Decomposition and Inertia Groups

In this chapter we will study extra structure in the case when $K$ is Galois over  $\mathbf {Q}$. We will learn about Frobenius elements, the Artin symbol, decomposition groups, and how the Galois group of $K$ is related to Galois groups of residue class fields. These are the basic structures needed to attach $L$-function to representations of $\Gal (\overline{\mathbf{Q}}/\mathbf{Q})$, which will play a central role in the next few chapters.