Galois Groups of Maximal ̂ -extensions

Let p be an odd prime and F a field of characteristic different from p containing a primitive p\h root of unity. Assume that the Galois group G of the maximal p-extension of F has a finite normal series with abelian factor groups. Then the commutator subgroup of G is abelian. Moreover, G has a normal abelian subgroup with pro-cyclic factor group. If, in addition, F contains a primitive p2th root of unity then G has generators {x,y¡}jeI with relations y¡yj = yjy¡ and xy¡x~l = y'+1 where q = 0 or q = p" for some n > 1 . This is used to calculate the cohomology ring of G , when G has finite rank. The field F is characterized in terms of the behavior of cyclic algebras (of degree p) over finite p-extensions. In what follows p will be a fixed odd prime and F will be a field of characteristic different from p containing a primitive pth root of unity a>. Let F(p) denote the maximal Galois extension of F whose Galois group Gf(p) = Ga\(F(p)/F) is a pro-p-group. An extension K/F is called a pextension if K ç F(p). Note that if K/F is a p-extension with [K : F] = p then K/F is Galois and K = F( HX(G) —> HX(G) TM HX(H) corresponds to Received by the editors February 6, 1990 and, in revised form, July 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 12F10, 20E18. This work was supported in part by NSA research grant no. MDA 904-88-H-2018. © 1992 American Mathematical Society 0002-9947/92 $1.00+ $.25 per page