A Remark on the Structure of Absolute Galois Groups

Let the field F contain all /?-power roots of unity for some prime p and suppose that the absolute Galois group G of F is a one-relator pro-p group. We use Merkurjev-Suslin's theorem on the power norm residue symbol to show that G is an extension of a Demushkin group by a free pro-/) group. Let F be a field, Fs its separable closure, and G = Ga\(Fs/F) its absolute Galois group. Let n be an integer not divisible by char(F) and denote by pn the G-module of nth roots of unity. Merkurjev and Suslin [1] have shown that the power norm residue symbol K2(F)/nK2(F) -* H2(G, ju,® ) is an isomorphism. It is a natural question to ask what bearing this theorem has on the structure of G. The present note is based on the fact that the symbol above factors through the homomorphism Hl(G, fi„)® -» H2(G, /í® ), induced by the cup product, which hence is surjective. To be modest, we specialize to Sylow subgroups of G, i.e., we let G itself be a pro-p group. Using well-known properties of such groups we then derive some information in the case where G has exactly one relation and F contains all p-power roots of unity. The notation will be standard: subgroups of a pro-p group G are understood to be closed; the Frattini subgroup is G* = GPG2, G2 denoting the commutator subgroup; subgroups generated by commutators are written as [X, Y]; cohomology groups with coefficients being Z/(p) are denoted by H'(G). Theorem. Let F be a field with separable closure Fs and absolute Galois group G = Gal(Fs/F). Suppose that G is a finitely generated one-relator pro-p group where the prime p is unequal to char( F ) and F contains all p-power roots of unity. Then there is a normal subgroup N of G which is pro-p free such that G/N is a Demushkin group and the inflation map H2(S/N,Z/(pn)) -> H2(S,Z/(p")) is an isomorphism for every subgroup S of G containing N, and for all integers n. Proof. (1) We begin by constructing a normal subgroup N of G such that, setting G = G/N, the inflation map H2(G) -* H2(G) is an isomorphism and the cup product _ H\G) X H\G) H2(G) is nondegenerate. Received by the editors December 6, 1984. 1980 Mathematics Subject Classification. Primary 12G05. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 353 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use