Profinite Groups and Hilbertian Fields

Zusammenfassung

Here we continue the discussion on profinite groups of Chapter 1 to further our investigation of Hilbertian fields. In particular we show that an abelian extension of a Hilbertian field K is Hilbertian, and that a closed normal subgroup of G (K) can be neither finitely generated nor prosolvable (Theorem 15.10). Central to this chapter is also a discussion on free profinite groups. In particular we prove that an open subgroup of a free profinite group is free (Proposition 15.27).