Abstract Class Field Theory

Abstract

Every field k is equipped with a distinguished Galois extension: the separable closure $\bar k|k$. Its Galois group ${G_k} = G(\bar k|k)$ is called the absolute Galois group of k. As a rule, this extension will have infinite degree. It does, however, have the advantage of collecting all finite Galois extensions of k. This is why it is reasonable to try to give it a prominent place in Galois theory. But such an attempt faces the difficulty that the main theorem of Galois theory does not remain true for infinite extensions. Let us explain this in the following