Random Polynomials, Probabilistic Galois Theory, and Finite Field Arithmetic

ALGEBRAIC GEOMETRY & NUMBER THEORY SEMINAR

We will discuss recent advances on the following two question: Let A(X) =Σ ±Xi be a random polynomial of degree n with coefficients taking the values -1, 1 independently each with probability 1/2.
Q1: What is the probability that A is irreducible as the degree goes to infinity
Q2: What is the typical Galois of A?
One believes that the answers are YES and THE FULL SYMMETRIC GROUP, respectively. These questions were studied extensively in recent years, and we will survey the tools developed to attack these problems and partial results.