The Absolute Galois Group of a Global Field

Abstract

AbstractIn this chapter we will deal with two fundamental questions in algebraic number theory. The first of these questions is the determination of all extensions of a fixed base field k (where the most important case is \(k = \mathbb{Q}\)), which means exploring how these extensions are built up over each other, how they are related and how they can be classified. In other words, we want to study the structure of the absolute Galois group G k of k as a profinite group. But in contrast to the local Galois groups we are far from a complete understanding of the global situation and there are many conjectures but only a few conceptual results. For example, there is a famous conjecture due to I. R. Šafarevič which asserts that the subgroup G k(μ) of G k is a free profinite group, where k(μ) is the field obtained from k by adjoining all roots of unity. This was proved by F. Pop [171] for function fields, but the conjecture is open in the number field case.Mathematics Subject Classification11Gxx11Rxx11Sxx12Gxx14Hxx20Jxx