Division algebras of prime degree and maximal galois p-extensions

Let p bean odd prime number, and let F be afield of characteristic not p and not containing the group mu(p) of p-th roots of unity. We consider cyclic p-algebras over F by descent from L = F(mu(p)). We generalize a theorem of Albert by showing that if mu(pn) subset of L, then a division algebra D of degree p(n) over F is a cyclic algebra if and only if there is d is an element of D with d(Pn) is an element of F - F-P. Let F(p) be the maximal p-extension of F. We show that F(p) has a noncyclic algebra of degree p if and only if a certain eigencomponent of the p-torsion of Br(F(p)(mu(p))) is nontrivial. To get a better understanding of F(p), we consider the valuations on F(p) with residue characteristic not p, and determine what residue fields and value groups can occur. Our results support the conjecture that the p torsion in Br(F(p)) is always trivial.