Counting Number Fields -- Conjectures, Results, and Methods

For a fixed integer d, how many number fields are there of fixed degree d of absolute discriminant less than X? Denote this number N(d, X). This talk will be a friendly foray through the known results, proofs, and main conjectures regarding this question. In particular, it is known that if d = 2,3,4,5 then N(d, X) = O(X). We will discuss the key components of the proofs, which involve the parametrizations of number fields and geometry of numbers. For n >= 6, we will discuss Malle's conjecture, which predicts the asymptotics of N(d, X) for general degree d. We will also discuss classical upper bounds on N(d, X).