Documenta Math. 517 Function Fields of One Variable over PAC Fields Moshe Jarden and Florian Pop Received: July 8, 2009 Communicated by Peter Schneider Abstract. We give evidence for a conjecture of Serre and a conjec- ture of Bogomolov. 2000 Mathematics Subject Classification: 12E30, 20G15 Keywords and Phrases: Function fields of one variable, PAC fields, Conjecture II of Serre, Conjecture II of Serre considers a field F of characteristic p with cd(Gal(F )) ≤ 2 such that either p = 0 or p > 0 and [F : F p ] ≤ p and predicts that H 1 (Gal(F ),G) = 1 (i.e. each principal homogeneous G-spaces has an F - rational point) for each simply connected semi-simple linear algebraic group G [Ser97, p. 139]. As Serre notes, the hypothesis of the conjecture holds in the case where F is a field of transcendence degree 1 over a perfect field K with cd(Gal(K)) ≤ 1. Indeed, in this case cd(Gal(F )) ≤ 2 [Ser97, p. 83, Prop. 11] and [F : F p ] ≤ p if p> 0 (by the theory of p-bases [FrJ08, Lemma 2.7.2]). We prove the conjecture for F in the special case, where K is PAC of characteristic 0 that contains all roots of unity. One of the main ingredients of the proof is the projectivity of Gal(K(x) ab ) (where x is transcendental over K and K(x) ab is the maximal Abelian ex- tension of K(x)). We also use the same ingredient to establish an analog to the wellknown open problem of Shafarevich that Gal(Q ab ) is free. Under the assumption that K is PAC and contains all roots of unity we prove that Gal(K(x) ab ) is not only projective but even free. This proves a stronger version of a conjecture of Bogomolov for a function field of one variable F over a PAC field that contains all roots of unity [Pos05, Conjecture 1.1]. 517