BRAUER GROUP ANALOGS OF RESULTS RELATING THE WITT RING TO VALUATIONS AND GALOIS THEORY

Let F be a field of characteristic different from p containing a primi tive p-th root of unity. This paper studies the cup product pairing H- 1 (F,p) x H-1 (F,p) --> H-2(F,p) and its relationship to valuation the ory and Galois theory. Sufficient conditions on the pairing which guar antee the existence of a valuation on the field are described. In the non p-adic case these results provide a converse to the well-known str ucture theory in this situation. In the p-adic case, the pairing is de scribed using the notion of ''relative rigidity''. These results are a nalogues of results in quadratic form theory developed in the past dec ade, which cover the special case p = 2. Applications to the maximal p ro-p Galois group of F are also described.