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This paper concerns cup product pairings in \'etale cohomology related towork of M. Kim and of W. McCallum and R. Sharifi. We will show that byconsidering Ext groups rather than cohomology groups, one arrives at a pairingwhich combines invariants defined by Kim with a pairing defined by McCallum andSharifi. We also prove a formula for Kim's invariant in terms of Artin maps inthe case of cyclic unramified Kummer extensions. One consequence is that forall $n > 1$, there are infinitely many number fields $F$ over which there areboth trivial and non-trivial Kim invariants associated to cyclic groups oforder $n$.

Cup products in the etale cohomology of number fields