## Olivetti Club

The Class Group of a Number Field $K$ is an invariant which underscores its arithmetic complexity, it's finite and its order is refered to as the class number of $K$. The inquiry into the divisibility properties of Class numbers of cyclotomic fields $\mathbb{Q}(e^{\frac{2\pi i}{p}})$ coincides with the evolution of the theory of Galois representations associated to elliptic curves and modular forms. We will see that divisibility properties of class numbers are encoded by upper triangular Galois representations

\[\rho:G_{\mathbb{Q}}\rightarrow GL_2(\mathbb{F}_p)\]

with special properties. To illustrate the effectivity of this perspective, we'll describe a proof of Kummer's criterion due to Herbrand and Ribet. Namely, an odd prime $p$ divides the class number of $\mathbb{Q}(e^{\frac{2\pi i}{p}})$ precisely when it divides the $k$th Bernoulli number $B_k$ for an even $k$ between $2$ and $p-3$.