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.

Nondefinability Of Rings Of Integers In Algebraic Extensions Of The Rationals