This article describes the finitely generated pro-\(p\) groups \(G\) that can occur as absolute Galois group of an algebraic extension \(F\) of \(K(t)\) where \(t\) is one indeterminate and \(K\) is a local field different from \(\mathbb R\) and \(\mathbb C\). In other words, the quest is for the finitely generated pro-\(p\) subgroups of the absolute Galois group of \(K(t)\). One may think of \(K(t)\) as a two-dimensional object. If \(K(t)\) is replaced by a one-dimensional object, that is, a local or global field, such descriptions are available: see the earlier work of the author and also the recent book of J. Neukirch, A. Schmidt, and K. Wingberg [Cohomology of number fields. Grundlehren der Math. Wiss. 323 Berlin: Springer-Verlag (2000) in particular §X.5]. Moreover, the case \(K={\mathbb Q}_l\) with \(l\not=p\) was treated in 1996 by C. U. Jensen and A. Prestel [Manuscr. Math. 90, 225-238 (1996; Zbl 0857.12001)].
One main point of the results, and also of the proofs, is a decomposition of the groups in question as a free product (in the category of pro-\(p\)-groups) of Galois groups \(G_i\) of suitable completions \(F_i\). These \(G_i\) can be described as semidirect products of \({\mathbb Z}\) with the absolute Galois group \(\overline G_i\) of the residue class field \(\overline F_i\) of \(F_i\). The groups \(\overline G_i\) are then known, since \(\overline F_i\) is algebraic over some local or global field.
The methods mostly come from valuation theory. At one point, a Hasse principle [due mainly to F. Pop, J. Reine Angew. Math. 392, 145-175 (1988; Zbl 0671.12005)] concerning a second Galois cohomology group is used, in conjunction with the “Main Lemma” of Jensen and Prestel (loc.cit.) The idea is, roughly speaking: If a certain global-to-local map on H\(^2\) is injective, then a certain Galois group is a free product of local Galois groups.