# Mathematical Sciences Research Institute

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# Seminar

Nondefinability of Rings of Integers in Algebraic Extensions of the Rationals August 03, 2022 (02:15 PM PDT - 03:00 PM PDT)
Parent Program: Definability, Decidability, and Computability in Number Theory, part 2 MSRI: Simons Auditorium, Online/Virtual
Speaker(s) Arno Fehm (TU Dresden)
Description No Description
Video

#### Nondefinability Of Rings Of Integers In Algebraic Extensions Of The Rationals

Abstract/Media

Complementing the various methods to define subrings of fields, I will state a few general principles that tell us a priori that subrings of certain fields cannot possibly be definable (or at least not existentially definable). I will then discuss joint work with Philip Dittmann in which we determine the common theory of (in a precise sense) almost all'' algebraic extensions of the field of rational numbers, in order to conclude that almost all such extensions have no definable proper subrings at all, strengthening a recent result by Eisentr\"ager, Miller, Springer and Westrick.