Kardos, Michael2023-06-052023-06-052023-052023-05-03May 2023http://hdl.handle.net/10342/12855We discuss the technical background and relevant research regarding the undecidability of $O_{\mathbb{Q}^{\text{ab}}}$. Given an algebraic extension $K/\mathbb{Q}$, we consider the subring defined by $$R_K=\{x\in O_K\,|\,\forall \varepsilon\in U_K\setminus\{1\}\,\exists\delta\in U_K:\delta-1\equiv x(\varepsilon-1)\bmod(\varepsilon-1)^2\}.$$ We later consider a similar construction over subrings of $\mathbb{Q}$ of characteristic $0$. In doing this, we hope to gain insight into the result of the construction of $R_K$ when $K=\mathbb{Q}^{\text{ab}}$.application/pdfennumber theorylogicsubringabelianextensionunitdefinabilityMaster's Thesis2023-06-02