Preview

In the next chapter we will study extra structure in the case when $K$ is Galois over  $\mathbf{Q}$; the results are nicely algebraic, beautiful, and have interesting ramifications. We'll 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 make any sense of representations of Galois groups, which is at the heart of much of number theory.