Decidability of the class of all the rings $Z/mZ$: A problem of Ax

Abstract: In his celebrated 1968 paper on the elementary theory of finite fields James Ax asked if the theory of the class of all the rings $Z/mZ$, for all $m>1$, is decidable. In that paper, Ax proved that the existential theory of this class is decidable using his result that the theory of all the rings $Z/p^nZ$ (with $p$ and $n$ varying) is decidable. This used Chebotarev’s density theorem and Ax's pioneering work and axiomatization of the theory of pseudo-finite fields. In that paper Ax proved that the theory of the class of all finite fields is decidable.

In this talk I will present joint work with Angus Macintyre giving a solution to Ax’s problem. Our solution uses some previous work of ours on the model theory of the ring of adeles. These are locally compact rings associated to number fields and have been of fundamental importance in number theory ever since they were introduced by Chevalley, Weil, Artin. Interestingly Ax’s problem can be reduced to the decidability of the ring of adeles of the rational numbers. So while the theory of pseudo-finite fields governs the theory of all finite fields as shown by Ax, the theory of all $Z/mZ$ is governed by the theory of the rational adele ring.

(This work is published in Forum of Mathematics, Sigma, 24 July 2023.)

logicnumber theory

Audience: researchers in the topic

Online logic seminar

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