2 - Class Field Theory

Summary

For a global function field F/F_q, all finite abelian extensions of F can be described by class field theory. For our purposes, we want to construct global function fields with many rational places by considering finite abelian extensions of a given global function field. In this chapter, we discuss some important results from class field theory and focus on ray class fields, narrow ray class fields, and class field towers. Some well-known results are presented without proof. The books of Cassels and Fröhlich [13], Koch [60], Neukirch [85], Serre [140], Weil [172], and Weiss [173] are suitable references for this chapter.

Local Fields

In order to obtain information about algebraic function fields over finite fields, it is often useful to study their various completions. It is a common technique in algebraic number theory and algebraic geometry to reduce problems to local fields.

Definition 2.1.1 A discrete valuation of a field E is a surjective map v: E → Gu{∞}, where G is a nonzero discrete subgroup of (R, +) and where v satisfies:

1. (i) v(x) = ∞ if and only if x = 0;

2. (ii) v(xy) = v(x) + v(y) for all _{x, y} ∈ E.

3. (iii) v(x + y) ≥ min(v(x), v(y)) for all x, y ∈ E.

The field E together with the discrete valuation v, or more precisely the ordered pair (E, v), is called a value d field. If v(E^*) = Z, then the discrete valuation v is called normalized.