Probabilistic Galois Theory studies the distribution of Galois groups in various natural ensembles of Galois extensions of global fields. This area goes back to the classical result of van der Waerden that most polynomials $X^n+a_{n-1}X^{n-1}+...+a_0$, $a_i$ integers, $|a_i|<H$ have Galois group $S_n$ over $Q$ (n fixed, $H \to infty$) and has seen much activity and progress in recent years. In the talk I will survey the history of Probabilistic Galois Theory and recent developments in the area and discuss recent work joint with Alexander Popov, Lior Bary-Soroker and Eilidh McKemmie on the distribution of Galois groups over $F_q(t)$ (where $q$ is a prime power) of polynomials of the form $X^n+a_{n-1}(t)X^{n-1}+...+a_0(t)$, $a_i(t) \in F_q[t]$ as well as additive polynomials of the form $X^{q^n}+a_{n-1}(t)X^{q^{n-1}}+...+a_0(t)X,$ $a_i(t) \in F_q[t].$ A novel feature of the latter family is that it involves polynomials with Galois group $GL_n(q)$ and variants, while previously studied families involved variants of $S_n$. The main results rely crucially on deep results on subgroups of $GL_n(q)$, combined with a variety of tools from algebra and analytic number theory.

# Cable Car Algebra Seminar: Probabilistic Galois theory in function fields

Abstract: