work of Alexander Perlis in \cite{perlis2004roots} on Cluster Size, we establish the existence of polynomials for given degree and cluster size over number fields which generalises a result of Perlis. We state the Strong cluster magnification problem and establish an equivalent criterion for that. We also discuss the notion of weak cluster magnification and prove some properties. We provide an important example answering a question about Cluster Towers. We introduce the concept of Root capacity and prove some of its properties. We also introduce the concept of unique descending and ascending chains for extensions and establish some properties and explicitly compute some interesting examples. Finally we establish results about all these phenomena under a particular type of base change. The thesis concludes with results about strong cluster magnification and unique chains and some properties of the ascending index for a field extension.