## Distribution of Galois groups

**Goal:**

Associated to an irreducible monic polynomial with rational coefficients we have its Galois group, which measures the symmetries of the complex solutions of this polynomial.

If instead of rational coefficients we use integral coefficients, we can consider those polynomials for which the absolute value of the coefficients is bounded above by a small constant, so that we have a finite set. If we compute all the Galois groups that appear for such polynomials, and let the bound grow higher and higher, we will observe some patterns.

The goal is to understand which pattern appears, and understand some of the group theory that goes into these patterns.

This project can be done without having done Algèbre first: you will learn what a Galois group is during the course. For more advanced students we can discuss some of the algorithmic aspects that arise.

**Supervisors: **
Pieter Belmans

**Difficulty level:**
any

**Tools:**