Genus numbers in families of number fields

Let $K$ be an algebraic number field. The genus field $K^$ of $K$ is the maximal abelian extension of $K$ that is unramified at all finite primes and can be written as $K^ = k^K$, where $k^$ is an abelian extension of $\mathbb{Q}$. The genus number $g_K$ is the degree $[K^* : K]$. In this talk, I will present some results on the distribution of genus numbers within a particular family of number fields, highlighting both classical techniques and recent developments.