Random Algebraic Extensions of $\mathbb{Q}$.

 

Abstract: What does a “typical” field look like? To answer this question, we explore a notion of algorithmic randomness for fields, making use of the Haar measure on the absolute Galois group. We prove the existence of algebraic extensions of $\mathbb{Q}$ which are random in this sense, and that this notion does not coincide with the set of noncomputable algebraic fields. Joint work with Valentina Harizanov and Alexandra Shlapentokh.