Which algebraic extensions of the rationals (AERs) have existentially or universally definable algebraic integers? Equipping the set of AERs with a natural topology, we show that only a meager subset have this property. An important tool is a new normal form theorem for existential definitions in AERs. As a corollary we construct countably many distinct computable AERs whose algebraic integers are neither existentially nor universally definable. Joint work with Kirsten Eisentraeger, Russell Miller, and Caleb Springer. This talk will be a more technically detailed presentation of the work described by Springer in his 10/20 talk in the DDC Junior Seminar.