On the decidability of finite extensions of decidable fields - Thanagopal, K (2018)
This paper is primarily concerned with the following question which first appeared in Koenigsmann’s On a Question of Abraham Robinson’s: Is a finite field extension of a decidable field always decidable? This offers a “twist” to a question that was originally posed by Abraham Robinson in 1973 which had asked whether every finitely generated extension of an undecidable field remains undecidable. The above-mentioned work of Koenigsmann from 2016 (and independently, a result of Cherlin, van den Dries, and Macintyre much earlier in the 1980s) showed that there are indeed undecidable fields which admit decidable finite extensions. This paper aims to show that one could, similarly, find examples of decidable fields which admit undecidable finite extensions, thereby answering the above-stated question negatively. This result is achieved by identifying a sufficient condition which a decidable field must satisfy in order for it to have an undecidable finite extension. In an earlier iteration of this work, we had pointed out what we had believed to be one such condition. Unfortunately, this turned out not to be the case, which we illustrate using an explicit example. Through this demonstration, we were able to accentuate the weakness of the formerly mentioned criterion, which we strengthen in this thesis. We provide justification that the strengthened criterion is indeed sufficient – any decidable field satisfying this strengthened criterion would form a counterexample to the above-mentioned question. We study one such class of decidable fields, known as the wonderful extensions of the rational numbers, first introduced by Ershov in the early 2000s, whose (sufficiently saturated) elementary extensions satisfy this strengthened criterion. This provides us with a concrete counterexample which shows that there are indeed decidable fields which admit undecidable finite extensions. We also point out various attempts at finding other counterexamples to the above-mentioned question, the difficulties faced in those instances, and some further questions in the flavour of the above-mentioned question that appear to be interesting in their own rights.