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SUMMARY:Jamshid Derakhshan (Oxford University)
DTSTART:20240208T190000Z
DTEND:20240208T200000Z
DTSTAMP:20260423T052837Z
UID:OLS/147
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OLS/147/">De
 cidability of the class of all the rings $Z/mZ$: A problem of Ax</a>\nby J
 amshid Derakhshan (Oxford University) as part of Online logic seminar\n\n\
 nAbstract\nIn 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 existen
 tial 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 axiomatizatio
 n of the theory of pseudo-finite fields. In that paper Ax proved that the 
 theory of the class of all finite fields is decidable.\n\nIn this talk I w
 ill present joint work with Angus Macintyre giving a solution to Ax’s pr
 oblem. 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 pr
 oblem can be reduced to the decidability of the ring of adeles of the rati
 onal numbers. So while the theory of pseudo-finite fields governs the theo
 ry of all finite fields as shown by Ax\, the theory of all $Z/mZ$ is gover
 ned by the theory of the rational adele ring.\n\n(This work is published i
 n Forum of Mathematics\, Sigma\, 24 July 2023.)\n
LOCATION:https://researchseminars.org/talk/OLS/147/
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