Decidability of the class of all the rings : A problem of Ax
-
Published:2023
Issue:
Volume:11
Page:
-
ISSN:2050-5094
-
Container-title:Forum of Mathematics, Sigma
-
language:en
-
Short-container-title:Forum of Mathematics, Sigma
Author:
Derakhshan Jamshid,Macintyre Angus
Abstract
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
$\mathbb {Q}$
.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
Reference8 articles.
1. Model theory of adeles I;Derakhshan;Ann. Pure Appl. Logic 173,2022
2. The Elementary Theory of Finite Fields