Decidability of the class of all the rings : A problem of Ax
Jamshid Derakhshan, Angus J. Macintyre · Forum of Mathematics Sigma · 2023
Abstract We prove that the class of all the rings $\mathbb {Z}/m\mathbb {Z}$ for all $m>1$ is decidable. This gives a positive solution to a problem of Ax asked in his celebrated 1968 paper on the elementary theory of finite fields [1, Problem 5, p. 270]. In our proof, we reduce the problem to the decidability of the ring of adeles $\mathbb {A}_{\mathbb {Q}}$ of $\mathbb {Q}$ .