THE ABSOLUTE GALOIS GROUPS OF FINITE EXTENSIONS OF R(t)∗

Let R be a real closed field and L be a finite extension of R(t). We prove that Gal(L) ∼= Gal(R(t)) if L is formally real and Gal(L) is the free profinite group of rank card(R) if L is not formally real. MR Classification: 12E30 Directory: \Jarden\Diary\RealFree 3 April, 2007 * Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. Introduction Let R be a real closed field, t an indeterminate, and K = R(t) the field of rational functions in t over R. In their work [KrN71], Krull and Neukirch consider the case where R is the field of real numbers R. For each finite set S of prime divisors of K/R they introduce the maximal extension KS of K unramified outside S and present Gal(KS/K) by generators and relations. Based on this description, they present the absolute Galois group Gal(K) as a semi-direct product of Gal(C/R) and Gal(C(t)) with an explicit action. Schuppar [Sch80] extends the results of [KrN71] to an arbitrary real closed field R. In [HaJ85] we apply the presentation of Gal(KS/K) by generators and relations to present Gal(K) (for an arbitrary real closed field R) as a free product C(X) ∗ F , where C(X) is a free product of groups of order 2 over an indexed profinite space X of weight m = card(R) and F is a free profinite group of rank m. In a letter to the second author, David Harbater asked about the isomorphism type of Gal(L), where L ranges over the finite extensions of K. In particular he asked whether Gal(L) depends on the number of the connected components of Γ(R), where Γ is a smooth model of K/R. The goal of this note is to prove that there are actually only two isomorphism types for Gal(L), either Gal(K) or a free profinite group of rank m = card(R). Indeed, we prove the following theorem. Main Theorem: Let R be a real closed field, K = R(t) the field of rational functions over R, and L a finite extension of K. Let C(X) be the free product on a constant sheaf of groups of order 2 over the profinite space X of orderings of K, and let F be the free profinite group of rank card(R). If L is formally real, then Gal(L) ∼= C(X) ∗ F ; if L is not formally real, then Gal(L) ∼= F . Our proof applies Kurosh Subgroup Theorem for free profinite product of finitely many profinite groups to reduce the main theorem to the case K = L. An essential ingredient in the proof is Proposition 1.4 which states that every non-empty open-closed subset of the space of orderings X(K) of K is homeomorphic to X(K).