In the talk, we will discuss the irreducibility and the Galois group of random polynomials over the integers. After giving motivation (coming from work of Breuillard--Varjú, Eberhard, Ferber--Jain--Sah--Sawhney, and others), I will present a result, conditional on the extended Riemann hypothesis, showing that the characteristic polynomial of certain random tridiagonal matrices is irreducible, with probability tending to 1 as the size of the matrices tends to infinity. 

The proof involves random walks in direct products of SL_2(p), where we use results of Breuillard--Gamburd and Golsefidy--Srinivas. 

Joint work with Lior Bary-Soroker and Sasha Sodin.