This is the introductory lecture of the course on class field theory in the Research school on Arithmetic Statistics in May 2023. It briefly reviews the necessary algebraic number theory, and presents class field theory as the analogue of the Kronecker-Weber theorem over number fields. In a similar way, the Chebotarev density theorem is treated as an analogue of the Dirichlet theorem on primes in arithmetic progressions.
Two further lectures dealt with idelic and cohomological reformulations of the main theorem of class field theory, and two more were devoted to power reciprocity laws and Redei reciprocity.[-]

Let $L/K$ be an extension of number fields. The norm map $N_{L/K} :L^{*}\to K^{*}$ extends to a norm map from the ideles of L to those of $K$. The Hasse norm principle is said to hold for $L/K$ if, for elements of $K^{*}$, being in the image of the idelic norm map is equivalent to being the norm of an element of L^{*}. The frequency of failure of the Hasse norm principle in families of abelian extensions is fairly well understood, thanks to previous work of Christopher Frei, Daniel Loughran and myself, as well as recent work of Peter Koymans and Nick Rome. In this talk, I will focus on the non-abelian setting and discuss joint work with Ila Varma on the statistics of the Hasse norm principle in field extensions with normal closure having Galois group $S_{4}$ or $S_{5}$.[-]