Let K be a number field containing the group it, of nth roots of unity, and let S be a set of primes of K including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal Sramified extension of K with coefficients in mu(n), which yields a pairing on a subgroup of K-x containing the S-units. In this general situation, we determine a formula for the cup product of two elements that pair trivially at all local places. Our primary focus is the case in which K = Q(mup) for n = p, an odd prime, and S consists of the unique prime above p in K. We describe a formula for this cup product in the case that one element is a pth root of unity. We explain a conjectural calculation of the restriction of the cup product to p-units for all p less than or equal to 10, 000 and conjecture its surjectivity for all p satisfying Vandiver's conjecture. We prove this for the smallest irregular prime p = 37 via a computation related to the Galois module structure of p-units in the unramified extension, of K of degree p. We describe a number of applications: to a product map in K-theory, to the structure of S-class groups in Kummer extensions of K, to relations in the Galois group of the maximal pro-p extension of Q(mup) unramified outside p, to relations in the graded Z(p)-Lie algebra associated to the representation of the absolute Galois group of Q in the outer automorphism group of the pro-p fundamental group of P-1 ((Q) over bar) - {0, 1, infinity}, and to Greenberg's pseudonullity conjecture. [References: 25]
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