Modul: MAT075 Zurich Graduate Colloquium

## What is... Chebotarev density?

Talk by Dr. Kaloyan Slavov

**Date:** 28.09.21 **Time:** 16.30 - 17.30 **Room:**

Take a polynomial map \(\mathbb{F}_q\to\mathbb{F}_q\) over a (large) finite field and compute the fraction of elements in its image. Most likely, you got \(\approx 0.632\). Once we explain why that is, we arrive at a group theory question: Suppose a subgroup $G$ of the symmetric group \(S_n\) has the same fraction of fixed-point-free elements as \(S_n\) itself. Does it follow that \(G=S_n$\)? The talk will be nontechnical. We will invoke a property of \(e=2.71...\).