First Order Definition of Rings Using Group of Units

Abstract

We discuss the technical background and relevant research regarding the undecidability of OQab. Given an algebraic extension K/Q, we consider the subring defined by RK={x [epsilon] OK \[for-all] [epsilon] UK \ {1\}\ [exists][delta]\in UK\[delta]-1\[equivalent] x([epsilon]-1) mod ([epsilon]-1)^²}.We later consider a similar construction over subrings of Q of characteristic 0. In doing this, we hope to gain insight into the result of the construction of RK when K=Qab.