In this seminar, I aim to go through the "main prequel" of the talk I gave during the first Advanced Class of this term, and provide a "simple" answer to Abraham Robinson's original question that he posed in 1973 regarding the (un)decidability of finitely generated extensions of undecidable fields. I will provide a quick introduction to, and some classical results from, the mathematical discipline of Field Arithmetic, and using these results show that one can construct undecidable (large) fields that have finitely generated extensions which are decidable. Of course, as I had mentioned in the advanced class, a counterexample to the "simple" question that I have been working on unfortunately does not seem to lie within this class of large fields. If time permits, I will provide a sneak peek into the possible "sequel" by briefly talking about what the main issue of solving the "simple" problem is, and how a "hide-and-seek" method might come in handy in tackling that problem.
