Reconstruction of Function Fields Notes by Valentijn Karemaker

Let ` be a prime number and let k be an algebraically closed field with char(k) 6= `. Let K/k be a function field of transcendence degree td(K/k) > 1 with absolute Galois group GK . Take G K = GK and for all i ≥ 1, let G (i+1) K = [G , G](G) ∞ . In other words, the groups {G K }i≥1 form the descending central `∞-series of GK . Then ΠK = G /G = Gal(K ′/K) is the Galois group of the maximal pro-` abelian extension K ′ of K, and ΠK = G (1) K /G (3) K = Gal(K ′′/K) is the Galois group of the maximal pro-` abelian-by-central extension K ′′ of K. Note that the projection map pr : ΠK → ΠK has kernel [ΠK ,ΠK ], and therefore ΠK can be group theoretically recovered from ΠK . We will prove the following theorem, following the results of Pop in [1–3].