Random Polynomials, Probabilistic Galois Theory, and Finite Field Arithmetic
Heilbronn Number Theory Seminar
14th October 2020, 4:00 pm – 5:00 pm
In the talk we will discuss recent advances on the following two questions:
Let A(X) = sum ±X^i 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 group 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.